3.196 \(\int \frac {(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx\)

Optimal. Leaf size=412 \[ \frac {a^3 x}{c^4}-\frac {(b c-a d) \left (a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+b^2 c^2 d \left (13 c^2+2 d^2\right )\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^3 (c \cos (e+f x)+d)}-\frac {\left (a^3 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )-a^2 b \left (6 c^7+9 c^5 d^2\right )+3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^4 f \sqrt {c-d} \sqrt {c+d} \left (c^2-d^2\right )^3}-\frac {d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^3}+\frac {(b c-a d)^2 \left (-8 a c^2 d+3 a d^3+3 b c^3+2 b c d^2\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)^2} \]

[Out]

a^3*x/c^4-1/3*d*(-a*d+b*c)*(b+a*cos(f*x+e))^2*sin(f*x+e)/c/(c^2-d^2)/f/(d+c*cos(f*x+e))^3+1/6*(-a*d+b*c)^2*(-8
*a*c^2*d+3*a*d^3+3*b*c^3+2*b*c*d^2)*sin(f*x+e)/c^3/(c^2-d^2)^2/f/(d+c*cos(f*x+e))^2-1/6*(-a*d+b*c)*(b^2*c^2*d*
(13*c^2+2*d^2)-a*b*c*(18*c^4+17*c^2*d^2-5*d^4)+a^2*(34*c^4*d-28*c^2*d^3+9*d^5))*sin(f*x+e)/c^3/(c^2-d^2)^3/f/(
d+c*cos(f*x+e))-(3*a*b^2*c^4*d*(4*c^2+d^2)-b^3*c^5*(c^2+4*d^2)-a^2*b*(6*c^7+9*c^5*d^2)+a^3*(8*c^6*d-8*c^4*d^3+
7*c^2*d^5-2*d^7))*arctanh((c-d)^(1/2)*tan(1/2*e+1/2*f*x)/(c+d)^(1/2))/c^4/(c^2-d^2)^3/f/(c-d)^(1/2)/(c+d)^(1/2
)

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Rubi [A]  time = 1.06, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3941, 2989, 3031, 3021, 2735, 2659, 208} \[ -\frac {(b c-a d) \left (a^2 \left (-28 c^2 d^3+34 c^4 d+9 d^5\right )-a b c \left (17 c^2 d^2+18 c^4-5 d^4\right )+b^2 c^2 d \left (13 c^2+2 d^2\right )\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^3 (c \cos (e+f x)+d)}-\frac {\left (-a^2 b \left (9 c^5 d^2+6 c^7\right )+a^3 \left (7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right )+3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^4 f \sqrt {c-d} \sqrt {c+d} \left (c^2-d^2\right )^3}+\frac {a^3 x}{c^4}+\frac {(b c-a d)^2 \left (-8 a c^2 d+3 a d^3+3 b c^3+2 b c d^2\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)^2}-\frac {d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^4,x]

[Out]

(a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d
- 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^4*Sqrt[c - d]*Sqrt[
c + d]*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e
+ f*x])^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(
d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*
(34*c^4*d - 28*c^2*d^3 + 9*d^5))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x]))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2989

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((b*c - a*d)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)
*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[
e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (
A*b + a*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)
*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2,
0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 3031

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3941

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Int[
((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(m + n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
&& NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n] && LeQ[-2, m + n, 0]

Rubi steps

\begin {align*} \int \frac {(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx &=\int \frac {\cos (e+f x) (b+a \cos (e+f x))^3}{(d+c \cos (e+f x))^4} \, dx\\ &=-\frac {d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac {\int \frac {(b+a \cos (e+f x)) \left ((3 b c-2 a d) (b c-a d)-\left (3 a^2 c d+2 b^2 c d-a b \left (6 c^2-d^2\right )\right ) \cos (e+f x)+3 a^2 \left (c^2-d^2\right ) \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^3} \, dx}{3 c \left (c^2-d^2\right )}\\ &=-\frac {d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac {(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac {\int \frac {-2 c (b c-a d) \left (9 a b c^3-8 a^2 c^2 d-5 b^2 c^2 d+a b c d^2+3 a^2 d^3\right )+\left (3 a b^2 c^2 d \left (6 c^2-d^2\right )-b^3 c^3 \left (3 c^2+2 d^2\right )-a^2 b c \left (18 c^4-7 c^2 d^2+4 d^4\right )+a^3 \left (12 c^4 d-10 c^2 d^3+3 d^5\right )\right ) \cos (e+f x)-6 a^3 c \left (c^2-d^2\right )^2 \cos ^2(e+f x)}{(d+c \cos (e+f x))^2} \, dx}{6 c^3 \left (c^2-d^2\right )^2}\\ &=-\frac {d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac {(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}-\frac {\int \frac {-3 c^2 (b c-a d) \left (6 a^2 c^4+b^2 c^4-11 a b c^3 d-2 a^2 c^2 d^2+4 b^2 c^2 d^2+a b c d^3+a^2 d^4\right )-6 a^3 c \left (c^2-d^2\right )^3 \cos (e+f x)}{d+c \cos (e+f x)} \, dx}{6 c^4 \left (c^2-d^2\right )^3}\\ &=\frac {a^3 x}{c^4}-\frac {d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac {(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}-\frac {\left (3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )-a^2 b \left (6 c^7+9 c^5 d^2\right )+a^3 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\right ) \int \frac {1}{d+c \cos (e+f x)} \, dx}{2 c^4 \left (c^2-d^2\right )^3}\\ &=\frac {a^3 x}{c^4}-\frac {d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac {(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}-\frac {\left (3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )-a^2 b \left (6 c^7+9 c^5 d^2\right )+a^3 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c+d+(-c+d) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{c^4 \left (c^2-d^2\right )^3 f}\\ &=\frac {a^3 x}{c^4}-\frac {\left (3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )-a^2 b \left (6 c^7+9 c^5 d^2\right )+a^3 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^4 \sqrt {c-d} \sqrt {c+d} \left (c^2-d^2\right )^3 f}-\frac {d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac {(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}\\ \end {align*}

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Mathematica [A]  time = 3.78, size = 459, normalized size = 1.11 \[ \frac {\sec (e+f x) (a+b \sec (e+f x))^3 (c \cos (e+f x)+d) \left (6 a^3 (e+f x) (c \cos (e+f x)+d)^3+\frac {c \left (a^3 \left (36 c^4 d^2-32 c^2 d^4+11 d^6\right )-3 a^2 b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )+3 a b^2 c^2 \left (6 c^4+10 c^2 d^2-d^4\right )-b^3 c^3 d \left (13 c^2+2 d^2\right )\right ) \sin (e+f x) (c \cos (e+f x)+d)^2}{\left (c^2-d^2\right )^3}-\frac {6 \left (a^3 \left (-8 c^6 d+8 c^4 d^3-7 c^2 d^5+2 d^7\right )+a^2 b \left (6 c^7+9 c^5 d^2\right )-3 a b^2 c^4 d \left (4 c^2+d^2\right )+b^3 c^5 \left (c^2+4 d^2\right )\right ) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left (\frac {(d-c) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}-\frac {2 c d (b c-a d)^3 \sin (e+f x)}{c^2-d^2}+\frac {c \left (-12 a c^2 d+7 a d^3+3 b c^3+2 b c d^2\right ) (b c-a d)^2 \sin (e+f x) (c \cos (e+f x)+d)}{\left (c^2-d^2\right )^2}\right )}{6 c^4 f (a \cos (e+f x)+b)^3 (c+d \sec (e+f x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^4,x]

[Out]

((d + c*Cos[e + f*x])*Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(6*a^3*(e + f*x)*(d + c*Cos[e + f*x])^3 - (6*(-3*a*b
^2*c^4*d*(4*c^2 + d^2) + b^3*c^5*(c^2 + 4*d^2) + a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(-8*c^6*d + 8*c^4*d^3 - 7*c^2
*d^5 + 2*d^7))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^3)/(c^2 - d^2)^(7/2)
- (2*c*d*(b*c - a*d)^3*Sin[e + f*x])/(c^2 - d^2) + (c*(b*c - a*d)^2*(3*b*c^3 - 12*a*c^2*d + 2*b*c*d^2 + 7*a*d^
3)*(d + c*Cos[e + f*x])*Sin[e + f*x])/(c^2 - d^2)^2 + (c*(-(b^3*c^3*d*(13*c^2 + 2*d^2)) + 3*a*b^2*c^2*(6*c^4 +
10*c^2*d^2 - d^4) - 3*a^2*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) + a^3*(36*c^4*d^2 - 32*c^2*d^4 + 11*d^6))*(d + c
*Cos[e + f*x])^2*Sin[e + f*x])/(c^2 - d^2)^3))/(6*c^4*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^4)

