3.2.94 \(\int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^4} \, dx\) [194]

Optimal. Leaf size=377 \[ \frac {a^2 x}{c^4}-\frac {\left (b^2 c^4 d \left (4 c^2+d^2\right )-a b \left (4 c^7+6 c^5 d^2\right )+a^2 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^4 (c-d)^{7/2} (c+d)^{7/2} f}+\frac {d^2 (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}-\frac {d (b c-a d) \left (6 b c^3-8 a c^2 d-b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac {\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))} \]

[Out]

a^2*x/c^4-(b^2*c^4*d*(4*c^2+d^2)-a*b*(4*c^7+6*c^5*d^2)+a^2*(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*f*x+1/2*e)/(c+d)^(1/2))/c^4/(c-d)^(7/2)/(c+d)^(7/2)/f+1/3*d^2*(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*d*(-a*d+b*c)*(-8*a*c^2*d+3*a*d^3+6*b*c^3-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*(2*a*b*c*d*(18*c^4-5*c^2*d^2+2*d^4)-a^2*d^2*(34*c^4-28*c^2*d^2+9*d^4)-b^2*(6*c^6+10*c
^4*d^2-c^2*d^4))*sin(f*x+e)/c^3/(c^2-d^2)^3/f/(d+c*cos(f*x+e))

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Rubi [A]
time = 1.37, antiderivative size = 377, 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 = {4026, 3127, 3110, 3100, 2814, 2738, 214} \begin {gather*} -\frac {\left (-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )+2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-\left (b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\right )\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 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )-a b \left (4 c^7+6 c^5 d^2\right )+b^2 c^4 d \left (4 c^2+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 (c-d)^{7/2} (c+d)^{7/2}}+\frac {a^2 x}{c^4}+\frac {d^2 \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 {d (b c-a d) \left (-8 a c^2 d+3 a d^3+6 b c^3-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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^4,x]

[Out]

