3.2.97 \(\int \frac {(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^5} \, dx\) [197]

Optimal. Leaf size=622 \[ \frac {a^3 x}{c^5}-\frac {\left (15 a b^2 c^6 d \left (4 c^2+3 d^2\right )-3 a^2 b c^5 \left (8 c^4+24 c^2 d^2+3 d^4\right )-b^3 c^5 \left (4 c^4+27 c^2 d^2+4 d^4\right )+a^3 \left (40 c^8 d-40 c^6 d^3+63 c^4 d^5-36 c^2 d^7+8 d^9\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{4 c^5 \sqrt {c-d} \sqrt {c+d} \left (c^2-d^2\right )^4 f}+\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}-\frac {d \left (8 b c^3-11 a c^2 d-b c d^2+4 a d^3\right ) (b+a \cos (e+f x))^2 \sin (e+f x)}{12 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^3}-\frac {(b c-a d) \left (2 a b c d \left (32 c^4+c^2 d^2+2 d^4\right )-a^2 d^2 \left (58 c^4-35 c^2 d^2+12 d^4\right )-b^2 \left (12 c^6+25 c^4 d^2-2 c^2 d^4\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))^2}-\frac {\left (b^3 c^3 d \left (68 c^4+39 c^2 d^2-2 d^4\right )+a^2 b c d \left (272 c^6+10 c^4 d^2+49 c^2 d^4-16 d^6\right )-3 a b^2 c^2 \left (24 c^6+84 c^4 d^2-5 c^2 d^4+2 d^6\right )-a^3 \left (212 c^6 d^2-210 c^4 d^4+139 c^2 d^6-36 d^8\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^4 f (d+c \cos (e+f x))} \]

[Out]

a^3*x/c^5+1/4*d^2*(b+a*cos(f*x+e))^3*sin(f*x+e)/c/(c^2-d^2)/f/(d+c*cos(f*x+e))^4-1/12*d*(-11*a*c^2*d+4*a*d^3+8
*b*c^3-b*c*d^2)*(b+a*cos(f*x+e))^2*sin(f*x+e)/c^2/(c^2-d^2)^2/f/(d+c*cos(f*x+e))^3-1/24*(-a*d+b*c)*(2*a*b*c*d*
(32*c^4+c^2*d^2+2*d^4)-a^2*d^2*(58*c^4-35*c^2*d^2+12*d^4)-b^2*(12*c^6+25*c^4*d^2-2*c^2*d^4))*sin(f*x+e)/c^4/(c
^2-d^2)^3/f/(d+c*cos(f*x+e))^2-1/24*(b^3*c^3*d*(68*c^4+39*c^2*d^2-2*d^4)+a^2*b*c*d*(272*c^6+10*c^4*d^2+49*c^2*
d^4-16*d^6)-3*a*b^2*c^2*(24*c^6+84*c^4*d^2-5*c^2*d^4+2*d^6)-a^3*(212*c^6*d^2-210*c^4*d^4+139*c^2*d^6-36*d^8))*
sin(f*x+e)/c^4/(c^2-d^2)^4/f/(d+c*cos(f*x+e))-1/4*(15*a*b^2*c^6*d*(4*c^2+3*d^2)-3*a^2*b*c^5*(8*c^4+24*c^2*d^2+
3*d^4)-b^3*c^5*(4*c^4+27*c^2*d^2+4*d^4)+a^3*(40*c^8*d-40*c^6*d^3+63*c^4*d^5-36*c^2*d^7+8*d^9))*arctanh((c-d)^(
1/2)*tan(1/2*f*x+1/2*e)/(c+d)^(1/2))/c^5/(c^2-d^2)^4/f/(c-d)^(1/2)/(c+d)^(1/2)

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Rubi [A]
time = 1.23, antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4026, 3127, 3126, 3110, 3100, 2814, 2738, 214} \begin {gather*} \frac {a^3 x}{c^5}-\frac {(b c-a d) \left (-a^2 d^2 \left (58 c^4-35 c^2 d^2+12 d^4\right )+2 a b c d \left (32 c^4+c^2 d^2+2 d^4\right )-\left (b^2 \left (12 c^6+25 c^4 d^2-2 c^2 d^4\right )\right )\right ) \sin (e+f x)}{24 c^4 f \left (c^2-d^2\right )^3 (c \cos (e+f x)+d)^2}-\frac {\left (-\left (a^3 \left (212 c^6 d^2-210 c^4 d^4+139 c^2 d^6-36 d^8\right )\right )+a^2 b c d \left (272 c^6+10 c^4 d^2+49 c^2 d^4-16 d^6\right )-3 a b^2 c^2 \left (24 c^6+84 c^4 d^2-5 c^2 d^4+2 d^6\right )+b^3 c^3 d \left (68 c^4+39 c^2 d^2-2 d^4\right )\right ) \sin (e+f x)}{24 c^4 f \left (c^2-d^2\right )^4 (c \cos (e+f x)+d)}-\frac {\left (a^3 \left (40 c^8 d-40 c^6 d^3+63 c^4 d^5-36 c^2 d^7+8 d^9\right )-3 a^2 b c^5 \left (8 c^4+24 c^2 d^2+3 d^4\right )+15 a b^2 c^6 d \left (4 c^2+3 d^2\right )-b^3 c^5 \left (4 c^4+27 c^2 d^2+4 d^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{4 c^5 f \sqrt {c-d} \sqrt {c+d} \left (c^2-d^2\right )^4}+\frac {d^2 \sin (e+f x) (a \cos (e+f x)+b)^3}{4 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^4}-\frac {d \left (-11 a c^2 d+4 a d^3+8 b c^3-b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^2}{12 c^2 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^5,x]

[Out]

(a^3*x)/c^5 - ((15*a*b^2*c^6*d*(4*c^2 + 3*d^2) - 3*a^2*b*c^5*(8*c^4 + 24*c^2*d^2 + 3*d^4) - b^3*c^5*(4*c^4 + 2
7*c^2*d^2 + 4*d^4) + a^3*(40*c^8*d - 40*c^6*d^3 + 63*c^4*d^5 - 36*c^2*d^7 + 8*d^9))*ArcTanh[(Sqrt[c - d]*Tan[(
e + f*x)/2])/Sqrt[c + d]])/(4*c^5*Sqrt[c - d]*Sqrt[c + d]*(c^2 - d^2)^4*f) + (d^2*(b + a*Cos[e + f*x])^3*Sin[e
+ f*x])/(4*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^4) - (d*(8*b*c^3 - 11*a*c^2*d - b*c*d^2 + 4*a*d^3)*(b + a*Cos
[e + f*x])^2*Sin[e + f*x])/(12*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^3) - ((b*c - a*d)*(2*a*b*c*d*(32*c^4 +
c^2*d^2 + 2*d^4) - a^2*d^2*(58*c^4 - 35*c^2*d^2 + 12*d^4) - b^2*(12*c^6 + 25*c^4*d^2 - 2*c^2*d^4))*Sin[e + f*
x])/(24*c^4*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x])^2) - ((b^3*c^3*d*(68*c^4 + 39*c^2*d^2 - 2*d^4) + a^2*b*c*d*(2
72*c^6 + 10*c^4*d^2 + 49*c^2*d^4 - 16*d^6) - 3*a*b^2*c^2*(24*c^6 + 84*c^4*d^2 - 5*c^2*d^4 + 2*d^6) - a^3*(212*
c^6*d^2 - 210*c^4*d^4 + 139*c^2*d^6 - 36*d^8))*Sin[e + f*x])/(24*c^4*(c^2 - d^2)^4*f*(d + c*Cos[e + f*x]))

Rule 214

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

Rule 2738

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 2814

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 3100

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 3110

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] &&
NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3126