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fricas [B]  time = 0.81, size = 2776, normalized size = 6.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x, algorithm="fricas")

[Out]

[1/12*(12*(a^3*c^11 - 4*a^3*c^9*d^2 + 6*a^3*c^7*d^4 - 4*a^3*c^5*d^6 + a^3*c^3*d^8)*f*x*cos(f*x + e)^3 + 36*(a^
3*c^10*d - 4*a^3*c^8*d^3 + 6*a^3*c^6*d^5 - 4*a^3*c^4*d^7 + a^3*c^2*d^9)*f*x*cos(f*x + e)^2 + 36*(a^3*c^9*d^2 -
4*a^3*c^7*d^4 + 6*a^3*c^5*d^6 - 4*a^3*c^3*d^8 + a^3*c*d^10)*f*x*cos(f*x + e) + 12*(a^3*c^8*d^3 - 4*a^3*c^6*d^
5 + 6*a^3*c^4*d^7 - 4*a^3*c^2*d^9 + a^3*d^11)*f*x + 3*(7*a^3*c^2*d^8 - 2*a^3*d^10 - (6*a^2*b + b^3)*c^7*d^3 +
4*(2*a^3 + 3*a*b^2)*c^6*d^4 - (9*a^2*b + 4*b^3)*c^5*d^5 - (8*a^3 - 3*a*b^2)*c^4*d^6 + (7*a^3*c^5*d^5 - 2*a^3*c
^3*d^7 - (6*a^2*b + b^3)*c^10 + 4*(2*a^3 + 3*a*b^2)*c^9*d - (9*a^2*b + 4*b^3)*c^8*d^2 - (8*a^3 - 3*a*b^2)*c^7*
d^3)*cos(f*x + e)^3 + 3*(7*a^3*c^4*d^6 - 2*a^3*c^2*d^8 - (6*a^2*b + b^3)*c^9*d + 4*(2*a^3 + 3*a*b^2)*c^8*d^2 -
(9*a^2*b + 4*b^3)*c^7*d^3 - (8*a^3 - 3*a*b^2)*c^6*d^4)*cos(f*x + e)^2 + 3*(7*a^3*c^3*d^7 - 2*a^3*c*d^9 - (6*a
^2*b + b^3)*c^8*d^2 + 4*(2*a^3 + 3*a*b^2)*c^7*d^3 - (9*a^2*b + 4*b^3)*c^6*d^4 - (8*a^3 - 3*a*b^2)*c^5*d^5)*cos
(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c^2 - d^2)*(d*cos(f
*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) + 2*(b^3*c^10*d + 6*
a*b^2*c^9*d^2 + 23*a^3*c^3*d^8 - 6*a^3*c*d^10 - 11*(3*a^2*b + b^3)*c^8*d^3 + (26*a^3 + 33*a*b^2)*c^7*d^4 + (21
*a^2*b + 4*b^3)*c^6*d^5 - (43*a^3 + 39*a*b^2)*c^5*d^6 + 6*(2*a^2*b + b^3)*c^4*d^7 + (18*a*b^2*c^11 + 6*a^2*b*c
^4*d^7 - 11*a^3*c^3*d^8 - (54*a^2*b + 13*b^3)*c^10*d + 12*(3*a^3 + a*b^2)*c^9*d^2 + (69*a^2*b + 11*b^3)*c^8*d^
3 - (68*a^3 + 33*a*b^2)*c^7*d^4 - (21*a^2*b - 2*b^3)*c^6*d^5 + (43*a^3 + 3*a*b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*
(b^3*c^11 + 6*a*b^2*c^10*d - 5*a^3*c^2*d^9 - (27*a^2*b + 10*b^3)*c^9*d^2 + (20*a^3 + 21*a*b^2)*c^8*d^3 + (24*a
^2*b + 7*b^3)*c^7*d^4 - 5*(7*a^3 + 6*a*b^2)*c^6*d^5 + (3*a^2*b + 2*b^3)*c^5*d^6 + (20*a^3 + 3*a*b^2)*c^4*d^7)*
cos(f*x + e))*sin(f*x + e))/((c^15 - 4*c^13*d^2 + 6*c^11*d^4 - 4*c^9*d^6 + c^7*d^8)*f*cos(f*x + e)^3 + 3*(c^14
*d - 4*c^12*d^3 + 6*c^10*d^5 - 4*c^8*d^7 + c^6*d^9)*f*cos(f*x + e)^2 + 3*(c^13*d^2 - 4*c^11*d^4 + 6*c^9*d^6 -
4*c^7*d^8 + c^5*d^10)*f*cos(f*x + e) + (c^12*d^3 - 4*c^10*d^5 + 6*c^8*d^7 - 4*c^6*d^9 + c^4*d^11)*f), 1/6*(6*(
a^3*c^11 - 4*a^3*c^9*d^2 + 6*a^3*c^7*d^4 - 4*a^3*c^5*d^6 + a^3*c^3*d^8)*f*x*cos(f*x + e)^3 + 18*(a^3*c^10*d -
4*a^3*c^8*d^3 + 6*a^3*c^6*d^5 - 4*a^3*c^4*d^7 + a^3*c^2*d^9)*f*x*cos(f*x + e)^2 + 18*(a^3*c^9*d^2 - 4*a^3*c^7*
d^4 + 6*a^3*c^5*d^6 - 4*a^3*c^3*d^8 + a^3*c*d^10)*f*x*cos(f*x + e) + 6*(a^3*c^8*d^3 - 4*a^3*c^6*d^5 + 6*a^3*c^
4*d^7 - 4*a^3*c^2*d^9 + a^3*d^11)*f*x - 3*(7*a^3*c^2*d^8 - 2*a^3*d^10 - (6*a^2*b + b^3)*c^7*d^3 + 4*(2*a^3 + 3
*a*b^2)*c^6*d^4 - (9*a^2*b + 4*b^3)*c^5*d^5 - (8*a^3 - 3*a*b^2)*c^4*d^6 + (7*a^3*c^5*d^5 - 2*a^3*c^3*d^7 - (6*
a^2*b + b^3)*c^10 + 4*(2*a^3 + 3*a*b^2)*c^9*d - (9*a^2*b + 4*b^3)*c^8*d^2 - (8*a^3 - 3*a*b^2)*c^7*d^3)*cos(f*x
+ e)^3 + 3*(7*a^3*c^4*d^6 - 2*a^3*c^2*d^8 - (6*a^2*b + b^3)*c^9*d + 4*(2*a^3 + 3*a*b^2)*c^8*d^2 - (9*a^2*b +
4*b^3)*c^7*d^3 - (8*a^3 - 3*a*b^2)*c^6*d^4)*cos(f*x + e)^2 + 3*(7*a^3*c^3*d^7 - 2*a^3*c*d^9 - (6*a^2*b + b^3)*
c^8*d^2 + 4*(2*a^3 + 3*a*b^2)*c^7*d^3 - (9*a^2*b + 4*b^3)*c^6*d^4 - (8*a^3 - 3*a*b^2)*c^5*d^5)*cos(f*x + e))*s
qrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) + (b^3*c^10*d + 6*a*
b^2*c^9*d^2 + 23*a^3*c^3*d^8 - 6*a^3*c*d^10 - 11*(3*a^2*b + b^3)*c^8*d^3 + (26*a^3 + 33*a*b^2)*c^7*d^4 + (21*a
^2*b + 4*b^3)*c^6*d^5 - (43*a^3 + 39*a*b^2)*c^5*d^6 + 6*(2*a^2*b + b^3)*c^4*d^7 + (18*a*b^2*c^11 + 6*a^2*b*c^4
*d^7 - 11*a^3*c^3*d^8 - (54*a^2*b + 13*b^3)*c^10*d + 12*(3*a^3 + a*b^2)*c^9*d^2 + (69*a^2*b + 11*b^3)*c^8*d^3
- (68*a^3 + 33*a*b^2)*c^7*d^4 - (21*a^2*b - 2*b^3)*c^6*d^5 + (43*a^3 + 3*a*b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*(b
^3*c^11 + 6*a*b^2*c^10*d - 5*a^3*c^2*d^9 - (27*a^2*b + 10*b^3)*c^9*d^2 + (20*a^3 + 21*a*b^2)*c^8*d^3 + (24*a^2
*b + 7*b^3)*c^7*d^4 - 5*(7*a^3 + 6*a*b^2)*c^6*d^5 + (3*a^2*b + 2*b^3)*c^5*d^6 + (20*a^3 + 3*a*b^2)*c^4*d^7)*co
s(f*x + e))*sin(f*x + e))/((c^15 - 4*c^13*d^2 + 6*c^11*d^4 - 4*c^9*d^6 + c^7*d^8)*f*cos(f*x + e)^3 + 3*(c^14*d
- 4*c^12*d^3 + 6*c^10*d^5 - 4*c^8*d^7 + c^6*d^9)*f*cos(f*x + e)^2 + 3*(c^13*d^2 - 4*c^11*d^4 + 6*c^9*d^6 - 4*
c^7*d^8 + c^5*d^10)*f*cos(f*x + e) + (c^12*d^3 - 4*c^10*d^5 + 6*c^8*d^7 - 4*c^6*d^9 + c^4*d^11)*f)]