(a^2*x)/c^4 - ((b^2*c^4*d*(4*c^2 + d^2) - a*b*(4*c^7 + 6*c^5*d^2) + a^2*(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*(c - d)^(7/2)*(c + d)^(7/2)*f) + (d^2*(b + a*Co
s[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) - (d*(b*c - a*d)*(6*b*c^3 - 8*a*c^2*d -
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) - ((2*a*b*c*d*(18*c^4 - 5*c^2
*d^2 + 2*d^4) - a^2*d^2*(34*c^4 - 28*c^2*d^2 + 9*d^4) - b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*Sin[e + f*x])/(6*c
^3*(c^2 - d^2)^3*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 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))^2}{(c+d \sec (e+f x))^4} \, dx &=\int \frac {\cos ^2(e+f x) (b+a \cos (e+f x))^2}{(d+c \cos (e+f x))^4} \, dx\\ &=\frac {d^2 (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 (-d (3 b c-2 a d)+\left (3 b c^2-3 a c d-b d^2\right ) \cos (e+f x)+3 a \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^2 (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 {d (b c-a d) \left (6 b c^3-8 a c^2 d-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 \left (a^2 d^2 \left (8 c^2-3 d^2\right )-2 a b c d \left (6 c^2-d^2\right )+b^2 \left (3 c^4+2 c^2 d^2\right )\right )+\left (b^2 c^2 d \left (6 c^2-d^2\right )-2 a b c \left (6 c^4-3 c^2 d^2+2 d^4\right )+a^2 \left (12 c^4 d-10 c^2 d^3+3 d^5\right )\right ) \cos (e+f x)-6 a^2 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^2 (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 {d (b c-a d) \left (6 b c^3-8 a c^2 d-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 {\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 \left (b^2 c^2 d \left (4 c^2+d^2\right )-2 a b c^3 \left (2 c^2+3 d^2\right )+a^2 \left (6 c^4 d-2 c^2 d^3+d^5\right )\right )-6 a^2 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^2 x}{c^4}+\frac {d^2 (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 {d (b c-a d) \left (6 b c^3-8 a c^2 d-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 {\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 (b^2 c^4 d \left (4 c^2+d^2\right )-2 a b c^5 \left (2 c^2+3 d^2\right )+a^2 \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^2 x}{c^4}+\frac {d^2 (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 {d (b c-a d) \left (6 b c^3-8 a c^2 d-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 {\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 (b^2 c^4 d \left (4 c^2+d^2\right )-2 a b c^5 \left (2 c^2+3 d^2\right )+a^2 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\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 )}{c^4 \left (c^2-d^2\right )^3 f}\\ &=\frac {a^2 x}{c^4}+\frac {\left (4 a b c^7-8 a^2 c^6 d-4 b^2 c^6 d+6 a b c^5 d^2+8 a^2 c^4 d^3-b^2 c^4 d^3-7 a^2 c^2 d^5+2 a^2 d^7\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^4 (c-d)^{7/2} (c+d)^{7/2} f}+\frac {d^2 (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 {d (b c-a d) \left (6 b c^3-8 a c^2 d-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 {\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 = 3.31, size = 438, normalized size = 1.16 \begin {gather*} \frac {(d+c \cos (e+f x)) \sec ^2(e+f x) (a+b \sec (e+f x))^2 \left (6 a^2 (e+f x) (d+c \cos (e+f x))^3+\frac {6 \left (b^2 c^4 d \left (4 c^2+d^2\right )-2 a b c^5 \left (2 c^2+3 d^2\right )+a^2 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\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))^3}{\left (c^2-d^2\right )^{7/2}}+\frac {2 c d^2 (b c-a d)^2 \sin (e+f x)}{c^2-d^2}-\frac {c d \left (a^2 d^2 \left (12 c^2-7 d^2\right )+b^2 \left (6 c^4-c^2 d^2\right )+a b \left (-18 c^3 d+8 c d^3\right )\right ) (d+c \cos (e+f x)) \sin (e+f x)}{\left (c^2-d^2\right )^2}+\frac {c \left (-2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )+a^2 d^2 \left (36 c^4-32 c^2 d^2+11 d^4\right )+b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\right )\right ) (d+c \cos (e+f x))^2 \sin (e+f x)}{\left (c^2-d^2\right )^3}\right )}{6 c^4 f (b+a \cos (e+f x))^2 (c+d \sec (e+f x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^4,x]

[Out]

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

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Maple [A]
time = 0.60, size = 635, normalized size = 1.68