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

Rule 3127

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

Rule 4026

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))^5} \, dx &=\int \frac {\cos ^2(e+f x) (b+a \cos (e+f x))^3}{(d+c \cos (e+f x))^5} \, dx\\ &=\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}+\frac {\int \frac {(b+a \cos (e+f x))^2 \left (-d (4 b c-3 a d)+\left (4 b c^2-4 a c d-b d^2\right ) \cos (e+f x)+4 a \left (c^2-d^2\right ) \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^4} \, dx}{4 c \left (c^2-d^2\right )}\\ &=\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}-\frac {d \left (8 b c^3-11 a c^2 d-b c d^2+4 a d^3\right ) (b+a \cos (e+f x))^2 \sin (e+f x)}{12 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^3}+\frac {\int \frac {(b+a \cos (e+f x)) \left (2 a^2 d^2 \left (11 c^2-4 d^2\right )-5 a b c d \left (8 c^2-d^2\right )+3 b^2 \left (4 c^4+3 c^2 d^2\right )-\left (2 b^2 c d \left (8 c^2-d^2\right )+3 a^2 \left (8 c^3 d-c d^3\right )-a b \left (24 c^4+7 c^2 d^2+4 d^4\right )\right ) \cos (e+f x)+12 a^2 \left (c^2-d^2\right )^2 \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^3} \, dx}{12 c^2 \left (c^2-d^2\right )^2}\\ &=\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}-\frac {d \left (8 b c^3-11 a c^2 d-b c d^2+4 a d^3\right ) (b+a \cos (e+f x))^2 \sin (e+f x)}{12 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^3}-\frac {(b c-a d) \left (2 a b c d \left (32 c^4+c^2 d^2+2 d^4\right )-a^2 d^2 \left (58 c^4-35 c^2 d^2+12 d^4\right )-b^2 \left (12 c^6+25 c^4 d^2-2 c^2 d^4\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))^2}-\frac {\int \frac {2 c \left (7 b^3 c^3 d \left (4 c^2+d^2\right )-3 a b^2 c^2 \left (12 c^4+24 c^2 d^2-d^4\right )+a^2 b c d \left (100 c^4-3 c^2 d^2+8 d^4\right )-a^3 \left (58 c^4 d^2-35 c^2 d^4+12 d^6\right )\right )-\left (b^3 c^3 \left (12 c^4+25 c^2 d^2-2 d^4\right )-3 a b^2 c^2 d \left (36 c^4-3 c^2 d^2+2 d^4\right )+a^2 b c \left (72 c^6+16 c^4 d^2+33 c^2 d^4-16 d^6\right )-a^3 \left (72 c^6 d-68 c^4 d^3+43 c^2 d^5-12 d^7\right )\right ) \cos (e+f x)-24 a^3 c \left (c^2-d^2\right )^3 \cos ^2(e+f x)}{(d+c \cos (e+f x))^2} \, dx}{24 c^4 \left (c^2-d^2\right )^3}\\ &=\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}-\frac {d \left (8 b c^3-11 a c^2 d-b c d^2+4 a d^3\right ) (b+a \cos (e+f x))^2 \sin (e+f x)}{12 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^3}-\frac {(b c-a d) \left (2 a b c d \left (32 c^4+c^2 d^2+2 d^4\right )-a^2 d^2 \left (58 c^4-35 c^2 d^2+12 d^4\right )-b^2 \left (12 c^6+25 c^4 d^2-2 c^2 d^4\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))^2}-\frac {\left (b^3 c^3 d \left (68 c^4+39 c^2 d^2-2 d^4\right )+a^2 b c d \left (272 c^6+10 c^4 d^2+49 c^2 d^4-16 d^6\right )-3 a b^2 c^2 \left (24 c^6+84 c^4 d^2-5 c^2 d^4+2 d^6\right )-a^3 \left (212 c^6 d^2-210 c^4 d^4+139 c^2 d^6-36 d^8\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^4 f (d+c \cos (e+f x))}-\frac {\int \frac {3 c \left (15 a b^2 c^5 d \left (4 c^2+3 d^2\right )-3 a^2 b c^4 \left (8 c^4+24 c^2 d^2+3 d^4\right )-b^3 c^4 \left (4 c^4+27 c^2 d^2+4 d^4\right )+a^3 \left (32 c^7 d-8 c^5 d^3+15 c^3 d^5-4 c d^7\right )\right )-24 a^3 c \left (c^2-d^2\right )^4 \cos (e+f x)}{d+c \cos (e+f x)} \, dx}{24 c^5 \left (c^2-d^2\right )^4}\\ &=\frac {a^3 x}{c^5}+\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}-\frac {d \left (8 b c^3-11 a c^2 d-b c d^2+4 a d^3\right ) (b+a \cos (e+f x))^2 \sin (e+f x)}{12 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^3}-\frac {(b c-a d) \left (2 a b c d \left (32 c^4+c^2 d^2+2 d^4\right )-a^2 d^2 \left (58 c^4-35 c^2 d^2+12 d^4\right )-b^2 \left (12 c^6+25 c^4 d^2-2 c^2 d^4\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))^2}-\frac {\left (b^3 c^3 d \left (68 c^4+39 c^2 d^2-2 d^4\right )+a^2 b c d \left (272 c^6+10 c^4 d^2+49 c^2 d^4-16 d^6\right )-3 a b^2 c^2 \left (24 c^6+84 c^4 d^2-5 c^2 d^4+2 d^6\right )-a^3 \left (212 c^6 d^2-210 c^4 d^4+139 c^2 d^6-36 d^8\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^4 f (d+c \cos (e+f x))}-\frac {\left (15 a b^2 c^6 d \left (4 c^2+3 d^2\right )-3 a^2 b c^5 \left (8 c^4+24 c^2 d^2+3 d^4\right )-b^3 c^5 \left (4 c^4+27 c^2 d^2+4 d^4\right )+a^3 \left (40 c^8 d-40 c^6 d^3+63 c^4 d^5-36 c^2 d^7+8 d^9\right )\right ) \int \frac {1}{d+c \cos (e+f x)} \, dx}{8 c^5 \left (c^2-d^2\right )^4}\\ &=\frac {a^3 x}{c^5}+\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}-\frac {d \left (8 b c^3-11 a c^2 d-b c d^2+4 a d^3\right ) (b+a \cos (e+f x))^2 \sin (e+f x)}{12 