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giac [B]  time = 0.93, size = 1639, normalized size = 3.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x, algorithm="giac")

[Out]

1/3*(3*(6*a^2*b*c^7 + b^3*c^7 - 8*a^3*c^6*d - 12*a*b^2*c^6*d + 9*a^2*b*c^5*d^2 + 4*b^3*c^5*d^2 + 8*a^3*c^4*d^3
- 3*a*b^2*c^4*d^3 - 7*a^3*c^2*d^5 + 2*a^3*d^7)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + arctan(-(c
*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((c^10 - 3*c^8*d^2 + 3*c^6*d^4 - c^4*d^6)*s
qrt(-c^2 + d^2)) + 3*(f*x + e)*a^3/c^4 - (18*a*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 3*b^3*c^8*tan(1/2*f*x + 1/2*e)
^5 - 54*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e)^5 - 18*a*b^2*c^7*d*tan(1/2*f*x + 1/2*e)^5 - 12*b^3*c^7*d*tan(1/2*f*x
+ 1/2*e)^5 + 36*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 81*a^2*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 36*a*b^2*c^6*d^
2*tan(1/2*f*x + 1/2*e)^5 + 27*b^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 - 60*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 18*
a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 81*a*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 12*b^3*c^5*d^3*tan(1/2*f*x +
1/2*e)^5 - 6*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 9*a^2*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 36*a*b^2*c^4*d^4*ta
n(1/2*f*x + 1/2*e)^5 + 6*b^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 45*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 18*a^2*b
*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 + 9*a*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 6*b^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^
5 - 6*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 15*a^3*c*d^7*tan(1/2*f*x + 1/2*e)^5 + 6*a^3*d^8*tan(1/2*f*x + 1/2*e
)^5 - 36*a*b^2*c^8*tan(1/2*f*x + 1/2*e)^3 + 108*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e)^3 + 28*b^3*c^7*d*tan(1/2*f*x
+ 1/2*e)^3 - 72*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 48*a*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 96*a^2*b*c^5*d^
3*tan(1/2*f*x + 1/2*e)^3 - 16*b^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 + 116*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 84
*a*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 12*a^2*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 12*b^3*c^3*d^5*tan(1/2*f*x +
1/2*e)^3 - 56*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 12*a^3*d^8*tan(1/2*f*x + 1/2*e)^3 + 18*a*b^2*c^8*tan(1/2*f
*x + 1/2*e) + 3*b^3*c^8*tan(1/2*f*x + 1/2*e) - 54*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e) + 18*a*b^2*c^7*d*tan(1/2*f*
x + 1/2*e) - 12*b^3*c^7*d*tan(1/2*f*x + 1/2*e) + 36*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e) - 81*a^2*b*c^6*d^2*tan(1/
2*f*x + 1/2*e) + 36*a*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e) - 27*b^3*c^6*d^2*tan(1/2*f*x + 1/2*e) + 60*a^3*c^5*d^3*
tan(1/2*f*x + 1/2*e) - 18*a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e) + 81*a*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e) - 12*b^3*
c^5*d^3*tan(1/2*f*x + 1/2*e) - 6*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e) - 9*a^2*b*c^4*d^4*tan(1/2*f*x + 1/2*e) + 36*
a*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e) - 6*b^3*c^4*d^4*tan(1/2*f*x + 1/2*e) - 45*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)
- 18*a^2*b*c^3*d^5*tan(1/2*f*x + 1/2*e) - 9*a*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e) - 6*b^3*c^3*d^5*tan(1/2*f*x + 1
/2*e) - 6*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e) + 15*a^3*c*d^7*tan(1/2*f*x + 1/2*e) + 6*a^3*d^8*tan(1/2*f*x + 1/2*e
))/((c^9 - 3*c^7*d^2 + 3*c^5*d^4 - c^3*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)^3))/
f

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maple [B]  time = 0.76, size = 4330, normalized size = 10.51 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x)

[Out]

-12/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*
x)*a^3*d^2-4/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*e
+1/2*f*x)^3*a^2*b*d^3+6/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3
)*tan(1/2*e+1/2*f*x)^5*a^2*b*d^3+24/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c^2-2*c*d+d^2)/
(c^2+2*c*d+d^2)*tan(1/2*e+1/2*f*x)^3*a^3*d^2-44/3/f/c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c
^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*e+1/2*f*x)^3*a^3*d^4+4/f/c^3/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^
2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*e+1/2*f*x)^3*a^3*d^6+3/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+
1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*a*b^2*d^3-3/f/(tan(1/2*e+1/2*f*x)^2*c-t
an(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a*b^2*d^3-6/f*c^3/(tan(1/2*e
+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*a*b^2+6/f*c^2/(
tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*b^3*d-
2/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)
*b^3*d^2-12/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*
e+1/2*f*x)^5*a^3*d^2+6/f/c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^
3)*tan(1/2*e+1/2*f*x)^5*a^3*d^4+1/f/c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2
*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a^3*d^5-2/f/c^3/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-
d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a^3*d^6-6/f*c^3/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2
*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a*b^2-28/3/f*c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1/
2*e+1/2*f*x)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*e+1/2*f*x)^3*b^3*d+6/f*c^2/(tan(1/2*e+1/2*f*x)
^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*b^3*d+2/f*c/(tan(1/2*e
+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*b^3*d^2+9/f*c
/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))*a^2*b
*d^2-12/f*c^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))
^(1/2))*a*b^2*d+12/f*c^3/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)
*tan(1/2*e+1/2*f*x)^3*a*b^2+6/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d
^2-d^3)*tan(1/2*e+1/2*f*x)*a^2*b*d^3+6/f/c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*
c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*a^3*d^4-1/f/c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c
+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*a^3*d^5-2/f/c^3/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*
d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*a^3*d^6+2/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*
f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*b^3*d^3-4/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2
*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a^3*d^3+2/f/(tan(1/2*e+1/2*f*x)^2*
c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*b^3*d^3-3/f/(c^6-3*c^4*d^
2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))*a*b^2*d^3-8/f*c^2/(
c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))*a^3*d+2
/f/c^4/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))
*a^3*d^7+1/f*c^3/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2
*e+1/2*f*x)^5*b^3-1/f*c^3/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3
)*tan(1/2*e+1/2*f*x)*b^3-7/f/c^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*
(c-d)/((c+d)*(c-d))^(1/2))*a^3*d^5+6/f*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e
+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))*a^2*b+4/f*c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan
(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))*b^3*d^2-4/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(
c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*e+1/2*f*x)^3*b^3*d^3+4/f/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d
-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*a^3*d^3+2/f*a^3/c^4*arctan(tan(1/2*e+1/2*f*x))+18/f
*c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*
a^2*b*d+28/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*e
+1/2*f*x)^3*a*b^2*d^2+6/f*c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2
-d^3)*tan(1/2*e+1/2*f*x)*a*b^2*d-18/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2
*d+3*c*d^2-d^3)*tan(1/2*e+1/2*f*x)*a*b^2*d^2-6/f*c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-
d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a*b^2*d-18/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*
d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a*b^2*d^2-36/f*c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1
/2*e+1/2*f*x)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*e+1/2*f*x)^3*a^2*b*d+8/f/(c^6-3*c^4*d^2+3*c^2
*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))*a^3*d^3+1/f*c^3/(c^6-3*c^4
*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*e+1/2*f*x)*(c-d)/((c+d)*(c-d))^(1/2))*b^3+9/f*c/(tan(1
/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*f*x)^5*a^2*b*d^2
+18/f*c^2/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*e+1/2*
f*x)^5*a^2*b*d-9/f*c/(tan(1/2*e+1/2*f*x)^2*c-tan(1/2*e+1/2*f*x)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan
(1/2*e+1/2*f*x)*a^2*b*d^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?`
for more details)Is 4*c^2-4*d^2 positive or negative?