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {\left (12 a^{2} c^{4} d^{2}+4 a^{2} c^{3} d^{3}-6 a^{2} d^{4} c^{2}-a^{2} c \,d^{5}+2 a^{2} d^{6}-12 a b d \,c^{5}-6 a b \,c^{4} d^{2}-4 a b \,d^{3} c^{3}+2 b^{2} c^{6}+2 b^{2} c^{5} d +6 b^{2} c^{4} d^{2}+b^{2} c^{3} d^{3}\right ) c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c -d \right ) \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}+\frac {2 \left (18 a^{2} c^{4} d^{2}-11 a^{2} d^{4} c^{2}+3 a^{2} d^{6}-18 a b d \,c^{5}-2 a b \,d^{3} c^{3}+3 b^{2} c^{6}+7 b^{2} c^{4} d^{2}\right ) c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 \left (c^{2}-2 c d +d^{2}\right ) \left (c^{2}+2 c d +d^{2}\right )}-\frac {\left (12 a^{2} c^{4} d^{2}-4 a^{2} c^{3} d^{3}-6 a^{2} d^{4} c^{2}+a^{2} c \,d^{5}+2 a^{2} d^{6}-12 a b d \,c^{5}+6 a b \,c^{4} d^{2}-4 a b \,d^{3} c^{3}+2 b^{2} c^{6}-2 b^{2} c^{5} d +6 b^{2} c^{4} d^{2}-b^{2} c^{3} d^{3}\right ) c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{3}}-\frac {\left (8 a^{2} c^{6} d -8 a^{2} c^{4} d^{3}+7 a^{2} c^{2} d^{5}-2 a^{2} d^{7}-4 a b \,c^{7}-6 a b \,c^{5} d^{2}+4 b^{2} c^{6} d +b^{2} c^{4} d^{3}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{4}}+\frac {2 a^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{4}}}{f}\)	\(635\)
default	\(\frac {\frac {\frac {2 \left (-\frac {\left (12 a^{2} c^{4} d^{2}+4 a^{2} c^{3} d^{3}-6 a^{2} d^{4} c^{2}-a^{2} c \,d^{5}+2 a^{2} d^{6}-12 a b d \,c^{5}-6 a b \,c^{4} d^{2}-4 a b \,d^{3} c^{3}+2 b^{2} c^{6}+2 b^{2} c^{5} d +6 b^{2} c^{4} d^{2}+b^{2} c^{3} d^{3}\right ) c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c -d \right ) \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}+\frac {2 \left (18 a^{2} c^{4} d^{2}-11 a^{2} d^{4} c^{2}+3 a^{2} d^{6}-18 a b d \,c^{5}-2 a b \,d^{3} c^{3}+3 b^{2} c^{6}+7 b^{2} c^{4} d^{2}\right ) c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 \left (c^{2}-2 c d +d^{2}\right ) \left (c^{2}+2 c d +d^{2}\right )}-\frac {\left (12 a^{2} c^{4} d^{2}-4 a^{2} c^{3} d^{3}-6 a^{2} d^{4} c^{2}+a^{2} c \,d^{5}+2 a^{2} d^{6}-12 a b d \,c^{5}+6 a b \,c^{4} d^{2}-4 a b \,d^{3} c^{3}+2 b^{2} c^{6}-2 b^{2} c^{5} d +6 b^{2} c^{4} d^{2}-b^{2} c^{3} d^{3}\right ) c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{3}}-\frac {\left (8 a^{2} c^{6} d -8 a^{2} c^{4} d^{3}+7 a^{2} c^{2} d^{5}-2 a^{2} d^{7}-4 a b \,c^{7}-6 a b \,c^{5} d^{2}+4 b^{2} c^{6} d +b^{2} c^{4} d^{3}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{4}}+\frac {2 a^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{4}}}{f}\)	\(635\)
risch	\(\text {Expression too large to display}\)	\(2571\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(f*x+e))^2/(c+d*sec(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

1/f*(2/c^4*((-1/2*(12*a^2*c^4*d^2+4*a^2*c^3*d^3-6*a^2*c^2*d^4-a^2*c*d^5+2*a^2*d^6-12*a*b*c^5*d-6*a*b*c^4*d^2-4
*a*b*c^3*d^3+2*b^2*c^6+2*b^2*c^5*d+6*b^2*c^4*d^2+b^2*c^3*d^3)*c/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/
2*e)^5+2/3*(18*a^2*c^4*d^2-11*a^2*c^2*d^4+3*a^2*d^6-18*a*b*c^5*d-2*a*b*c^3*d^3+3*b^2*c^6+7*b^2*c^4*d^2)*c/(c^2
-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3-1/2*(12*a^2*c^4*d^2-4*a^2*c^3*d^3-6*a^2*c^2*d^4+a^2*c*d^5+2*a
^2*d^6-12*a*b*c^5*d+6*a*b*c^4*d^2-4*a*b*c^3*d^3+2*b^2*c^6-2*b^2*c^5*d+6*b^2*c^4*d^2-b^2*c^3*d^3)*c/(c+d)/(c^3-
3*c^2*d+3*c*d^2-d^3)*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)^3-1/2*(8*a^2*c^6*
d-8*a^2*c^4*d^3+7*a^2*c^2*d^5-2*a^2*d^7-4*a*b*c^7-6*a*b*c^5*d^2+4*b^2*c^6*d+b^2*c^4*d^3)/(c^6-3*c^4*d^2+3*c^2*
d^4-d^6)/((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^2/c^4*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))^2/(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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1166 vs. \(2 (370) = 740\).
time = 4.92, size = 2394, normalized size = 6.35 \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))^2/(c+d*sec(f*x+e))^4,x, algorithm="fricas")