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^3}-\frac {(b c-a d) \left (2 a b c d \left (32 c^4+c^2 d^2+2 d^4\right )-a^2 d^2 \left (58 c^4-35 c^2 d^2+12 d^4\right )-b^2 \left (12 c^6+25 c^4 d^2-2 c^2 d^4\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))^2}-\frac {\left (b^3 c^3 d \left (68 c^4+39 c^2 d^2-2 d^4\right )+a^2 b c d \left (272 c^6+10 c^4 d^2+49 c^2 d^4-16 d^6\right )-3 a b^2 c^2 \left (24 c^6+84 c^4 d^2-5 c^2 d^4+2 d^6\right )-a^3 \left (212 c^6 d^2-210 c^4 d^4+139 c^2 d^6-36 d^8\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^4 f (d+c \cos (e+f x))}-\frac {\left (15 a b^2 c^6 d \left (4 c^2+3 d^2\right )-3 a^2 b c^5 \left (8 c^4+24 c^2 d^2+3 d^4\right )-b^3 c^5 \left (4 c^4+27 c^2 d^2+4 d^4\right )+a^3 \left (40 c^8 d-40 c^6 d^3+63 c^4 d^5-36 c^2 d^7+8 d^9\right )\right ) \text {Subst}\left (\int \frac {1}{c+d+(-c+d) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{4 c^5 \left (c^2-d^2\right )^4 f}\\ &=\frac {a^3 x}{c^5}-\frac {\left (15 a b^2 c^6 d \left (4 c^2+3 d^2\right )-3 a^2 b c^5 \left (8 c^4+24 c^2 d^2+3 d^4\right )-b^3 c^5 \left (4 c^4+27 c^2 d^2+4 d^4\right )+a^3 \left (40 c^8 d-40 c^6 d^3+63 c^4 d^5-36 c^2 d^7+8 d^9\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{4 c^5 \sqrt {c-d} \sqrt {c+d} \left (c^2-d^2\right )^4 f}+\frac {d^2 (b+a \cos (e+f x))^3 \sin (e+f x)}{4 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^4}-\frac {d \left (8 b c^3-11 a c^2 d-b c d^2+4 a d^3\right ) (b+a \cos (e+f x))^2 \sin (e+f x)}{12 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^3}-\frac {(b c-a d) \left (2 a b c d \left (32 c^4+c^2 d^2+2 d^4\right )-a^2 d^2 \left (58 c^4-35 c^2 d^2+12 d^4\right )-b^2 \left (12 c^6+25 c^4 d^2-2 c^2 d^4\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))^2}-\frac {\left (b^3 c^3 d \left (68 c^4+39 c^2 d^2-2 d^4\right )+a^2 b c d \left (272 c^6+10 c^4 d^2+49 c^2 d^4-16 d^6\right )-3 a b^2 c^2 \left (24 c^6+84 c^4 d^2-5 c^2 d^4+2 d^6\right )-a^3 \left (212 c^6 d^2-210 c^4 d^4+139 c^2 d^6-36 d^8\right )\right ) \sin (e+f x)}{24 c^4 \left (c^2-d^2\right )^4 f (d+c \cos (e+f x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1285\) vs. \(2(622)=1244\).
time = 6.82, size = 1285, normalized size = 2.07 \begin {gather*} \frac {a^3 (e+f x) (d+c \cos (e+f x))^5 \sec ^2(e+f x) (a+b \sec (e+f x))^3}{c^5 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac {\left (-24 a^2 b c^9-4 b^3 c^9+40 a^3 c^8 d+60 a b^2 c^8 d-72 a^2 b c^7 d^2-27 b^3 c^7 d^2-40 a^3 c^6 d^3+45 a b^2 c^6 d^3-9 a^2 b c^5 d^4-4 b^3 c^5 d^4+63 a^3 c^4 d^5-36 a^3 c^2 d^7+8 a^3 d^9\right ) \tanh ^{-1}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) (d+c \cos (e+f x))^5 \sec ^2(e+f x) (a+b \sec (e+f x))^3}{4 c^5 \sqrt {c^2-d^2} \left (-c^2+d^2\right )^4 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac {(d+c \cos (e+f x)) \sec ^2(e+f x) (a+b \sec (e+f x))^3 \left (b^3 c^3 d^2 \sin (e+f x)-3 a b^2 c^2 d^3 \sin (e+f x)+3 a^2 b c d^4 \sin (e+f x)-a^3 d^5 \sin (e+f x)\right )}{4 c^4 \left (c^2-d^2\right ) f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac {(d+c \cos (e+f x))^2 \sec ^2(e+f x) (a+b \sec (e+f x))^3 \left (-8 b^3 c^5 d \sin (e+f x)+36 a b^2 c^4 d^2 \sin (e+f x)-48 a^2 b c^3 d^3 \sin (e+f x)+b^3 c^3 d^3 \sin (e+f x)+20 a^3 c^2 d^4 \sin (e+f x)-15 a b^2 c^2 d^4 \sin (e+f x)+27 a^2 b c d^5 \sin (e+f x)-13 a^3 d^6 \sin (e+f x)\right )}{12 c^4 \left (c^2-d^2\right )^2 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac {(d+c \cos (e+f x))^3 \sec ^2(e+f x) (a+b \sec (e+f x))^3 \left (12 b^3 c^7 \sin (e+f x)-108 a b^2 c^6 d \sin (e+f x)+216 a^2 b c^5 d^2 \sin (e+f x)+25 b^3 c^5 d^2 \sin (e+f x)-120 a^3 c^4 d^3 \sin (e+f x)+9 a b^2 c^4 d^3 \sin (e+f x)-165 a^2 b c^3 d^4 \sin (e+f x)-2 b^3 c^3 d^4 \sin (e+f x)+131 a^3 c^2 d^5 \sin (e+f x)-6 a b^2 c^2 d^5 \sin (e+f x)+54 a^2 b c d^6 \sin (e+f x)-46 a^3 d^7 \sin (e+f x)\right )}{24 c^4 \left (c^2-d^2\right )^3 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5}+\frac {(d+c \cos (e+f x))^4 \sec ^2(e+f x) (a+b \sec (e+f x))^3 \left (72 a b^2 c^8 \sin (e+f x)-288 a^2 b c^7 d \sin (e+f x)-68 b^3 c^7 d \sin (e+f x)+240 a^3 c^6 d^2 \sin (e+f x)+252 a b^2 c^6 d^2 \sin (e+f x)+24 a^2 b c^5 d^3 \sin (e+f x)-39 b^3 c^5 d^3 \sin (e+f x)-280 a^3 c^4 d^4 \sin (e+f x)-15 a b^2 c^4 d^4 \sin (e+f x)-69 a^2 b c^3 d^5 \sin (e+f x)+2 b^3 c^3 d^5 \sin (e+f x)+195 a^3 c^2 d^6 \sin (e+f x)+6 a b^2 c^2 d^6 \sin (e+f x)+18 a^2 b c d^7 \sin (e+f x)-50 a^3 d^8 \sin (e+f x)\right )}{24 c^4 \left (c^2-d^2\right )^4 f (b+a \cos (e+f x))^3 (c+d \sec (e+f x))^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^5,x]