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mupad [B]  time = 16.09, size = 15647, normalized size = 37.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(e + f*x))^3/(c + d/cos(e + f*x))^4,x)

[Out]

((tan(e/2 + (f*x)/2)^5*(b^3*c^6 - 2*a^3*d^6 - 6*a*b^2*c^6 + a^3*c*d^5 + 6*b^3*c^5*d + 6*a^3*c^2*d^4 - 4*a^3*c^
3*d^3 - 12*a^3*c^4*d^2 + 2*b^3*c^3*d^3 + 2*b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 - 18*a*b^2*c^4*d^2 + 6*a^2*b*c^3*d^3
+ 9*a^2*b*c^4*d^2 - 6*a*b^2*c^5*d + 18*a^2*b*c^5*d))/((c^3*d - c^4)*(c + d)^3) + (4*tan(e/2 + (f*x)/2)^3*(7*b^
3*c^5*d - 9*a*b^2*c^6 - 3*a^3*d^6 + 11*a^3*c^2*d^4 - 18*a^3*c^4*d^2 + 3*b^3*c^3*d^3 - 21*a*b^2*c^4*d^2 + 3*a^2
*b*c^3*d^3 + 27*a^2*b*c^5*d))/(3*(c + d)^2*(c^5 - 2*c^4*d + c^3*d^2)) - (tan(e/2 + (f*x)/2)*(2*a^3*d^6 + b^3*c
^6 + 6*a*b^2*c^6 + a^3*c*d^5 - 6*b^3*c^5*d - 6*a^3*c^2*d^4 - 4*a^3*c^3*d^3 + 12*a^3*c^4*d^2 - 2*b^3*c^3*d^3 +
2*b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 18*a*b^2*c^4*d^2 - 6*a^2*b*c^3*d^3 + 9*a^2*b*c^4*d^2 - 6*a*b^2*c^5*d - 18*a^
2*b*c^5*d))/((c + d)*(3*c^5*d - c^6 + c^3*d^3 - 3*c^4*d^2)))/(f*(tan(e/2 + (f*x)/2)^2*(3*c*d^2 - 3*c^2*d - 3*c
^3 + 3*d^3) - tan(e/2 + (f*x)/2)^4*(3*c*d^2 + 3*c^2*d - 3*c^3 - 3*d^3) + 3*c*d^2 + 3*c^2*d + c^3 + d^3 - tan(e
/2 + (f*x)/2)^6*(3*c*d^2 - 3*c^2*d + c^3 - d^3))) - (2*a^3*atan(((a^3*((a^3*((8*(4*a^3*c^21 + 2*b^3*c^21 + 12*
a^2*b*c^21 - 16*a^3*c^20*d - 2*b^3*c^20*d - 4*a^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10*d^11 - 14*a^3*c^11*d
^10 - 70*a^3*c^12*d^9 + 30*a^3*c^13*d^8 + 110*a^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^16*d^5 + 20*a^3*c^17*
d^4 + 64*a^3*c^18*d^3 - 12*a^3*c^19*d^2 + 8*b^3*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d^7 + 22*b^3*c^15*d^6
+ 18*b^3*c^16*d^5 - 18*b^3*c^17*d^4 - 2*b^3*c^18*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10 + 6*a*b^2*c^12*d^9 -
6*a*b^2*c^13*d^8 + 6*a*b^2*c^14*d^7 + 54*a*b^2*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c^17*d^4 + 66*a*b^2*c^
18*d^3 + 24*a*b^2*c^19*d^2 + 18*a^2*b*c^12*d^9 - 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 + 42*a^2*b*c^15*d^6 + 1
8*a^2*b*c^16*d^5 - 18*a^2*b*c^17*d^4 + 18*a^2*b*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c^20*d - 12*a^2*b*c^20
*d))/(c^19*d + c^20 - c^9*d^11 - c^10*d^10 + 5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5
+ 10*c^16*d^4 - 5*c^17*d^3 - 5*c^18*d^2) - (a^3*tan(e/2 + (f*x)/2)*(8*c^21*d - 8*c^8*d^14 + 8*c^9*d^13 + 48*c
^10*d^12 - 48*c^11*d^11 - 120*c^12*d^10 + 120*c^13*d^9 + 160*c^14*d^8 - 160*c^15*d^7 - 120*c^16*d^6 + 120*c^17
*d^5 + 48*c^18*d^4 - 48*c^19*d^3 - 8*c^20*d^2)*8i)/(c^4*(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c
^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*1i)/c^4 + (8*tan(e
/2 + (f*x)/2)*(4*a^6*c^14 + 8*a^6*d^14 + b^6*c^14 - 8*a^6*c*d^13 - 8*a^6*c^13*d + 12*a^2*b^4*c^14 + 36*a^4*b^2
*c^14 - 48*a^6*c^2*d^12 + 48*a^6*c^3*d^11 + 117*a^6*c^4*d^10 - 120*a^6*c^5*d^9 - 164*a^6*c^6*d^8 + 160*a^6*c^7
*d^7 + 156*a^6*c^8*d^6 - 120*a^6*c^9*d^5 - 92*a^6*c^10*d^4 + 48*a^6*c^11*d^3 + 44*a^6*c^12*d^2 + 16*b^6*c^10*d
^4 + 8*b^6*c^12*d^2 - 24*a*b^5*c^9*d^5 - 102*a*b^5*c^11*d^3 - 160*a^3*b^3*c^13*d + 36*a^5*b*c^5*d^9 - 102*a^5*
b*c^7*d^7 + 60*a^5*b*c^9*d^5 - 48*a^5*b*c^11*d^3 + 9*a^2*b^4*c^8*d^6 + 144*a^2*b^4*c^10*d^4 + 210*a^2*b^4*c^12
*d^2 + 16*a^3*b^3*c^5*d^9 - 52*a^3*b^3*c^7*d^7 - 4*a^3*b^3*c^9*d^5 - 300*a^3*b^3*c^11*d^3 - 12*a^4*b^2*c^4*d^1
0 - 6*a^4*b^2*c^6*d^8 + 120*a^4*b^2*c^8*d^6 - 63*a^4*b^2*c^10*d^4 + 300*a^4*b^2*c^12*d^2 - 24*a*b^5*c^13*d - 9
6*a^5*b*c^13*d))/(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10
*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))/c^4 - (a^3*((a^3*((8*(4*a^3*c^21 + 2*b^3*c^21 + 12*a^2*b*
c^21 - 16*a^3*c^20*d - 2*b^3*c^20*d - 4*a^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10*d^11 - 14*a^3*c^11*d^10 -
70*a^3*c^12*d^9 + 30*a^3*c^13*d^8 + 110*a^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^16*d^5 + 20*a^3*c^17*d^4 +
64*a^3*c^18*d^3 - 12*a^3*c^19*d^2 + 8*b^3*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d^7 + 22*b^3*c^15*d^6 + 18*b
^3*c^16*d^5 - 18*b^3*c^17*d^4 - 2*b^3*c^18*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10 + 6*a*b^2*c^12*d^9 - 6*a*b
^2*c^13*d^8 + 6*a*b^2*c^14*d^7 + 54*a*b^2*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c^17*d^4 + 66*a*b^2*c^18*d^3
+ 24*a*b^2*c^19*d^2 + 18*a^2*b*c^12*d^9 - 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 + 42*a^2*b*c^15*d^6 + 18*a^2*
b*c^16*d^5 - 18*a^2*b*c^17*d^4 + 18*a^2*b*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c^20*d - 12*a^2*b*c^20*d))/(
c^19*d + c^20 - c^9*d^11 - c^10*d^10 + 5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5 + 10*
c^16*d^4 - 5*c^17*d^3 - 5*c^18*d^2) + (a^3*tan(e/2 + (f*x)/2)*(8*c^21*d - 8*c^8*d^14 + 8*c^9*d^13 + 48*c^10*d^
12 - 48*c^11*d^11 - 120*c^12*d^10 + 120*c^13*d^9 + 160*c^14*d^8 - 160*c^15*d^7 - 120*c^16*d^6 + 120*c^17*d^5 +
48*c^18*d^4 - 48*c^19*d^3 - 8*c^20*d^2)*8i)/(c^4*(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8
- 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*1i)/c^4 - (8*tan(e/2 + (
