3.12.35 \(\int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx\) [1135]

Optimal. Leaf size=351 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 (c-i d)^{5/2} f}+\frac {\left (2 i c^2-14 c d-47 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{9/2} f}+\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c-i d) (c+i d)^3 f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c-i d)^2 (c+i d)^4 f \sqrt {c+d \tan (e+f x)}} \]

[Out]

-1/4*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^2/(c-I*d)^(5/2)/f+1/8*(2*I*c^2-14*c*d-47*I*d^2)*arctanh
((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^2/(c+I*d)^(9/2)/f+1/8*d*(2*c^3+9*I*c^2*d+88*c*d^2-45*I*d^3)/a^2/(c-I*
d)^2/(c+I*d)^4/f/(c+d*tan(f*x+e))^(1/2)+1/24*d*(6*c^2+27*I*c*d+49*d^2)/a^2/(c-I*d)/(c+I*d)^3/f/(c+d*tan(f*x+e)
)^(3/2)+1/8*(2*I*c-9*d)/a^2/(c+I*d)^2/f/(1+I*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2)-1/4/(I*c-d)/f/(a+I*a*tan(f*x+e
))^2/(c+d*tan(f*x+e))^(3/2)

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Rubi [A]
time = 0.70, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3640, 3677, 3610, 3620, 3618, 65, 214} \begin {gather*} \frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))^{3/2}}+\frac {\left (2 i c^2-14 c d-47 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 f (c+i d)^{9/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 f (c-i d)^2 (c+i d)^4 \sqrt {c+d \tan (e+f x)}}+\frac {-9 d+2 i c}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 f (c-i d)^{5/2}}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

((-1/4*I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*(c - I*d)^(5/2)*f) + (((2*I)*c^2 - 14*c*d - (4
7*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(9/2)*f) + (d*(6*c^2 + (27*I)*c*d
+ 49*d^2))/(24*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*I)*c - 9*d)/(8*a^2*(c + I*d)^2*f*
(1 + I*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*
x])^(3/2)) + (d*(2*c^3 + (9*I)*c^2*d + 88*c*d^2 - (45*I)*d^3))/(8*a^2*(c - I*d)^2*(c + I*d)^4*f*Sqrt[c + d*Tan
[e + f*x]])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3640