[Out]

[1/12*(12*(a^2*c^11 - 4*a^2*c^9*d^2 + 6*a^2*c^7*d^4 - 4*a^2*c^5*d^6 + a^2*c^3*d^8)*f*x*cos(f*x + e)^3 + 36*(a^
2*c^10*d - 4*a^2*c^8*d^3 + 6*a^2*c^6*d^5 - 4*a^2*c^4*d^7 + a^2*c^2*d^9)*f*x*cos(f*x + e)^2 + 36*(a^2*c^9*d^2 -
4*a^2*c^7*d^4 + 6*a^2*c^5*d^6 - 4*a^2*c^3*d^8 + a^2*c*d^10)*f*x*cos(f*x + e) + 12*(a^2*c^8*d^3 - 4*a^2*c^6*d^
5 + 6*a^2*c^4*d^7 - 4*a^2*c^2*d^9 + a^2*d^11)*f*x - 3*(4*a*b*c^7*d^3 + 6*a*b*c^5*d^5 - 7*a^2*c^2*d^8 + 2*a^2*d
^10 - 4*(2*a^2 + b^2)*c^6*d^4 + (8*a^2 - b^2)*c^4*d^6 + (4*a*b*c^10 + 6*a*b*c^8*d^2 - 7*a^2*c^5*d^5 + 2*a^2*c^
3*d^7 - 4*(2*a^2 + b^2)*c^9*d + (8*a^2 - b^2)*c^7*d^3)*cos(f*x + e)^3 + 3*(4*a*b*c^9*d + 6*a*b*c^7*d^3 - 7*a^2
*c^4*d^6 + 2*a^2*c^2*d^8 - 4*(2*a^2 + b^2)*c^8*d^2 + (8*a^2 - b^2)*c^6*d^4)*cos(f*x + e)^2 + 3*(4*a*b*c^8*d^2
+ 6*a*b*c^6*d^4 - 7*a^2*c^3*d^7 + 2*a^2*c*d^9 - 4*(2*a^2 + b^2)*c^7*d^3 + (8*a^2 - 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*(2*b^2*c^9*d^2 - 22*a*b*c^
8*d^3 + 14*a*b*c^6*d^5 + 8*a*b*c^4*d^7 + 23*a^2*c^3*d^8 - 6*a^2*c*d^10 + (26*a^2 + 11*b^2)*c^7*d^4 - (43*a^2 +
13*b^2)*c^5*d^6 + (6*b^2*c^11 - 36*a*b*c^10*d + 46*a*b*c^8*d^3 - 14*a*b*c^6*d^5 + 4*a*b*c^4*d^7 - 11*a^2*c^3*
d^8 + 4*(9*a^2 + b^2)*c^9*d^2 - (68*a^2 + 11*b^2)*c^7*d^4 + (43*a^2 + b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*(2*b^2*
c^10*d - 18*a*b*c^9*d^2 + 16*a*b*c^7*d^4 + 2*a*b*c^5*d^6 - 5*a^2*c^2*d^9 + (20*a^2 + 7*b^2)*c^8*d^3 - 5*(7*a^2
+ 2*b^2)*c^6*d^5 + (20*a^2 + 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^2*c^11 - 4*a^2*c^9*d^2 + 6*a^2*c^7*d^4 - 4*a^2*c^5*d^6 + a^2*c^3