f*x)/2)*(4*a^6*c^14 + 8*a^6*d^14 + b^6*c^14 - 8*a^6*c*d^13 - 8*a^6*c^13*d + 12*a^2*b^4*c^14 + 36*a^4*b^2*c^14
- 48*a^6*c^2*d^12 + 48*a^6*c^3*d^11 + 117*a^6*c^4*d^10 - 120*a^6*c^5*d^9 - 164*a^6*c^6*d^8 + 160*a^6*c^7*d^7 +
156*a^6*c^8*d^6 - 120*a^6*c^9*d^5 - 92*a^6*c^10*d^4 + 48*a^6*c^11*d^3 + 44*a^6*c^12*d^2 + 16*b^6*c^10*d^4 + 8
*b^6*c^12*d^2 - 24*a*b^5*c^9*d^5 - 102*a*b^5*c^11*d^3 - 160*a^3*b^3*c^13*d + 36*a^5*b*c^5*d^9 - 102*a^5*b*c^7*
d^7 + 60*a^5*b*c^9*d^5 - 48*a^5*b*c^11*d^3 + 9*a^2*b^4*c^8*d^6 + 144*a^2*b^4*c^10*d^4 + 210*a^2*b^4*c^12*d^2 +
16*a^3*b^3*c^5*d^9 - 52*a^3*b^3*c^7*d^7 - 4*a^3*b^3*c^9*d^5 - 300*a^3*b^3*c^11*d^3 - 12*a^4*b^2*c^4*d^10 - 6*
a^4*b^2*c^6*d^8 + 120*a^4*b^2*c^8*d^6 - 63*a^4*b^2*c^10*d^4 + 300*a^4*b^2*c^12*d^2 - 24*a*b^5*c^13*d - 96*a^5*
b*c^13*d))/(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*
d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))/c^4)/((a^3*((a^3*((8*(4*a^3*c^21 + 2*b^3*c^21 + 12*a^2*b*c^21 -
16*a^3*c^20*d - 2*b^3*c^20*d - 4*a^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10*d^11 - 14*a^3*c^11*d^10 - 70*a^3
*c^12*d^9 + 30*a^3*c^13*d^8 + 110*a^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^16*d^5 + 20*a^3*c^17*d^4 + 64*a^3
*c^18*d^3 - 12*a^3*c^19*d^2 + 8*b^3*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d^7 + 22*b^3*c^15*d^6 + 18*b^3*c^1
6*d^5 - 18*b^3*c^17*d^4 - 2*b^3*c^18*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10 + 6*a*b^2*c^12*d^9 - 6*a*b^2*c^1
3*d^8 + 6*a*b^2*c^14*d^7 + 54*a*b^2*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c^17*d^4 + 66*a*b^2*c^18*d^3 + 24*
a*b^2*c^19*d^2 + 18*a^2*b*c^12*d^9 - 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 + 42*a^2*b*c^15*d^6 + 18*a^2*b*c^16
*d^5 - 18*a^2*b*c^17*d^4 + 18*a^2*b*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c^20*d - 12*a^2*b*c^20*d))/(c^19*d
+ c^20 - c^9*d^11 - c^10*d^10 + 5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5 + 10*c^16*d
^4 - 5*c^17*d^3 - 5*c^18*d^2) - (a^3*tan(e/2 + (f*x)/2)*(8*c^21*d - 8*c^8*d^14 + 8*c^9*d^13 + 48*c^10*d^12 - 4
8*c^11*d^11 - 120*c^12*d^10 + 120*c^13*d^9 + 160*c^14*d^8 - 160*c^15*d^7 - 120*c^16*d^6 + 120*c^17*d^5 + 48*c^
18*d^4 - 48*c^19*d^3 - 8*c^20*d^2)*8i)/(c^4*(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*
c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*1i)/c^4 + (8*tan(e/2 + (f*x)/2
)*(4*a^6*c^14 + 8*a^6*d^14 + b^6*c^14 - 8*a^6*c*d^13 - 8*a^6*c^13*d + 12*a^2*b^4*c^14 + 36*a^4*b^2*c^14 - 48*a
^6*c^2*d^12 + 48*a^6*c^3*d^11 + 117*a^6*c^4*d^10 - 120*a^6*c^5*d^9 - 164*a^6*c^6*d^8 + 160*a^6*c^7*d^7 + 156*a
^6*c^8*d^6 - 120*a^6*c^9*d^5 - 92*a^6*c^10*d^4 + 48*a^6*c^11*d^3 + 44*a^6*c^12*d^2 + 16*b^6*c^10*d^4 + 8*b^6*c
^12*d^2 - 24*a*b^5*c^9*d^5 - 102*a*b^5*c^11*d^3 - 160*a^3*b^3*c^13*d + 36*a^5*b*c^5*d^9 - 102*a^5*b*c^7*d^7 +
60*a^5*b*c^9*d^5 - 48*a^5*b*c^11*d^3 + 9*a^2*b^4*c^8*d^6 + 144*a^2*b^4*c^10*d^4 + 210*a^2*b^4*c^12*d^2 + 16*a^
3*b^3*c^5*d^9 - 52*a^3*b^3*c^7*d^7 - 4*a^3*b^3*c^9*d^5 - 300*a^3*b^3*c^11*d^3 - 12*a^4*b^2*c^4*d^10 - 6*a^4*b^
2*c^6*d^8 + 120*a^4*b^2*c^8*d^6 - 63*a^4*b^2*c^10*d^4 + 300*a^4*b^2*c^12*d^2 - 24*a*b^5*c^13*d - 96*a^5*b*c^13
*d))/(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 +
10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2))*1i)/c^4 - (16*(4*a^9*d^13 - 12*a^8*b*c^13 - 2*a^9*c*d^12 + 16*a^9*c^12
*d + a^3*b^6*c^13 + 12*a^5*b^4*c^13 - 2*a^6*b^3*c^13 + 36*a^7*b^2*c^13 - 26*a^9*c^2*d^11 + 11*a^9*c^3*d^10 + 7
0*a^9*c^4*d^9 - 34*a^9*c^5*d^8 - 110*a^9*c^6*d^7 + 66*a^9*c^7*d^6 + 110*a^9*c^8*d^5 - 64*a^9*c^9*d^4 - 64*a^9*
c^10*d^3 + 48*a^9*c^11*d^2 - 24*a^4*b^5*c^12*d - 158*a^6*b^3*c^12*d + 24*a^7*b^2*c^12*d + 18*a^8*b*c^4*d^9 + 1
8*a^8*b*c^5*d^8 - 60*a^8*b*c^6*d^7 - 42*a^8*b*c^7*d^6 + 42*a^8*b*c^8*d^5 + 18*a^8*b*c^9*d^4 - 66*a^8*b*c^10*d^
3 + 18*a^8*b*c^11*d^2 + 16*a^3*b^6*c^9*d^4 + 8*a^3*b^6*c^11*d^2 - 24*a^4*b^5*c^8*d^5 - 102*a^4*b^5*c^10*d^3 +
9*a^5*b^4*c^7*d^6 + 144*a^5*b^4*c^9*d^4 + 210*a^5*b^4*c^11*d^2 + 8*a^6*b^3*c^4*d^9 + 8*a^6*b^3*c^5*d^8 - 30*a^
6*b^3*c^6*d^7 - 22*a^6*b^3*c^7*d^6 - 22*a^6*b^3*c^8*d^5 + 18*a^6*b^3*c^9*d^4 - 298*a^6*b^3*c^10*d^3 - 2*a^6*b^
3*c^11*d^2 - 6*a^7*b^2*c^3*d^10 - 6*a^7*b^2*c^4*d^9 - 6*a^7*b^2*c^6*d^7 + 66*a^7*b^2*c^7*d^6 + 54*a^7*b^2*c^8*
d^5 + 3*a^7*b^2*c^9*d^4 - 66*a^7*b^2*c^10*d^3 + 276*a^7*b^2*c^11*d^2 - 84*a^8*b*c^12*d))/(c^19*d + c^20 - c^9*
d^11 - c^10*d^10 + 5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5 + 10*c^16*d^4 - 5*c^17*d^
3 - 5*c^18*d^2) + (a^3*((a^3*((8*(4*a^3*c^21 + 2*b^3*c^21 + 12*a^2*b*c^21 - 16*a^3*c^20*d - 2*b^3*c^20*d - 4*a
^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10*d^11 - 14*a^3*c^11*d^10 - 70*a^3*c^12*d^9 + 30*a^3*c^13*d^8 + 110*a
^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^16*d^5 + 20*a^3*c^17*d^4 + 64*a^3*c^18*d^3 - 12*a^3*c^19*d^2 + 8*b^3
*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d^7 + 22*b^3*c^15*d^6 + 18*b^3*c^16*d^5 - 18*b^3*c^17*d^4 - 2*b^3*c^1
8*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10 + 6*a*b^2*c^12*d^9 - 6*a*b^2*c^13*d^8 + 6*a*b^2*c^14*d^7 + 54*a*b^2
*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c^17*d^4 + 66*a*b^2*c^18*d^3 + 24*a*b^2*c^19*d^2 + 18*a^2*b*c^12*d^9