[Out]

(a^3*(e + f*x)*(d + c*Cos[e + f*x])^5*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3)/(c^5*f*(b + a*Cos[e + f*x])^3*(c
+ d*Sec[e + f*x])^5) + ((-24*a^2*b*c^9 - 4*b^3*c^9 + 40*a^3*c^8*d + 60*a*b^2*c^8*d - 72*a^2*b*c^7*d^2 - 27*b^3
*c^7*d^2 - 40*a^3*c^6*d^3 + 45*a*b^2*c^6*d^3 - 9*a^2*b*c^5*d^4 - 4*b^3*c^5*d^4 + 63*a^3*c^4*d^5 - 36*a^3*c^2*d
^7 + 8*a^3*d^9)*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^5*Sec[e + f*x]^2*(a
+ b*Sec[e + f*x])^3)/(4*c^5*Sqrt[c^2 - d^2]*(-c^2 + d^2)^4*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5) +
((d + c*Cos[e + f*x])*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3*(b^3*c^3*d^2*Sin[e + f*x] - 3*a*b^2*c^2*d^3*Sin[e
+ f*x] + 3*a^2*b*c*d^4*Sin[e + f*x] - a^3*d^5*Sin[e + f*x]))/(4*c^4*(c^2 - d^2)*f*(b + a*Cos[e + f*x])^3*(c +
d*Sec[e + f*x])^5) + ((d + c*Cos[e + f*x])^2*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3*(-8*b^3*c^5*d*Sin[e + f*x]
+ 36*a*b^2*c^4*d^2*Sin[e + f*x] - 48*a^2*b*c^3*d^3*Sin[e + f*x] + b^3*c^3*d^3*Sin[e + f*x] + 20*a^3*c^2*d^4*Si
n[e + f*x] - 15*a*b^2*c^2*d^4*Sin[e + f*x] + 27*a^2*b*c*d^5*Sin[e + f*x] - 13*a^3*d^6*Sin[e + f*x]))/(12*c^4*(
c^2 - d^2)^2*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5) + ((d + c*Cos[e + f*x])^3*Sec[e + f*x]^2*(a + b*
Sec[e + f*x])^3*(12*b^3*c^7*Sin[e + f*x] - 108*a*b^2*c^6*d*Sin[e + f*x] + 216*a^2*b*c^5*d^2*Sin[e + f*x] + 25*
b^3*c^5*d^2*Sin[e + f*x] - 120*a^3*c^4*d^3*Sin[e + f*x] + 9*a*b^2*c^4*d^3*Sin[e + f*x] - 165*a^2*b*c^3*d^4*Sin
[e + f*x] - 2*b^3*c^3*d^4*Sin[e + f*x] + 131*a^3*c^2*d^5*Sin[e + f*x] - 6*a*b^2*c^2*d^5*Sin[e + f*x] + 54*a^2*
b*c*d^6*Sin[e + f*x] - 46*a^3*d^7*Sin[e + f*x]))/(24*c^4*(c^2 - d^2)^3*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e +
f*x])^5) + ((d + c*Cos[e + f*x])^4*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^3*(72*a*b^2*c^8*Sin[e + f*x] - 288*a^2
*b*c^7*d*Sin[e + f*x] - 68*b^3*c^7*d*Sin[e + f*x] + 240*a^3*c^6*d^2*Sin[e + f*x] + 252*a*b^2*c^6*d^2*Sin[e + f
*x] + 24*a^2*b*c^5*d^3*Sin[e + f*x] - 39*b^3*c^5*d^3*Sin[e + f*x] - 280*a^3*c^4*d^4*Sin[e + f*x] - 15*a*b^2*c^
4*d^4*Sin[e + f*x] - 69*a^2*b*c^3*d^5*Sin[e + f*x] + 2*b^3*c^3*d^5*Sin[e + f*x] + 195*a^3*c^2*d^6*Sin[e + f*x]
+ 6*a*b^2*c^2*d^6*Sin[e + f*x] + 18*a^2*b*c*d^7*Sin[e + f*x] - 50*a^3*d^8*Sin[e + f*x]))/(24*c^4*(c^2 - d^2)^
4*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1344\) vs. \(2(603)=1206\).
time = 1.08, size = 1345, normalized size = 2.16 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))^5,x,method=_RETURNVERBOSE)