*d^8)*f*x*cos(f*x + e)^3 + 18*(a^2*c^10*d - 4*a^2*c^8*d^3 + 6*a^2*c^6*d^5 - 4*a^2*c^4*d^7 + a^2*c^2*d^9)*f*x*c
os(f*x + e)^2 + 18*(a^2*c^9*d^2 - 4*a^2*c^7*d^4 + 6*a^2*c^5*d^6 - 4*a^2*c^3*d^8 + a^2*c*d^10)*f*x*cos(f*x + e)
+ 6*(a^2*c^8*d^3 - 4*a^2*c^6*d^5 + 6*a^2*c^4*d^7 - 4*a^2*c^2*d^9 + a^2*d^11)*f*x + 3*(4*a*b*c^7*d^3 + 6*a*b*c
^5*d^5 - 7*a^2*c^2*d^8 + 2*a^2*d^10 - 4*(2*a^2 + b^2)*c^6*d^4 + (8*a^2 - b^2)*c^4*d^6 + (4*a*b*c^10 + 6*a*b*c^
8*d^2 - 7*a^2*c^5*d^5 + 2*a^2*c^3*d^7 - 4*(2*a^2 + b^2)*c^9*d + (8*a^2 - b^2)*c^7*d^3)*cos(f*x + e)^3 + 3*(4*a
*b*c^9*d + 6*a*b*c^7*d^3 - 7*a^2*c^4*d^6 + 2*a^2*c^2*d^8 - 4*(2*a^2 + b^2)*c^8*d^2 + (8*a^2 - b^2)*c^6*d^4)*co
s(f*x + e)^2 + 3*(4*a*b*c^8*d^2 + 6*a*b*c^6*d^4 - 7*a^2*c^3*d^7 + 2*a^2*c*d^9 - 4*(2*a^2 + b^2)*c^7*d^3 + (8*a
^2 - b^2)*c^5*d^5)*cos(f*x + e))*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*s
in(f*x + e))) + (2*b^2*c^9*d^2 - 22*a*b*c^8*d^3 + 14*a*b*c^6*d^5 + 8*a*b*c^4*d^7 + 23*a^2*c^3*d^8 - 6*a^2*c*d^
10 + (26*a^2 + 11*b^2)*c^7*d^4 - (43*a^2 + 13*b^2)*c^5*d^6 + (6*b^2*c^11 - 36*a*b*c^10*d + 46*a*b*c^8*d^3 - 14
*a*b*c^6*d^5 + 4*a*b*c^4*d^7 - 11*a^2*c^3*d^8 + 4*(9*a^2 + b^2)*c^9*d^2 - (68*a^2 + 11*b^2)*c^7*d^4 + (43*a^2
+ b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*(2*b^2*c^10*d - 18*a*b*c^9*d^2 + 16*a*b*c^7*d^4 + 2*a*b*c^5*d^6 - 5*a^2*c^2
*d^9 + (20*a^2 + 7*b^2)*c^8*d^3 - 5*(7*a^2 + 2*b^2)*c^6*d^5 + (20*a^2 + 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)]