- 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 + 42*a^2*b*c^15*d^6 + 18*a^2*b*c^16*d^5 - 18*a^2*b*c^17*d^4 + 18*a^2*b
*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c^20*d - 12*a^2*b*c^20*d))/(c^19*d + c^20 - c^9*d^11 - c^10*d^10 + 5*
c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5 + 10*c^16*d^4 - 5*c^17*d^3 - 5*c^18*d^2) + (a^
3*tan(e/2 + (f*x)/2)*(8*c^21*d - 8*c^8*d^14 + 8*c^9*d^13 + 48*c^10*d^12 - 48*c^11*d^11 - 120*c^12*d^10 + 120*c
^13*d^9 + 160*c^14*d^8 - 160*c^15*d^7 - 120*c^16*d^6 + 120*c^17*d^5 + 48*c^18*d^4 - 48*c^19*d^3 - 8*c^20*d^2)*
8i)/(c^4*(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^
5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*1i)/c^4 - (8*tan(e/2 + (f*x)/2)*(4*a^6*c^14 + 8*a^6*d^14 + b^6*c^
14 - 8*a^6*c*d^13 - 8*a^6*c^13*d + 12*a^2*b^4*c^14 + 36*a^4*b^2*c^14 - 48*a^6*c^2*d^12 + 48*a^6*c^3*d^11 + 117
*a^6*c^4*d^10 - 120*a^6*c^5*d^9 - 164*a^6*c^6*d^8 + 160*a^6*c^7*d^7 + 156*a^6*c^8*d^6 - 120*a^6*c^9*d^5 - 92*a
^6*c^10*d^4 + 48*a^6*c^11*d^3 + 44*a^6*c^12*d^2 + 16*b^6*c^10*d^4 + 8*b^6*c^12*d^2 - 24*a*b^5*c^9*d^5 - 102*a*
b^5*c^11*d^3 - 160*a^3*b^3*c^13*d + 36*a^5*b*c^5*d^9 - 102*a^5*b*c^7*d^7 + 60*a^5*b*c^9*d^5 - 48*a^5*b*c^11*d^
3 + 9*a^2*b^4*c^8*d^6 + 144*a^2*b^4*c^10*d^4 + 210*a^2*b^4*c^12*d^2 + 16*a^3*b^3*c^5*d^9 - 52*a^3*b^3*c^7*d^7
- 4*a^3*b^3*c^9*d^5 - 300*a^3*b^3*c^11*d^3 - 12*a^4*b^2*c^4*d^10 - 6*a^4*b^2*c^6*d^8 + 120*a^4*b^2*c^8*d^6 - 6
3*a^4*b^2*c^10*d^4 + 300*a^4*b^2*c^12*d^2 - 24*a*b^5*c^13*d - 96*a^5*b*c^13*d))/(c^16*d + c^17 - c^6*d^11 - c^
7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d
^2))*1i)/c^4)))/(c^4*f) + (atan(((((8*tan(e/2 + (f*x)/2)*(4*a^6*c^14 + 8*a^6*d^14 + b^6*c^14 - 8*a^6*c*d^13 -
8*a^6*c^13*d + 12*a^2*b^4*c^14 + 36*a^4*b^2*c^14 - 48*a^6*c^2*d^12 + 48*a^6*c^3*d^11 + 117*a^6*c^4*d^10 - 120*
a^6*c^5*d^9 - 164*a^6*c^6*d^8 + 160*a^6*c^7*d^7 + 156*a^6*c^8*d^6 - 120*a^6*c^9*d^5 - 92*a^6*c^10*d^4 + 48*a^6
*c^11*d^3 + 44*a^6*c^12*d^2 + 16*b^6*c^10*d^4 + 8*b^6*c^12*d^2 - 24*a*b^5*c^9*d^5 - 102*a*b^5*c^11*d^3 - 160*a
^3*b^3*c^13*d + 36*a^5*b*c^5*d^9 - 102*a^5*b*c^7*d^7 + 60*a^5*b*c^9*d^5 - 48*a^5*b*c^11*d^3 + 9*a^2*b^4*c^8*d^
6 + 144*a^2*b^4*c^10*d^4 + 210*a^2*b^4*c^12*d^2 + 16*a^3*b^3*c^5*d^9 - 52*a^3*b^3*c^7*d^7 - 4*a^3*b^3*c^9*d^5
- 300*a^3*b^3*c^11*d^3 - 12*a^4*b^2*c^4*d^10 - 6*a^4*b^2*c^6*d^8 + 120*a^4*b^2*c^8*d^6 - 63*a^4*b^2*c^10*d^4 +
300*a^4*b^2*c^12*d^2 - 24*a*b^5*c^13*d - 96*a^5*b*c^13*d))/(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 +
5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2) + (((8*(4*a^3*c^
21 + 2*b^3*c^21 + 12*a^2*b*c^21 - 16*a^3*c^20*d - 2*b^3*c^20*d - 4*a^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10
*d^11 - 14*a^3*c^11*d^10 - 70*a^3*c^12*d^9 + 30*a^3*c^13*d^8 + 110*a^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^
16*d^5 + 20*a^3*c^17*d^4 + 64*a^3*c^18*d^3 - 12*a^3*c^19*d^2 + 8*b^3*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d
^7 + 22*b^3*c^15*d^6 + 18*b^3*c^16*d^5 - 18*b^3*c^17*d^4 - 2*b^3*c^18*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10
+ 6*a*b^2*c^12*d^9 - 6*a*b^2*c^13*d^8 + 6*a*b^2*c^14*d^7 + 54*a*b^2*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c
^17*d^4 + 66*a*b^2*c^18*d^3 + 24*a*b^2*c^19*d^2 + 18*a^2*b*c^12*d^9 - 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 +
42*a^2*b*c^15*d^6 + 18*a^2*b*c^16*d^5 - 18*a^2*b*c^17*d^4 + 18*a^2*b*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c
^20*d - 12*a^2*b*c^20*d))/(c^19*d + c^20 - c^9*d^11 - c^10*d^10 + 5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c
^14*d^6 + 10*c^15*d^5 + 10*c^16*d^4 - 5*c^17*d^3 - 5*c^18*d^2) - (4*tan(e/2 + (f*x)/2)*((c + d)^7*(c - d)^7)^(
1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^
2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d)*(8*c^21*d - 8*c^8*d^14 + 8*c^9*d^13 + 48*c^10*d^12 - 48*c^11*d^1
1 - 120*c^12*d^10 + 120*c^13*d^9 + 160*c^14*d^8 - 160*c^15*d^7 - 120*c^16*d^6 + 120*c^17*d^5 + 48*c^18*d^4 - 4
8*c^19*d^3 - 8*c^20*d^2))/((c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d
^4 - 7*c^16*d^2)*(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10
*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*
b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a
*b^2*c^6*d))/(2*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16
*d^2)))*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c
^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d)*1i)/(2*(c^18 - c^4*d^14 + 7*c^6*d
^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d^2)) + (((8*tan(e/2 + (f*x)/2)*(4*a^6*c^
14 + 8*a^6*d^14 + b^6*c^14 - 8*a^6*c*d^13 - 8*a^6*c^13*d + 12*a^2*b^4*c^14 + 36*a^4*b^2*c^14 - 48*a^6*c^2*d^12
+ 48*a^6*c^3*d^11 + 117*a^6*c^4*d^10 - 120*a^6*c^5*d^9 - 164*a^6*c^6*d^8 + 160*a^6*c^7*d^7 + 156*a^6*c^8*d^6
- 120*a^6*c^9*d^5 - 92*a^6*c^10*d^4 + 48*a^6*c^11*d^3 + 44*a^6*c^12*d^2 + 16*b^6*c^10*d^4 + 8*b^6*c^12*d^2 - 2
4*a*b^5*c^9*d^5 - 102*a*b^5*c^11*d^3 - 160*a^3*b^3*c^13*d + 36*a^5*b*c^5*d^9 - 102*a^5*b*c^7*d^7 + 60*a^5*b*c^
9*d^5 - 48*a^5*b*c^11*d^3 + 9*a^2*b^4*c^8*d^6 + 144*a^2*b^4*c^10*d^4 + 210*a^2*b^4*c^12*d^2 + 16*a^3*b^3*c^5*d