[Out]

1/f*(2/c^5*((-1/8*(80*a^3*c^6*d^2+40*a^3*c^5*d^3-40*a^3*c^4*d^4-15*a^3*c^3*d^5+32*a^3*c^2*d^6+4*a^3*c*d^7-8*a^
3*d^8-96*a^2*b*c^7*d-72*a^2*b*c^6*d^2-96*a^2*b*c^5*d^3-15*a^2*b*c^4*d^4+24*a*b^2*c^8+36*a*b^2*c^7*d+144*a*b^2*
c^6*d^2+51*a*b^2*c^5*d^3+24*a*b^2*c^4*d^4-4*b^3*c^8-32*b^3*c^7*d-21*b^3*c^6*d^2-32*b^3*c^5*d^3-4*b^3*c^4*d^4)*
c/(c-d)/(c^4+4*c^3*d+6*c^2*d^2+4*c*d^3+d^4)*tan(1/2*f*x+1/2*e)^7+1/24*c*(720*a^3*c^6*d^2+120*a^3*c^5*d^3-520*a
^3*c^4*d^4-69*a^3*c^3*d^5+320*a^3*c^2*d^6+12*a^3*c*d^7-72*a^3*d^8-864*a^2*b*c^7*d-216*a^2*b*c^6*d^2-480*a^2*b*
c^5*d^3+27*a^2*b*c^4*d^4+216*a*b^2*c^8+108*a*b^2*c^7*d+1008*a*b^2*c^6*d^2+81*a*b^2*c^5*d^3+120*a*b^2*c^4*d^4-1
2*b^3*c^8-224*b^3*c^7*d-39*b^3*c^6*d^2-224*b^3*c^5*d^3-12*b^3*c^4*d^4)/(c^3+3*c^2*d+3*c*d^2+d^3)/(c-d)^2*tan(1
/2*f*x+1/2*e)^5-1/24*c*(720*a^3*c^6*d^2-120*a^3*c^5*d^3-520*a^3*c^4*d^4+69*a^3*c^3*d^5+320*a^3*c^2*d^6-12*a^3*
c*d^7-72*a^3*d^8-864*a^2*b*c^7*d+216*a^2*b*c^6*d^2-480*a^2*b*c^5*d^3-27*a^2*b*c^4*d^4+216*a*b^2*c^8-108*a*b^2*
c^7*d+1008*a*b^2*c^6*d^2-81*a*b^2*c^5*d^3+120*a*b^2*c^4*d^4+12*b^3*c^8-224*b^3*c^7*d+39*b^3*c^6*d^2-224*b^3*c^
5*d^3+12*b^3*c^4*d^4)/(c-d)^3/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+1/8*(80*a^3*c^6*d^2-40*a^3*c^5*d^3-40*a^3*c
^4*d^4+15*a^3*c^3*d^5+32*a^3*c^2*d^6-4*a^3*c*d^7-8*a^3*d^8-96*a^2*b*c^7*d+72*a^2*b*c^6*d^2-96*a^2*b*c^5*d^3+15
*a^2*b*c^4*d^4+24*a*b^2*c^8-36*a*b^2*c^7*d+144*a*b^2*c^6*d^2-51*a*b^2*c^5*d^3+24*a*b^2*c^4*d^4+4*b^3*c^8-32*b^
3*c^7*d+21*b^3*c^6*d^2-32*b^3*c^5*d^3+4*b^3*c^4*d^4)*c/(c+d)/(c^4-4*c^3*d+6*c^2*d^2-4*c*d^3+d^4)*tan(1/2*f*x+1
/2*e))/(c*tan(1/2*f*x+1/2*e)^2-d*tan(1/2*f*x+1/2*e)^2-c-d)^4-1/8*(40*a^3*c^8*d-40*a^3*c^6*d^3+63*a^3*c^4*d^5-3
6*a^3*c^2*d^7+8*a^3*d^9-24*a^2*b*c^9-72*a^2*b*c^7*d^2-9*a^2*b*c^5*d^4+60*a*b^2*c^8*d+45*a*b^2*c^6*d^3-4*b^3*c^
9-27*b^3*c^7*d^2-4*b^3*c^5*d^4)/(c^8-4*c^6*d^2+6*c^4*d^4-4*c^2*d^6+d^8)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(
1/2*f*x+1/2*e)/((c+d)*(c-d))^(1/2)))+2*a^3/c^5*arctan(tan(1/2*f*x+1/2*e)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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))^5,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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2162 vs. \(2 (614) = 1228\).
time = 4.83, size = 4386, normalized size = 7.05 \begin {gather*} \text {Too large to display} \end {gather*}

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))^5,x, algorithm="fricas")

[Out]