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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 )^{2}}{\left (c + d \sec {\left (e + f x \right )}\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))**2/(c+d*sec(f*x+e))**4,x)

[Out]

Integral((a + b*sec(e + f*x))**2/(c + d*sec(e + f*x))**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1201 vs. \(2 (362) = 724\).
time = 0.64, size = 1201, normalized size = 3.19 \begin {gather*} \frac {\frac {3 \, {\left (4 \, a b c^{7} - 8 \, a^{2} c^{6} d - 4 \, b^{2} c^{6} d + 6 \, a b c^{5} d^{2} + 8 \, a^{2} c^{4} d^{3} - b^{2} c^{4} d^{3} - 7 \, a^{2} c^{2} d^{5} + 2 \, a^{2} d^{7}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c^{10} - 3 \, c^{8} d^{2} + 3 \, c^{6} d^{4} - c^{4} d^{6}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {3 \, {\left (f x + e\right )} a^{2}}{c^{4}} - \frac {6 \, b^{2} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 36 \, a b c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, b^{2} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 36 \, a^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 54 \, a b c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, b^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 60 \, a^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a b c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 27 \, b^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, a^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a b c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, b^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 45 \, a^{2} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a b c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, b^{2} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, a^{2} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, a^{2} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a^{2} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, b^{2} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 72 \, a b c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 72 \, a^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 16 \, b^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 64 \, a b c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 116 \, a^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 28 \, b^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, a b c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 56 \, a^{2} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, a^{2} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, b^{2} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, a b c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b^{2} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 36 \, a^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 54 \, a b c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, b^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, a^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a b c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 27 \, b^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a b c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, b^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 45 \, a^{2} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a b c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, b^{2} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a^{2} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, a^{2} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{9} - 3 \, c^{7} d^{2} + 3 \, c^{5} d^{4} - c^{3} d^{6}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{3}}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))^2/(c+d*sec(f*x+e))^4,x, algorithm="giac")

[Out]

1/3*(3*(4*a*b*c^7 - 8*a^2*c^6*d - 4*b^2*c^6*d + 6*a*b*c^5*d^2 + 8*a^2*c^4*d^3 - b^2*c^4*d^3 - 7*a^2*c^2*d^5 +
2*a^2*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)*sqrt(-c^2 + d^2)) + 3*(f*x + e)*a^2/c^
4 - (6*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 36*a*b*c^7*d*tan(1/2*f*x + 1/2*e)^5 - 6*b^2*c^7*d*tan(1/2*f*x + 1/2*e)
^5 + 36*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 54*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 12*b^2*c^6*d^2*tan(1/2*f*
x + 1/2*e)^5 - 60*a^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 12*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 27*b^2*c^5*d^3*
tan(1/2*f*x + 1/2*e)^5 - 6*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 6*a*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 12*b^2*
c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 45*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 12*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^5
+ 3*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 6*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 15*a^2*c*d^7*tan(1/2*f*x + 1/
2*e)^5 + 6*a^2*d^8*tan(1/2*f*x + 1/2*e)^5 - 12*b^2*c^8*tan(1/2*f*x + 1/2*e)^3 + 72*a*b*c^7*d*tan(1/2*f*x + 1/2
*e)^3 - 72*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 16*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 64*a*b*c^5*d^3*tan(1/2
*f*x + 1/2*e)^3 + 116*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 28*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 8*a*b*c^3*d
^5*tan(1/2*f*x + 1/2*e)^3 - 56*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 12*a^2*d^8*tan(1/2*f*x + 1/2*e)^3 + 6*b^2*
c^8*tan(1/2*f*x + 1/2*e) - 36*a*b*c^7*d*tan(1/2*f*x + 1/2*e) + 6*b^2*c^7*d*tan(1/2*f*x + 1/2*e) + 36*a^2*c^6*d
^2*tan(1/2*f*x + 1/2*e) - 54*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e) + 12*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e) + 60*a^2*c
^5*d^3*tan(1/2*f*x + 1/2*e) - 12*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e) + 27*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e) - 6*a^
2*c^4*d^4*tan(1/2*f*x + 1/2*e) - 6*a*b*c^4*d^4*tan(1/2*f*x + 1/2*e) + 12*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e) - 45
*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e) - 12*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e) - 3*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e) -
6*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e) + 15*a^2*c*d^7*tan(1/2*f*x + 1/2*e) + 6*a^2*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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Mupad [B]
time = 15.07, size = 2500, normalized size = 6.63 \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))^2/(c + d/cos(e + f*x))^4,x)

[Out]