^9 - 52*a^3*b^3*c^7*d^7 - 4*a^3*b^3*c^9*d^5 - 300*a^3*b^3*c^11*d^3 - 12*a^4*b^2*c^4*d^10 - 6*a^4*b^2*c^6*d^8 +
120*a^4*b^2*c^8*d^6 - 63*a^4*b^2*c^10*d^4 + 300*a^4*b^2*c^12*d^2 - 24*a*b^5*c^13*d - 96*a^5*b*c^13*d))/(c^16*
d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4
- 5*c^14*d^3 - 5*c^15*d^2) - (((8*(4*a^3*c^21 + 2*b^3*c^21 + 12*a^2*b*c^21 - 16*a^3*c^20*d - 2*b^3*c^20*d - 4
*a^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10*d^11 - 14*a^3*c^11*d^10 - 70*a^3*c^12*d^9 + 30*a^3*c^13*d^8 + 110
*a^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^16*d^5 + 20*a^3*c^17*d^4 + 64*a^3*c^18*d^3 - 12*a^3*c^19*d^2 + 8*b
^3*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d^7 + 22*b^3*c^15*d^6 + 18*b^3*c^16*d^5 - 18*b^3*c^17*d^4 - 2*b^3*c
^18*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10 + 6*a*b^2*c^12*d^9 - 6*a*b^2*c^13*d^8 + 6*a*b^2*c^14*d^7 + 54*a*b
^2*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c^17*d^4 + 66*a*b^2*c^18*d^3 + 24*a*b^2*c^19*d^2 + 18*a^2*b*c^12*d^
9 - 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 + 42*a^2*b*c^15*d^6 + 18*a^2*b*c^16*d^5 - 18*a^2*b*c^17*d^4 + 18*a^2
*b*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c^20*d - 12*a^2*b*c^20*d))/(c^19*d + c^20 - c^9*d^11 - c^10*d^10 +
5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5 + 10*c^16*d^4 - 5*c^17*d^3 - 5*c^18*d^2) + (
4*tan(e/2 + (f*x)/2)*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*
d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d)*(8*c^21*d - 8*c^8*d^
14 + 8*c^9*d^13 + 48*c^10*d^12 - 48*c^11*d^11 - 120*c^12*d^10 + 120*c^13*d^9 + 160*c^14*d^8 - 160*c^15*d^7 - 1
20*c^16*d^6 + 120*c^17*d^5 + 48*c^18*d^4 - 48*c^19*d^3 - 8*c^20*d^2))/((c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*
d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d^2)*(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9
+ 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*((c + d)^7*(c
- d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^
2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d))/(2*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c
^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d^2)))*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b
*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*
b^2*c^6*d)*1i)/(2*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^
16*d^2)))/((16*(4*a^9*d^13 - 12*a^8*b*c^13 - 2*a^9*c*d^12 + 16*a^9*c^12*d + a^3*b^6*c^13 + 12*a^5*b^4*c^13 - 2
*a^6*b^3*c^13 + 36*a^7*b^2*c^13 - 26*a^9*c^2*d^11 + 11*a^9*c^3*d^10 + 70*a^9*c^4*d^9 - 34*a^9*c^5*d^8 - 110*a^
9*c^6*d^7 + 66*a^9*c^7*d^6 + 110*a^9*c^8*d^5 - 64*a^9*c^9*d^4 - 64*a^9*c^10*d^3 + 48*a^9*c^11*d^2 - 24*a^4*b^5
*c^12*d - 158*a^6*b^3*c^12*d + 24*a^7*b^2*c^12*d + 18*a^8*b*c^4*d^9 + 18*a^8*b*c^5*d^8 - 60*a^8*b*c^6*d^7 - 42
*a^8*b*c^7*d^6 + 42*a^8*b*c^8*d^5 + 18*a^8*b*c^9*d^4 - 66*a^8*b*c^10*d^3 + 18*a^8*b*c^11*d^2 + 16*a^3*b^6*c^9*
d^4 + 8*a^3*b^6*c^11*d^2 - 24*a^4*b^5*c^8*d^5 - 102*a^4*b^5*c^10*d^3 + 9*a^5*b^4*c^7*d^6 + 144*a^5*b^4*c^9*d^4
+ 210*a^5*b^4*c^11*d^2 + 8*a^6*b^3*c^4*d^9 + 8*a^6*b^3*c^5*d^8 - 30*a^6*b^3*c^6*d^7 - 22*a^6*b^3*c^7*d^6 - 22
*a^6*b^3*c^8*d^5 + 18*a^6*b^3*c^9*d^4 - 298*a^6*b^3*c^10*d^3 - 2*a^6*b^3*c^11*d^2 - 6*a^7*b^2*c^3*d^10 - 6*a^7
*b^2*c^4*d^9 - 6*a^7*b^2*c^6*d^7 + 66*a^7*b^2*c^7*d^6 + 54*a^7*b^2*c^8*d^5 + 3*a^7*b^2*c^9*d^4 - 66*a^7*b^2*c^
10*d^3 + 276*a^7*b^2*c^11*d^2 - 84*a^8*b*c^12*d))/(c^19*d + c^20 - c^9*d^11 - c^10*d^10 + 5*c^11*d^9 + 5*c^12*
d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5 + 10*c^16*d^4 - 5*c^17*d^3 - 5*c^18*d^2) - (((8*tan(e/2 + (f*x)/
2)*(4*a^6*c^14 + 8*a^6*d^14 + b^6*c^14 - 8*a^6*c*d^13 - 8*a^6*c^13*d + 12*a^2*b^4*c^14 + 36*a^4*b^2*c^14 - 48*
a^6*c^2*d^12 + 48*a^6*c^3*d^11 + 117*a^6*c^4*d^10 - 120*a^6*c^5*d^9 - 164*a^6*c^6*d^8 + 160*a^6*c^7*d^7 + 156*
a^6*c^8*d^6 - 120*a^6*c^9*d^5 - 92*a^6*c^10*d^4 + 48*a^6*c^11*d^3 + 44*a^6*c^12*d^2 + 16*b^6*c^10*d^4 + 8*b^6*
c^12*d^2 - 24*a*b^5*c^9*d^5 - 102*a*b^5*c^11*d^3 - 160*a^3*b^3*c^13*d + 36*a^5*b*c^5*d^9 - 102*a^5*b*c^7*d^7 +
60*a^5*b*c^9*d^5 - 48*a^5*b*c^11*d^3 + 9*a^2*b^4*c^8*d^6 + 144*a^2*b^4*c^10*d^4 + 210*a^2*b^4*c^12*d^2 + 16*a
^3*b^3*c^5*d^9 - 52*a^3*b^3*c^7*d^7 - 4*a^3*b^3*c^9*d^5 - 300*a^3*b^3*c^11*d^3 - 12*a^4*b^2*c^4*d^10 - 6*a^4*b
^2*c^6*d^8 + 120*a^4*b^2*c^8*d^6 - 63*a^4*b^2*c^10*d^4 + 300*a^4*b^2*c^12*d^2 - 24*a*b^5*c^13*d - 96*a^5*b*c^1
3*d))/(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 +
10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2) + (((8*(4*a^3*c^21 + 2*b^3*c^21 + 12*a^2*b*c^21 - 16*a^3*c^20*d - 2*b^
3*c^20*d - 4*a^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10*d^11 - 14*a^3*c^11*d^10 - 70*a^3*c^12*d^9 + 30*a^3*c^
13*d^8 + 110*a^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^16*d^5 + 20*a^3*c^17*d^4 + 64*a^3*c^18*d^3 - 12*a^3*c^
19*d^2 + 8*b^3*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d^7 + 22*b^3*c^15*d^6 + 18*b^3*c^16*d^5 - 18*b^3*c^17*d
^4 - 2*b^3*c^18*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10 + 6*a*b^2*c^12*d^9 - 6*a*b^2*c^13*d^8 + 6*a*b^2*c^14*