[1/48*(48*(a^3*c^14 - 5*a^3*c^12*d^2 + 10*a^3*c^10*d^4 - 10*a^3*c^8*d^6 + 5*a^3*c^6*d^8 - a^3*c^4*d^10)*f*x*co
s(f*x + e)^4 + 192*(a^3*c^13*d - 5*a^3*c^11*d^3 + 10*a^3*c^9*d^5 - 10*a^3*c^7*d^7 + 5*a^3*c^5*d^9 - a^3*c^3*d^
11)*f*x*cos(f*x + e)^3 + 288*(a^3*c^12*d^2 - 5*a^3*c^10*d^4 + 10*a^3*c^8*d^6 - 10*a^3*c^6*d^8 + 5*a^3*c^4*d^10
- a^3*c^2*d^12)*f*x*cos(f*x + e)^2 + 192*(a^3*c^11*d^3 - 5*a^3*c^9*d^5 + 10*a^3*c^7*d^7 - 10*a^3*c^5*d^9 + 5*
a^3*c^3*d^11 - a^3*c*d^13)*f*x*cos(f*x + e) + 48*(a^3*c^10*d^4 - 5*a^3*c^8*d^6 + 10*a^3*c^6*d^8 - 10*a^3*c^4*d
^10 + 5*a^3*c^2*d^12 - a^3*d^14)*f*x + 3*(63*a^3*c^4*d^9 - 36*a^3*c^2*d^11 + 8*a^3*d^13 - 4*(6*a^2*b + b^3)*c^
9*d^4 + 20*(2*a^3 + 3*a*b^2)*c^8*d^5 - 9*(8*a^2*b + 3*b^3)*c^7*d^6 - 5*(8*a^3 - 9*a*b^2)*c^6*d^7 - (9*a^2*b +
4*b^3)*c^5*d^8 + (63*a^3*c^8*d^5 - 36*a^3*c^6*d^7 + 8*a^3*c^4*d^9 - 4*(6*a^2*b + b^3)*c^13 + 20*(2*a^3 + 3*a*b
^2)*c^12*d - 9*(8*a^2*b + 3*b^3)*c^11*d^2 - 5*(8*a^3 - 9*a*b^2)*c^10*d^3 - (9*a^2*b + 4*b^3)*c^9*d^4)*cos(f*x
+ e)^4 + 4*(63*a^3*c^7*d^6 - 36*a^3*c^5*d^8 + 8*a^3*c^3*d^10 - 4*(6*a^2*b + b^3)*c^12*d + 20*(2*a^3 + 3*a*b^2)
*c^11*d^2 - 9*(8*a^2*b + 3*b^3)*c^10*d^3 - 5*(8*a^3 - 9*a*b^2)*c^9*d^4 - (9*a^2*b + 4*b^3)*c^8*d^5)*cos(f*x +
e)^3 + 6*(63*a^3*c^6*d^7 - 36*a^3*c^4*d^9 + 8*a^3*c^2*d^11 - 4*(6*a^2*b + b^3)*c^11*d^2 + 20*(2*a^3 + 3*a*b^2)
*c^10*d^3 - 9*(8*a^2*b + 3*b^3)*c^9*d^4 - 5*(8*a^3 - 9*a*b^2)*c^8*d^5 - (9*a^2*b + 4*b^3)*c^7*d^6)*cos(f*x + e
)^2 + 4*(63*a^3*c^5*d^8 - 36*a^3*c^3*d^10 + 8*a^3*c*d^12 - 4*(6*a^2*b + b^3)*c^10*d^3 + 20*(2*a^3 + 3*a*b^2)*c
^9*d^4 - 9*(8*a^2*b + 3*b^3)*c^8*d^5 - 5*(8*a^3 - 9*a*b^2)*c^7*d^6 - (9*a^2*b + 4*b^3)*c^6*d^7)*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*(2*b^3*c^12*d^2 + 18*a*b^2*
c^11*d^3 - 116*a^3*c^3*d^11 + 24*a^3*c*d^13 - (150*a^2*b + 41*b^3)*c^10*d^4 + 77*(2*a^3 + 3*a*b^2)*c^9*d^5 - (
15*a^2*b + 29*b^3)*c^8*d^6 - (271*a^3 + 201*a*b^2)*c^7*d^7 + (165*a^2*b + 68*b^3)*c^6*d^8 + (209*a^3 - 48*a*b^
2)*c^5*d^9 + (72*a*b^2*c^14 - 18*a^2*b*c^5*d^9 + 50*a^3*c^4*d^10 - 4*(72*a^2*b + 17*b^3)*c^13*d + 60*(4*a^3 +
3*a*b^2)*c^12*d^2 + (312*a^2*b + 29*b^3)*c^11*d^3 - (520*a^3 + 267*a*b^2)*c^10*d^4 - (93*a^2*b - 41*b^3)*c^9*d
^5 + (475*a^3 + 21*a*b^2)*c^8*d^6 + (87*a^2*b - 2*b^3)*c^7*d^7 - (245*a^3 + 6*a*b^2)*c^6*d^8)*cos(f*x + e)^3 +
(12*b^3*c^14 + 108*a*b^2*c^13*d + 104*a^3*c^3*d^11 - (648*a^2*b + 203*b^3)*c^12*d^2 + 15*(40*a^3 + 51*a*b^2)*
c^11*d^3 + (339*a^2*b + 47*b^3)*c^10*d^4 - (1189*a^3 + 933*a*b^2)*c^9*d^5 + (321*a^2*b + 152*b^3)*c^8*d^6 + (9
97*a^3 + 84*a*b^2)*c^7*d^7 - 4*(3*a^2*b + 2*b^3)*c^6*d^8 - 8*(64*a^3 + 3*a*b^2)*c^5*d^9)*cos(f*x + e)^2 + (8*b
^3*c^13*d + 72*a*b^2*c^12*d^2 - 407*a^3*c^4*d^10 + 84*a^3*c^2*d^12 - 8*(66*a^2*b + 19*b^3)*c^11*d^3 + 8*(65*a^
3 + 93*a*b^2)*c^10*d^4 + (84*a^2*b - 47*b^3)*c^9*d^5 - (964*a^3 + 759*a*b^2)*c^8*d^6 + (471*a^2*b + 203*b^3)*c
^7*d^7 + (767*a^3 - 57*a*b^2)*c^6*d^8 - 3*(9*a^2*b + 4*b^3)*c^5*d^9)*cos(f*x + e))*sin(f*x + e))/((c^19 - 5*c^
17*d^2 + 10*c^15*d^4 - 10*c^13*d^6 + 5*c^11*d^8 - c^9*d^10)*f*cos(f*x + e)^4 + 4*(c^18*d - 5*c^16*d^3 + 10*c^1
4*d^5 - 10*c^12*d^7 + 5*c^10*d^9 - c^8*d^11)*f*cos(f*x + e)^3 + 6*(c^17*d^2 - 5*c^15*d^4 + 10*c^13*d^6 - 10*c^
11*d^8 + 5*c^9*d^10 - c^7*d^12)*f*cos(f*x + e)^2 + 4*(c^16*d^3 - 5*c^14*d^5 + 10*c^12*d^7 - 10*c^10*d^9 + 5*c^
8*d^11 - c^6*d^13)*f*cos(f*x + e) + (c^15*d^4 - 5*c^13*d^6 + 10*c^11*d^8 - 10*c^9*d^10 + 5*c^7*d^12 - c^5*d^14
)*f), 1/24*(24*(a^3*c^14 - 5*a^3*c^12*d^2 + 10*a^3*c^10*d^4 - 10*a^3*c^8*d^6 + 5*a^3*c^6*d^8 - a^3*c^4*d^10)*f
*x*cos(f*x + e)^4 + 96*(a^3*c^13*d - 5*a^3*c^11*d^3 + 10*a^3*c^9*d^5 - 10*a^3*c^7*d^7 + 5*a^3*c^5*d^9 - a^3*c^
3*d^11)*f*x*cos(f*x + e)^3 + 144*(a^3*c^12*d^2 - 5*a^3*c^10*d^4 + 10*a^3*c^8*d^6 - 10*a^3*c^6*d^8 + 5*a^3*c^4*
d^10 - a^3*c^2*d^12)*f*x*cos(f*x + e)^2 + 96*(a^3*c^11*d^3 - 5*a^3*c^9*d^5 + 10*a^3*c^7*d^7 - 10*a^3*c^5*d^9 +
5*a^3*c^3*d^11 - a^3*c*d^13)*f*x*cos(f*x + e) + 24*(a^3*c^10*d^4 - 5*a^3*c^8*d^6 + 10*a^3*c^6*d^8 - 10*a^3*c^
4*d^10 + 5*a^3*c^2*d^12 - a^3*d^14)*f*x - 3*(63*a^3*c^4*d^9 - 36*a^3*c^2*d^11 + 8*a^3*d^13 - 4*(6*a^2*b + b^3)
*c^9*d^4 + 20*(2*a^3 + 3*a*b^2)*c^8*d^5 - 9*(8*a^2*b + 3*b^3)*c^7*d^6 - 5*(8*a^3 - 9*a*b^2)*c^6*d^7 - (9*a^2*b
+ 4*b^3)*c^5*d^8 + (63*a^3*c^8*d^5 - 36*a^3*c^6*d^7 + 8*a^3*c^4*d^9 - 4*(6*a^2*b + b^3)*c^13 + 20*(2*a^3 + 3*
a*b^2)*c^12*d - 9*(8*a^2*b + 3*b^3)*c^11*d^2 - 5*(8*a^3 - 9*a*b^2)*c^10*d^3 - (9*a^2*b + 4*b^3)*c^9*d^4)*cos(f
*x + e)^4 + 4*(63*a^3*c^7*d^6 - 36*a^3*c^5*d^8 + 8*a^3*c^3*d^10 - 4*(6*a^2*b + b^3)*c^12*d + 20*(2*a^3 + 3*a*b
^2)*c^11*d^2 - 9*(8*a^2*b + 3*b^3)*c^10*d^3 - 5*(8*a^3 - 9*a*b^2)*c^9*d^4 - (9*a^2*b + 4*b^3)*c^8*d^5)*cos(f*x
+ e)^3 + 6*(63*a^3*c^6*d^7 - 36*a^3*c^4*d^9 + 8*a^3*c^2*d^11 - 4*(6*a^2*b + b^3)*c^11*d^2 + 20*(2*a^3 + 3*a*b
^2)*c^10*d^3 - 9*(8*a^2*b + 3*b^3)*c^9*d^4 - 5*(8*a^3 - 9*a*b^2)*c^8*d^5 - (9*a^2*b + 4*b^3)*c^7*d^6)*cos(f*x
+ e)^2 + 4*(63*a^3*c^5*d^8 - 36*a^3*c^3*d^10 + ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{3}}{\left (c + d \sec {\left (e + f x \right )}\right )^{5}}\, dx \end {gather*}