(2*a^2*atan(((a^2*((8*tan(e/2 + (f*x)/2)*(4*a^4*c^14 + 8*a^4*d^14 - 8*a^4*c*d^13 - 8*a^4*c^13*d + 16*a^2*b^2*c
^14 - 48*a^4*c^2*d^12 + 48*a^4*c^3*d^11 + 117*a^4*c^4*d^10 - 120*a^4*c^5*d^9 - 164*a^4*c^6*d^8 + 160*a^4*c^7*d
^7 + 156*a^4*c^8*d^6 - 120*a^4*c^9*d^5 - 92*a^4*c^10*d^4 + 48*a^4*c^11*d^3 + 44*a^4*c^12*d^2 + b^4*c^8*d^6 + 8
*b^4*c^10*d^4 + 16*b^4*c^12*d^2 - 12*a*b^3*c^9*d^5 - 56*a*b^3*c^11*d^3 + 24*a^3*b*c^5*d^9 - 68*a^3*b*c^7*d^7 +
40*a^3*b*c^9*d^5 - 32*a^3*b*c^11*d^3 - 4*a^2*b^2*c^4*d^10 - 2*a^2*b^2*c^6*d^8 + 40*a^2*b^2*c^8*d^6 - 12*a^2*b
^2*c^10*d^4 + 112*a^2*b^2*c^12*d^2 - 32*a*b^3*c^13*d - 64*a^3*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) + (
a^2*((8*(4*a^2*c^21 - 16*a^2*c^20*d - 8*b^2*c^20*d - 4*a^2*c^8*d^13 + 2*a^2*c^9*d^12 + 26*a^2*c^10*d^11 - 14*a
^2*c^11*d^10 - 70*a^2*c^12*d^9 + 30*a^2*c^13*d^8 + 110*a^2*c^14*d^7 - 30*a^2*c^15*d^6 - 110*a^2*c^16*d^5 + 20*
a^2*c^17*d^4 + 64*a^2*c^18*d^3 - 12*a^2*c^19*d^2 - 2*b^2*c^11*d^10 + 2*b^2*c^12*d^9 - 2*b^2*c^13*d^8 + 2*b^2*c
^14*d^7 + 18*b^2*c^15*d^6 - 18*b^2*c^16*d^5 - 22*b^2*c^17*d^4 + 22*b^2*c^18*d^3 + 8*b^2*c^19*d^2 + 8*a*b*c^21
- 8*a*b*c^20*d + 12*a*b*c^12*d^9 - 12*a*b*c^13*d^8 - 28*a*b*c^14*d^7 + 28*a*b*c^15*d^6 + 12*a*b*c^16*d^5 - 12*
a*b*c^17*d^4 + 12*a*b*c^18*d^3 - 12*a*b*c^19*d^2))/(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^2*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^1
4*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))/c^4 + (a^2*((8*tan(e/2 + (f*x)/2)*(4*a^4*c^14 + 8*a^4*d^14 - 8*a^4*c*d^1
3 - 8*a^4*c^13*d + 16*a^2*b^2*c^14 - 48*a^4*c^2*d^12 + 48*a^4*c^3*d^11 + 117*a^4*c^4*d^10 - 120*a^4*c^5*d^9 -
164*a^4*c^6*d^8 + 160*a^4*c^7*d^7 + 156*a^4*c^8*d^6 - 120*a^4*c^9*d^5 - 92*a^4*c^10*d^4 + 48*a^4*c^11*d^3 + 44
*a^4*c^12*d^2 + b^4*c^8*d^6 + 8*b^4*c^10*d^4 + 16*b^4*c^12*d^2 - 12*a*b^3*c^9*d^5 - 56*a*b^3*c^11*d^3 + 24*a^3
*b*c^5*d^9 - 68*a^3*b*c^7*d^7 + 40*a^3*b*c^9*d^5 - 32*a^3*b*c^11*d^3 - 4*a^2*b^2*c^4*d^10 - 2*a^2*b^2*c^6*d^8