d^7 + 54*a*b^2*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c^17*d^4 + 66*a*b^2*c^18*d^3 + 24*a*b^2*c^19*d^2 + 18*a
^2*b*c^12*d^9 - 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 + 42*a^2*b*c^15*d^6 + 18*a^2*b*c^16*d^5 - 18*a^2*b*c^17*
d^4 + 18*a^2*b*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c^20*d - 12*a^2*b*c^20*d))/(c^19*d + c^20 - c^9*d^11 -
c^10*d^10 + 5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*c^15*d^5 + 10*c^16*d^4 - 5*c^17*d^3 - 5*c
^18*d^2) - (4*tan(e/2 + (f*x)/2)*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d
- 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d)*(8*c^21*
d - 8*c^8*d^14 + 8*c^9*d^13 + 48*c^10*d^12 - 48*c^11*d^11 - 120*c^12*d^10 + 120*c^13*d^9 + 160*c^14*d^8 - 160*
c^15*d^7 - 120*c^16*d^6 + 120*c^17*d^5 + 48*c^18*d^4 - 48*c^19*d^3 - 8*c^20*d^2))/((c^18 - c^4*d^14 + 7*c^6*d^
12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d^2)*(c^16*d + c^17 - c^6*d^11 - c^7*d^10
+ 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*(
(c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 +
4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d))/(2*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8
*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d^2)))*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c
^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5
*d^2 - 12*a*b^2*c^6*d))/(2*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d
^4 - 7*c^16*d^2)) + (((8*tan(e/2 + (f*x)/2)*(4*a^6*c^14 + 8*a^6*d^14 + b^6*c^14 - 8*a^6*c*d^13 - 8*a^6*c^13*d
+ 12*a^2*b^4*c^14 + 36*a^4*b^2*c^14 - 48*a^6*c^2*d^12 + 48*a^6*c^3*d^11 + 117*a^6*c^4*d^10 - 120*a^6*c^5*d^9 -
164*a^6*c^6*d^8 + 160*a^6*c^7*d^7 + 156*a^6*c^8*d^6 - 120*a^6*c^9*d^5 - 92*a^6*c^10*d^4 + 48*a^6*c^11*d^3 + 4
4*a^6*c^12*d^2 + 16*b^6*c^10*d^4 + 8*b^6*c^12*d^2 - 24*a*b^5*c^9*d^5 - 102*a*b^5*c^11*d^3 - 160*a^3*b^3*c^13*d
+ 36*a^5*b*c^5*d^9 - 102*a^5*b*c^7*d^7 + 60*a^5*b*c^9*d^5 - 48*a^5*b*c^11*d^3 + 9*a^2*b^4*c^8*d^6 + 144*a^2*b
^4*c^10*d^4 + 210*a^2*b^4*c^12*d^2 + 16*a^3*b^3*c^5*d^9 - 52*a^3*b^3*c^7*d^7 - 4*a^3*b^3*c^9*d^5 - 300*a^3*b^3
*c^11*d^3 - 12*a^4*b^2*c^4*d^10 - 6*a^4*b^2*c^6*d^8 + 120*a^4*b^2*c^8*d^6 - 63*a^4*b^2*c^10*d^4 + 300*a^4*b^2*
c^12*d^2 - 24*a*b^5*c^13*d - 96*a^5*b*c^13*d))/(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 -
10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 10*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2) - (((8*(4*a^3*c^21 + 2*b^3*c^
21 + 12*a^2*b*c^21 - 16*a^3*c^20*d - 2*b^3*c^20*d - 4*a^3*c^8*d^13 + 2*a^3*c^9*d^12 + 26*a^3*c^10*d^11 - 14*a^
3*c^11*d^10 - 70*a^3*c^12*d^9 + 30*a^3*c^13*d^8 + 110*a^3*c^14*d^7 - 30*a^3*c^15*d^6 - 110*a^3*c^16*d^5 + 20*a
^3*c^17*d^4 + 64*a^3*c^18*d^3 - 12*a^3*c^19*d^2 + 8*b^3*c^12*d^9 - 8*b^3*c^13*d^8 - 22*b^3*c^14*d^7 + 22*b^3*c
^15*d^6 + 18*b^3*c^16*d^5 - 18*b^3*c^17*d^4 - 2*b^3*c^18*d^3 + 2*b^3*c^19*d^2 - 6*a*b^2*c^11*d^10 + 6*a*b^2*c^
12*d^9 - 6*a*b^2*c^13*d^8 + 6*a*b^2*c^14*d^7 + 54*a*b^2*c^15*d^6 - 54*a*b^2*c^16*d^5 - 66*a*b^2*c^17*d^4 + 66*
a*b^2*c^18*d^3 + 24*a*b^2*c^19*d^2 + 18*a^2*b*c^12*d^9 - 18*a^2*b*c^13*d^8 - 42*a^2*b*c^14*d^7 + 42*a^2*b*c^15
*d^6 + 18*a^2*b*c^16*d^5 - 18*a^2*b*c^17*d^4 + 18*a^2*b*c^18*d^3 - 18*a^2*b*c^19*d^2 - 24*a*b^2*c^20*d - 12*a^
2*b*c^20*d))/(c^19*d + c^20 - c^9*d^11 - c^10*d^10 + 5*c^11*d^9 + 5*c^12*d^8 - 10*c^13*d^7 - 10*c^14*d^6 + 10*
c^15*d^5 + 10*c^16*d^4 - 5*c^17*d^3 - 5*c^18*d^2) + (4*tan(e/2 + (f*x)/2)*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d
^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9
*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d)*(8*c^21*d - 8*c^8*d^14 + 8*c^9*d^13 + 48*c^10*d^12 - 48*c^11*d^11 - 120*c^12*
d^10 + 120*c^13*d^9 + 160*c^14*d^8 - 160*c^15*d^7 - 120*c^16*d^6 + 120*c^17*d^5 + 48*c^18*d^4 - 48*c^19*d^3 -
8*c^20*d^2))/((c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d
^2)*(c^16*d + c^17 - c^6*d^11 - c^7*d^10 + 5*c^8*d^9 + 5*c^9*d^8 - 10*c^10*d^7 - 10*c^11*d^6 + 10*c^12*d^5 + 1
0*c^13*d^4 - 5*c^14*d^3 - 5*c^15*d^2)))*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3
*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d))/
(2*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d^2)))*((c +
d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7 + 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^
3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^2 - 12*a*b^2*c^6*d))/(2*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^1
0 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d^4 - 7*c^16*d^2))))*((c + d)^7*(c - d)^7)^(1/2)*(2*a^3*d^7 + b^3*c^7
+ 6*a^2*b*c^7 - 8*a^3*c^6*d - 7*a^3*c^2*d^5 + 8*a^3*c^4*d^3 + 4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 9*a^2*b*c^5*d^
2 - 12*a*b^2*c^6*d)*1i)/(f*(c^18 - c^4*d^14 + 7*c^6*d^12 - 21*c^8*d^10 + 35*c^10*d^8 - 35*c^12*d^6 + 21*c^14*d
^4 - 7*c^16*d^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{3}}{\left (c + d \sec {\left (e + f x \right )}\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))**3/(c+d*sec(f*x+e))**4,x)

[Out]

Integral((a + b*sec(e + f*x))**3/(c + d*sec(e + f*x))**4, x)

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