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))**5,x)

[Out]

Integral((a + b*sec(e + f*x))**3/(c + d*sec(e + f*x))**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3173 vs. \(2 (603) = 1206\).
time = 0.77, size = 3173, normalized size = 5.10 \begin {gather*} \text {Too large to display} \end {gather*}

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))^5,x, algorithm="giac")

[Out]

1/12*(3*(24*a^2*b*c^9 + 4*b^3*c^9 - 40*a^3*c^8*d - 60*a*b^2*c^8*d + 72*a^2*b*c^7*d^2 + 27*b^3*c^7*d^2 + 40*a^3
*c^6*d^3 - 45*a*b^2*c^6*d^3 + 9*a^2*b*c^5*d^4 + 4*b^3*c^5*d^4 - 63*a^3*c^4*d^5 + 36*a^3*c^2*d^7 - 8*a^3*d^9)*(
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))/s
qrt(-c^2 + d^2)))/((c^13 - 4*c^11*d^2 + 6*c^9*d^4 - 4*c^7*d^6 + c^5*d^8)*sqrt(-c^2 + d^2)) + 12*(f*x + e)*a^3/
c^5 - (72*a*b^2*c^11*tan(1/2*f*x + 1/2*e)^7 - 12*b^3*c^11*tan(1/2*f*x + 1/2*e)^7 - 288*a^2*b*c^10*d*tan(1/2*f*
x + 1/2*e)^7 - 108*a*b^2*c^10*d*tan(1/2*f*x + 1/2*e)^7 - 60*b^3*c^10*d*tan(1/2*f*x + 1/2*e)^7 + 240*a^3*c^9*d^
2*tan(1/2*f*x + 1/2*e)^7 + 648*a^2*b*c^9*d^2*tan(1/2*f*x + 1/2*e)^7 + 324*a*b^2*c^9*d^2*tan(1/2*f*x + 1/2*e)^7
+ 189*b^3*c^9*d^2*tan(1/2*f*x + 1/2*e)^7 - 600*a^3*c^8*d^3*tan(1/2*f*x + 1/2*e)^7 - 504*a^2*b*c^8*d^3*tan(1/2
*f*x + 1/2*e)^7 - 891*a*b^2*c^8*d^3*tan(1/2*f*x + 1/2*e)^7 - 183*b^3*c^8*d^3*tan(1/2*f*x + 1/2*e)^7 + 240*a^3*
c^7*d^4*tan(1/2*f*x + 1/2*e)^7 + 459*a^2*b*c^7*d^4*tan(1/2*f*x + 1/2*e)^7 + 801*a*b^2*c^7*d^4*tan(1/2*f*x + 1/
2*e)^7 + 183*b^3*c^7*d^4*tan(1/2*f*x + 1/2*e)^7 + 435*a^3*c^6*d^5*tan(1/2*f*x + 1/2*e)^7 - 513*a^2*b*c^6*d^5*t
an(1/2*f*x + 1/2*e)^7 - 189*a*b^2*c^6*d^5*tan(1/2*f*x + 1/2*e)^7 - 189*b^3*c^6*d^5*tan(1/2*f*x + 1/2*e)^7 - 24
9*a^3*c^5*d^6*tan(1/2*f*x + 1/2*e)^7 + 153*a^2*b*c^5*d^6*tan(1/2*f*x + 1/2*e)^7 + 63*a*b^2*c^5*d^6*tan(1/2*f*x
+ 1/2*e)^7 + 60*b^3*c^5*d^6*tan(1/2*f*x + 1/2*e)^7 - 291*a^3*c^4*d^7*tan(1/2*f*x + 1/2*e)^7 + 45*a^2*b*c^4*d^
7*tan(1/2*f*x + 1/2*e)^7 - 72*a*b^2*c^4*d^7*tan(1/2*f*x + 1/2*e)^7 + 12*b^3*c^4*d^7*tan(1/2*f*x + 1/2*e)^7 + 2
73*a^3*c^3*d^8*tan(1/2*f*x + 1/2*e)^7 + 12*a^3*c^2*d^9*tan(1/2*f*x + 1/2*e)^7 - 84*a^3*c*d^10*tan(1/2*f*x + 1/
2*e)^7 + 24*a^3*d^11*tan(1/2*f*x + 1/2*e)^7 - 216*a*b^2*c^11*tan(1/2*f*x + 1/2*e)^5 + 12*b^3*c^11*tan(1/2*f*x
+ 1/2*e)^5 + 864*a^2*b*c^10*d*tan(1/2*f*x + 1/2*e)^5 + 108*a*b^2*c^10*d*tan(1/2*f*x + 1/2*e)^5 + 212*b^3*c^10*
d*tan(1/2*f*x + 1/2*e)^5 - 720*a^3*c^9*d^2*tan(1/2*f*x + 1/2*e)^5 - 648*a^2*b*c^9*d^2*tan(1/2*f*x + 1/2*e)^5 -
684*a*b^2*c^9*d^2*tan(1/2*f*x + 1/2*e)^5 - 197*b^3*c^9*d^2*tan(1/2*f*x + 1/2*e)^5 + 600*a^3*c^8*d^3*tan(1/2*f
*x + 1/2*e)^5 - 600*a^2*b*c^8*d^3*tan(1/2*f*x + 1/2*e)^5 + 819*a*b^2*c^8*d^3*tan(1/2*f*x + 1/2*e)^5 - 27*b^3*c
^8*d^3*tan(1/2*f*x + 1/2*e)^5 + 1360*a^3*c^7*d^4*tan(1/2*f*x + 1/2*e)^5 + 141*a^2*b*c^7*d^4*tan(1/2*f*x + 1/2*
e)^5 + 861*a*b^2*c^7*d^4*tan(1/2*f*x + 1/2*e)^5 - 27*b^3*c^7*d^4*tan(1/2*f*x + 1/2*e)^5 - 1051*a^3*c^6*d^5*tan
(1/2*f*x + 1/2*e)^5 - 237*a^2*b*c^6*d^5*tan(1/2*f*x + 1/2*e)^5 - 807*a*b^2*c^6*d^5*tan(1/2*f*x + 1/2*e)^5 - 19
7*b^3*c^6*d^5*tan(1/2*f*x + 1/2*e)^5 - 1029*a^3*c^5*d^6*tan(1/2*f*x + 1/2*e)^5 + 507*a^2*b*c^5*d^6*tan(1/2*f*x
+ 1/2*e)^5 + 39*a*b^2*c^5*d^6*tan(1/2*f*x + 1/2*e)^5 + 212*b^3*c^5*d^6*tan(1/2*f*x + 1/2*e)^5 + 759*a^3*c^4*d
^7*tan(1/2*f*x + 1/2*e)^5 - 27*a^2*b*c^4*d^7*tan(1/2*f*x + 1/2*e)^5 - 120*a*b^2*c^4*d^7*tan(1/2*f*x + 1/2*e)^5
+ 12*b^3*c^4*d^7*tan(1/2*f*x + 1/2*e)^5 + 473*a^3*c^3*d^8*tan(1/2*f*x + 1/2*e)^5 - 380*a^3*c^2*d^9*tan(1/2*f*
x + 1/2*e)^5 - 84*a^3*c*d^10*tan(1/2*f*x + 1/2*e)^5 + 72*a^3*d^11*tan(1/2*f*x + 1/2*e)^5 + 216*a*b^2*c^11*tan(
1/2*f*x + 1/2*e)^3 + 12*b^3*c^11*tan(1/2*f*x + 1/2*e)^3 - 864*a^2*b*c^10*d*tan(1/2*f*x + 1/2*e)^3 + 108*a*b^2*
c^10*d*tan(1/2*f*x + 1/2*e)^3 - 212*b^3*c^10*d*tan(1/2*f*x + 1/2*e)^3 + 720*a^3*c^9*d^2*tan(1/2*f*x + 1/2*e)^3
- 648*a^2*b*c^9*d^2*tan(1/2*f*x + 1/2*e)^3 + 684*a*b^2*c^9*d^2*tan(1/2*f*x + 1/2*e)^3 - 197*b^3*c^9*d^2*tan(1
/2*f*x + 1/2*e)^3 + 600*a^3*c^8*d^3*tan(1/2*f*x + 1/2*e)^3 + 600*a^2*b*c^8*d^3*tan(1/2*f*x + 1/2*e)^3 + 819*a*
b^2*c^8*d^3*tan(1/2*f*x + 1/2*e)^3 + 27*b^3*c^8*d^3*tan(1/2*f*x + 1/2*e)^3 - 1360*a^3*c^7*d^4*tan(1/2*f*x + 1/
2*e)^3 + 141*a^2*b*c^7*d^4*tan(1/2*f*x + 1/2*e)^3 - 861*a*b^2*c^7*d^4*tan(1/2*f*x + 1/2*e)^3 - 27*b^3*c^7*d^4*
tan(1/2*f*x + 1/2*e)^3 - 1051*a^3*c^6*d^5*tan(1/2*f*x + 1/2*e)^3 + 237*a^2*b*c^6*d^5*tan(1/2*f*x + 1/2*e)^3 -
807*a*b^2*c^6*d^5*tan(1/2*f*x + 1/2*e)^3 + 197*b^3*c^6*d^5*tan(1/2*f*x + 1/2*e)^3 + 1029*a^3*c^5*d^6*tan(1/2*f
*x + 1/2*e)^3 + 507*a^2*b*c^5*d^6*tan(1/2*f*x + 1/2*e)^3 - 39*a*b^2*c^5*d^6*tan(1/2*f*x + 1/2*e)^3 + 212*b^3*c
^5*d^6*tan(1/2*f*x + 1/2*e)^3 + 759*a^3*c^4*d^7*tan(1/2*f*x + 1/2*e)^3 + 27*a^2*b*c^4*d^7*tan(1/2*f*x + 1/2*e)
^3 - 120*a*b^2*c^4*d^7*tan(1/2*f*x + 1/2*e)^3 - 12*b^3*c^4*d^7*tan(1/2*f*x + 1/2*e)^3 - 473*a^3*c^3*d^8*tan(1/
2*f*x + 1/2*e)^3 - 380*a^3*c^2*d^9*tan(1/2*f*x + 1/2*e)^3 + 84*a^3*c*d^10*tan(1/2*f*x + 1/2*e)^3 + 72*a^3*d^11
*tan(1/2*f*x + 1/2*e)^3 - 72*a*b^2*c^11*tan(1/2*f*x + 1/2*e) - 12*b^3*c^11*tan(1/2*f*x + 1/2*e) + 288*a^2*b*c^
10*d*tan(1/2*f*x + 1/2*e) - 108*a*b^2*c^10*d*tan(1/2*f*x + 1/2*e) + 60*b^3*c^10*d*tan(1/2*f*x + 1/2*e) - 240*a
^3*c^9*d^2*tan(1/2*f*x + 1/2*e) + 648*a^2*b*c^9*d^2*tan(1/2*f*x + 1/2*e) - 324*a*b^2*c^9*d^2*tan(1/2*f*x + 1/2
*e) + 189*b^3*c^9*d^2*tan(1/2*f*x + 1/2*e) - 600*a^3*c^8*d^3*tan(1/2*f*x + 1/2*e) + 504*a^2*b*c^8*d^3*tan(1/2*
f*x + 1/2*e) - 891*a*b^2*c^8*d^3*tan(1/2*f*x + ...