+ 40*a^2*b^2*c^8*d^6 - 12*a^2*b^2*c^10*d^4 + 112*a^2*b^2*c^12*d^2 - 32*a*b^3*c^13*d - 64*a^3*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) - (a^2*((8*(4*a^2*c^21 - 16*a^2*c^20*d - 8*b^2*c^20*d - 4*a^2*c^8*d^13 + 2*a^2*c^9
*d^12 + 26*a^2*c^10*d^11 - 14*a^2*c^11*d^10 - 70*a^2*c^12*d^9 + 30*a^2*c^13*d^8 + 110*a^2*c^14*d^7 - 30*a^2*c^
15*d^6 - 110*a^2*c^16*d^5 + 20*a^2*c^17*d^4 + 64*a^2*c^18*d^3 - 12*a^2*c^19*d^2 - 2*b^2*c^11*d^10 + 2*b^2*c^12
*d^9 - 2*b^2*c^13*d^8 + 2*b^2*c^14*d^7 + 18*b^2*c^15*d^6 - 18*b^2*c^16*d^5 - 22*b^2*c^17*d^4 + 22*b^2*c^18*d^3
+ 8*b^2*c^19*d^2 + 8*a*b*c^21 - 8*a*b*c^20*d + 12*a*b*c^12*d^9 - 12*a*b*c^13*d^8 - 28*a*b*c^14*d^7 + 28*a*b*c
^15*d^6 + 12*a*b*c^16*d^5 - 12*a*b*c^17*d^4 + 12*a*b*c^18*d^3 - 12*a*b*c^19*d^2))/(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^2*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^1
2*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))/c^4)/((16*(4*a^6*d^13 - 8*a^5*b*c^13 - 2*
a^6*c*d^12 + 16*a^6*c^12*d + 16*a^4*b^2*c^13 - 26*a^6*c^2*d^11 + 11*a^6*c^3*d^10 + 70*a^6*c^4*d^9 - 34*a^6*c^5
*d^8 - 110*a^6*c^6*d^7 + 66*a^6*c^7*d^6 + 110*a^6*c^8*d^5 - 64*a^6*c^9*d^4 - 64*a^6*c^10*d^3 + 48*a^6*c^11*d^2
- 32*a^3*b^3*c^12*d + 8*a^4*b^2*c^12*d + 12*a^5*b*c^4*d^9 + 12*a^5*b*c^5*d^8 - 40*a^5*b*c^6*d^7 - 28*a^5*b*c^
7*d^6 + 28*a^5*b*c^8*d^5 + 12*a^5*b*c^9*d^4 - 44*a^5*b*c^10*d^3 + 12*a^5*b*c^11*d^2 + a^2*b^4*c^7*d^6 + 8*a^2*
b^4*c^9*d^4 + 16*a^2*b^4*c^11*d^2 - 12*a^3*b^3*c^8*d^5 - 56*a^3*b^3*c^10*d^3 - 2*a^4*b^2*c^3*d^10 - 2*a^4*b^2*
c^4*d^9 - 2*a^4*b^2*c^6*d^7 + 22*a^4*b^2*c^7*d^6 + 18*a^4*b^2*c^8*d^5 + 10*a^4*b^2*c^9*d^4 - 22*a^4*b^2*c^10*d
^3 + 104*a^4*b^2*c^11*d^2 - 56*a^5*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^2*((8*tan(e/2 + (f*x)/
2)*(4*a^4*c^14 + 8*a^4*d^14 - 8*a^4*c*d^13 - 8*a^4*c^13*d + 16*a^2*b^2*c^14 - 48*a^4*c^2*d^12 + 48*a^4*c^3*d^1
1 + 117*a^4*c^4*d^10 - 120*a^4*c^5*d^9 - 164*a^4*c^6*d^8 + 160*a^4*c^7*d^7 + 156*a^4*c^8*d^6 - 120*a^4*c^9*d^5
- 92*a^4*c^10*d^4 + 48*a^4*c^11*d^3 + 44*a^4*c...

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