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Mupad [B]
time = 16.35, size = 2500, normalized size = 4.02 \begin {gather*} \text {Too large to display} \end {gather*}

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))^5,x)

[Out]

(atan(((((c + d)^9*(c - d)^9)^(1/2)*((tan(e/2 + (f*x)/2)*(64*a^6*c^18 + 128*a^6*d^18 + 16*b^6*c^18 - 128*a^6*c
*d^17 - 128*a^6*c^17*d + 192*a^2*b^4*c^18 + 576*a^4*b^2*c^18 - 1024*a^6*c^2*d^16 + 1024*a^6*c^3*d^15 + 3584*a^
6*c^4*d^14 - 3584*a^6*c^5*d^13 - 6968*a^6*c^6*d^12 + 7168*a^6*c^7*d^11 + 8385*a^6*c^8*d^10 - 8960*a^6*c^9*d^9
- 7024*a^6*c^10*d^8 + 7168*a^6*c^11*d^7 + 4848*a^6*c^12*d^6 - 3584*a^6*c^13*d^5 - 1920*a^6*c^14*d^4 + 1024*a^6
*c^15*d^3 + 1152*a^6*c^16*d^2 + 16*b^6*c^10*d^8 + 216*b^6*c^12*d^6 + 761*b^6*c^14*d^4 + 216*b^6*c^16*d^2 - 360
*a*b^5*c^11*d^7 - 2910*a*b^5*c^13*d^5 - 3600*a*b^5*c^15*d^3 - 3200*a^3*b^3*c^17*d - 144*a^5*b*c^5*d^13 - 504*a
^5*b*c^7*d^11 + 3666*a^5*b*c^9*d^9 - 6624*a^5*b*c^11*d^7 + 2016*a^5*b*c^13*d^5 - 3840*a^5*b*c^15*d^3 + 72*a^2*
b^4*c^10*d^8 + 3087*a^2*b^4*c^12*d^6 + 9552*a^2*b^4*c^14*d^4 + 5472*a^2*b^4*c^16*d^2 - 64*a^3*b^3*c^5*d^13 - 1
44*a^3*b^3*c^7*d^11 + 1376*a^3*b^3*c^9*d^9 - 3604*a^3*b^3*c^11*d^7 - 6224*a^3*b^3*c^13*d^5 - 12640*a^3*b^3*c^1
5*d^3 + 720*a^4*b^2*c^6*d^12 - 2280*a^4*b^2*c^8*d^10 + 1431*a^4*b^2*c^10*d^8 + 5256*a^4*b^2*c^12*d^6 + 4416*a^
4*b^2*c^14*d^4 + 8256*a^4*b^2*c^16*d^2 - 480*a*b^5*c^17*d - 1920*a^5*b*c^17*d))/(2*(c^22*d + c^23 - c^8*d^15 -
c^9*d^14 + 7*c^10*d^13 + 7*c^11*d^12 - 21*c^12*d^11 - 21*c^13*d^10 + 35*c^14*d^9 + 35*c^15*d^8 - 35*c^16*d^7
- 35*c^17*d^6 + 21*c^18*d^5 + 21*c^19*d^4 - 7*c^20*d^3 - 7*c^21*d^2)) + (((32*a^3*c^27 + 16*b^3*c^27 + 96*a^2*
b*c^27 - 160*a^3*c^26*d - 16*b^3*c^26*d - 32*a^3*c^10*d^17 + 16*a^3*c^11*d^16 + 272*a^3*c^12*d^15 - 132*a^3*c^
13*d^14 - 1020*a^3*c^14*d^13 + 528*a^3*c^15*d^12 + 2160*a^3*c^16*d^11 - 1112*a^3*c^17*d^10 - 2920*a^3*c^18*d^9
+ 1280*a^3*c^19*d^8 + 2752*a^3*c^20*d^7 - 836*a^3*c^21*d^6 - 1852*a^3*c^22*d^5 + 352*a^3*c^23*d^4 + 800*a^3*c
^24*d^3 - 128*a^3*c^25*d^2 - 16*b^3*c^14*d^13 + 16*b^3*c^15*d^12 - 44*b^3*c^16*d^11 + 44*b^3*c^17*d^10 + 320*b
^3*c^18*d^9 - 320*b^3*c^19*d^8 - 520*b^3*c^20*d^7 + 520*b^3*c^21*d^6 + 320*b^3*c^22*d^5 - 320*b^3*c^23*d^4 - 4
4*b^3*c^24*d^3 + 44*b^3*c^25*d^2 + 180*a*b^2*c^15*d^12 - 180*a*b^2*c^16*d^11 - 480*a*b^2*c^17*d^10 + 480*a*b^2
*c^18*d^9 + 120*a*b^2*c^19*d^8 - 120*a*b^2*c^20*d^7 + 720*a*b^2*c^21*d^6 - 720*a*b^2*c^22*d^5 - 780*a*b^2*c^23
*d^4 + 780*a*b^2*c^24*d^3 + 240*a*b^2*c^25*d^2 - 36*a^2*b*c^14*d^13 + 36*a^2*b*c^15*d^12 - 144*a^2*b*c^16*d^11
+ 144*a^2*b*c^17*d^10 + 840*a^2*b*c^18*d^9 - 840*a^2*b*c^19*d^8 - 1200*a^2*b*c^20*d^7 + 1200*a^2*b*c^21*d^6 +
540*a^2*b*c^22*d^5 - 540*a^2*b*c^23*d^4 + 96*a^2*b*c^24*d^3 - 96*a^2*b*c^25*d^2 - 240*a*b^2*c^26*d - 96*a^2*b
*c^26*d)/(c^26*d + c^27 - c^12*d^15 - c^13*d^14 + 7*c^14*d^13 + 7*c^15*d^12 - 21*c^16*d^11 - 21*c^17*d^10 + 35
*c^18*d^9 + 35*c^19*d^8 - 35*c^20*d^7 - 35*c^21*d^6 + 21*c^22*d^5 + 21*c^23*d^4 - 7*c^24*d^3 - 7*c^25*d^2) - (
tan(e/2 + (f*x)/2)*((c + d)^9*(c - d)^9)^(1/2)*(4*b^3*c^9 - 8*a^3*d^9 + 24*a^2*b*c^9 - 40*a^3*c^8*d + 36*a^3*c
^2*d^7 - 63*a^3*c^4*d^5 + 40*a^3*c^6*d^3 + 4*b^3*c^5*d^4 + 27*b^3*c^7*d^2 - 45*a*b^2*c^6*d^3 + 9*a^2*b*c^5*d^4
+ 72*a^2*b*c^7*d^2 - 60*a*b^2*c^8*d)*(128*c^27*d - 128*c^10*d^18 + 128*c^11*d^17 + 1024*c^12*d^16 - 1024*c^13
*d^15 - 3584*c^14*d^14 + 3584*c^15*d^13 + 7168*c^16*d^12 - 7168*c^17*d^11 - 8960*c^18*d^10 + 8960*c^19*d^9 + 7
168*c^20*d^8 - 7168*c^21*d^7 - 3584*c^22*d^6 + 3584*c^23*d^5 + 1024*c^24*d^4 - 1024*c^25*d^3 - 128*c^26*d^2))/
(16*(c^23 - c^5*d^18 + 9*c^7*d^16 - 36*c^9*d^14 + 84*c^11*d^12 - 126*c^13*d^10 + 126*c^15*d^8 - 84*c^17*d^6 +
36*c^19*d^4 - 9*c^21*d^2)*(c^22*d + c^23 - c^8*d^15 - c^9*d^14 + 7*c^10*d^13 + 7*c^11*d^12 - 21*c^12*d^11 - 21
*c^13*d^10 + 35*c^14*d^9 + 35*c^15*d^8 - 35*c^16*d^7 - 35*c^17*d^6 + 21*c^18*d^5 + 21*c^19*d^4 - 7*c^20*d^3 -
7*c^21*d^2)))*((c + d)^9*(c - d)^9)^(1/2)*(4*b^3*c^9 - 8*a^3*d^9 + 24*a^2*b*c^9 - 40*a^3*c^8*d + 36*a^3*c^2*d^
7 - 63*a^3*c^4*d^5 + 40*a^3*c^6*d^3 + 4*b^3*c^5*d^4 + 27*b^3*c^7*d^2 - 45*a*b^2*c^6*d^3 + 9*a^2*b*c^5*d^4 + 72
*a^2*b*c^7*d^2 - 60*a*b^2*c^8*d))/(8*(c^23 - c^5*d^18 + 9*c^7*d^16 - 36*c^9*d^14 + 84*c^11*d^12 - 126*c^13*d^1
0 + 126*c^15*d^8 - 84*c^17*d^6 + 36*c^19*d^4 - 9*c^21*d^2)))*(4*b^3*c^9 - 8*a^3*d^9 + 24*a^2*b*c^9 - 40*a^3*c^
8*d + 36*a^3*c^2*d^7 - 63*a^3*c^4*d^5 + 40*a^3*c^6*d^3 + 4*b^3*c^5*d^4 + 27*b^3*c^7*d^2 - 45*a*b^2*c^6*d^3 + 9
*a^2*b*c^5*d^4 + 72*a^2*b*c^7*d^2 - 60*a*b^2*c^8*d)*1i)/(8*(c^23 - c^5*d^18 + 9*c^7*d^16 - 36*c^9*d^14 + 84*c^
11*d^12 - 126*c^13*d^10 + 126*c^15*d^8 - 84*c^17*d^6 + 36*c^19*d^4 - 9*c^21*d^2)) + (((c + d)^9*(c - d)^9)^(1/
2)*((tan(e/2 + (f*x)/2)*(64*a^6*c^18 + 128*a^6*d^18 + 16*b^6*c^18 - 128*a^6*c*d^17 - 128*a^6*c^17*d + 192*a^2*
b^4*c^18 + 576*a^4*b^2*c^18 - 1024*a^6*c^2*d^16 + 1024*a^6*c^3*d^15 + 3584*a^6*c^4*d^14 - 3584*a^6*c^5*d^13 -
6968*a^6*c^6*d^12 + 7168*a^6*c^7*d^11 + 8385*a^6*c^8*d^10 - 8960*a^6*c^9*d^9 - 7024*a^6*c^10*d^8 + 7168*a^6*c^
11*d^7 + 4848*a^6*c^12*d^6 - 3584*a^6*c^13*d^5 - 1920*a^6*c^14*d^4 + 1024*a^6*c^15*d^3 + 1152*a^6*c^16*d^2 + 1
6*b^6*c^10*d^8 + 216*b^6*c^12*d^6 + 761*b^6*c^14*d^4 + 216*b^6*c^16*d^2 - 360*a*b^5*c^11*d^7 - 2910*a*b^5*c^13
*d^5 - 3600*a*b^5*c^15*d^3 - 3200*a^3*b^3*c^17*...

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