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3677

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

Rubi steps

\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{2} a (4 i c-11 d)-\frac {7}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx}{4 a^2 (i c-d)}\\ &=\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{2} a^2 \left (4 c^2+18 i c d-49 d^2\right )-\frac {5}{2} a^2 (2 c+9 i d) d \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx}{8 a^4 (c+i d)^2}\\ &=\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{2} a^2 \left (4 c^3+18 i c^2 d-39 c d^2+45 i d^3\right )-\frac {1}{2} a^2 d \left (6 c^2+27 i c d+49 d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{8 a^4 (c+i d)^2 \left (c^2+d^2\right )}\\ &=\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {-\frac {1}{2} a^2 \left (4 c^4+18 i c^3 d-33 c^2 d^2+72 i c d^3+49 d^4\right )-\frac {1}{2} a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^4 (c+i d)^2 \left (c^2+d^2\right )^2}\\ &=\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^2 (c-i d)^2}+\frac {\left (2 c^2+14 i c d-47 d^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^2 (c+i d)^4}\\ &=\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{8 a^2 (c-i d)^2 f}-\frac {\left (i \left (2 c^2+14 i c d-47 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{16 a^2 (c+i d)^4 f}\\ &=\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{4 a^2 (c-i d)^2 d f}-\frac {\left (2 c^2+14 i c d-47 d^2\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{8 a^2 (c+i d)^4 d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 (c-i d)^{5/2} f}-\frac {\left (14 c d-i \left (2 c^2-47 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{9/2} f}+\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c+i d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1004\) vs. \(2(351)=702\).
time = 9.87, size = 1004, normalized size = 2.86 \begin {gather*} \frac {\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac {i (4 c+15 i d) \cos (2 f x)}{16 (c+i d)^4}+\frac {\left (9 i c^4 \cos (e)-24 c^3 d \cos (e)+75 i c^2 d^2 \cos (e)+458 c d^3 \cos (e)-192 i d^4 \cos (e)+9 i c^3 d \sin (e)-24 c^2 d^2 \sin (e)+75 i c d^3 \sin (e)+10 d^4 \sin (e)\right ) \left (\frac {1}{48} \cos (2 e)+\frac {1}{48} i \sin (2 e)\right )}{(c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e))}+\frac {\cos (4 f x) \left (\frac {1}{16} i \cos (2 e)+\frac {1}{16} \sin (2 e)\right )}{(c+i d)^3}+\frac {(4 c+15 i d) \sin (2 f x)}{16 (c+i d)^4}+\frac {\left (\frac {1}{16} \cos (2 e)-\frac {1}{16} i \sin (2 e)\right ) \sin (4 f x)}{(c+i d)^3}+\frac {\frac {2}{3} d^5 \cos (2 e)+\frac {2}{3} i d^5 \sin (2 e)}{(c-i d)^2 (c+i d)^4 (c \cos (e+f x)+d \sin (e+f x))^2}-\frac {4 \left (\frac {7}{2} i c d^4 \cos (2 e-f x)+\frac {3}{2} d^5 \cos (2 e-f x)-\frac {7}{2} i c d^4 \cos (2 e+f x)-\frac {3}{2} d^5 \cos (2 e+f x)-\frac {7}{2} c d^4 \sin (2 e-f x)+\frac {3}{2} i d^5 \sin (2 e-f x)+\frac {7}{2} c d^4 \sin (2 e+f x)-\frac {3}{2} i d^5 \sin (2 e+f x)\right )}{3 (c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}\right )}{f (a+i a \tan (e+f x))^2}+\frac {\sec ^2(e+f x) (\cos (2 e)+i \sin (2 e)) (\cos (f x)+i \sin (f x))^2 \left (-\frac {i \left (4 c^4+18 i c^3 d-33 c^2 d^2+72 i c d^3+49 d^4\right ) \left (\frac {\text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )}{\sqrt {-c-i d}}-\frac {\text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )}{\sqrt {-c+i d}}\right ) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left (1+\tan ^2(e+f x)\right )}+\frac {2 \left (2 c^3 d+9 i c^2 d^2+88 c d^3-45 i d^4\right ) \left (\frac {\text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )}{2 \sqrt {-c-i d}}+\frac {\text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )}{2 \sqrt {-c+i d}}\right ) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left (1+\tan ^2(e+f x)\right )}\right )}{16 (c-i d)^2 (c+i d)^4 f (a+i a \tan (e+f x))^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((I/16)*(4*c +
(15*I)*d)*Cos[2*f*x])/(c + I*d)^4 + (((9*I)*c^4*Cos[e] - 24*c^3*d*Cos[e] + (75*I)*c^2*d^2*Cos[e] + 458*c*d^3*
Cos[e] - (192*I)*d^4*Cos[e] + (9*I)*c^3*d*Sin[e] - 24*c^2*d^2*Sin[e] + (75*I)*c*d^3*Sin[e] + 10*d^4*Sin[e])*(C
os[2*e]/48 + (I/48)*Sin[2*e]))/((c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + (Cos[4*f*x]*((I/16)*Cos[2*e]
+ Sin[2*e]/16))/(c + I*d)^3 + ((4*c + (15*I)*d)*Sin[2*f*x])/(16*(c + I*d)^4) + ((Cos[2*e]/16 - (I/16)*Sin[2*e]
)*Sin[4*f*x])/(c + I*d)^3 + ((2*d^5*Cos[2*e])/3 + ((2*I)/3)*d^5*Sin[2*e])/((c - I*d)^2*(c + I*d)^4*(c*Cos[e +
f*x] + d*Sin[e + f*x])^2) - (4*(((7*I)/2)*c*d^4*Cos[2*e - f*x] + (3*d^5*Cos[2*e - f*x])/2 - ((7*I)/2)*c*d^4*Co
s[2*e + f*x] - (3*d^5*Cos[2*e + f*x])/2 - (7*c*d^4*Sin[2*e - f*x])/2 + ((3*I)/2)*d^5*Sin[2*e - f*x] + (7*c*d^4
*Sin[2*e + f*x])/2 - ((3*I)/2)*d^5*Sin[2*e + f*x]))/(3*(c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])*(c*Cos[e
+ f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]^2*(Cos[2*e] + I*Sin[2*e])*(Cos[f*x] +
I*Sin[f*x])^2*(((-I)*(4*c^4 + (18*I)*c^3*d - 33*c^2*d^2 + (72*I)*c*d^3 + 49*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f
*x]]/Sqrt[-c - I*d]]/Sqrt[-c - I*d] - ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/Sqrt[-c + I*d])*Sec[e +
f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2)) + (2*(2*c^3*d + (9*I)*c^2*
d^2 + 88*c*d^3 - (45*I)*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/(2*Sqrt[-c - I*d]) + ArcTan[Sqrt
[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/(2*Sqrt[-c + I*d]))*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] +
d*Sin[e + f*x])*(1 + Tan[e + f*x]^2))))/(16*(c - I*d)^2*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])^2)

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Maple [A]
time = 0.48, size = 518, normalized size = 1.48

method	result	size
derivativedivides	\(\frac {2 d^{3} \left (\frac {i \left (\frac {-\frac {d \left (2 i c^{6}-9 i c^{4} d^{2}-24 i c^{2} d^{4}-13 i d^{6}-15 c^{5} d -30 c^{3} d^{3}-15 c \,d^{5}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2 \left (2 i c d +c^{2}-d^{2}\right )}+\frac {d \left (2 i c^{7}-28 i c^{5} d^{2}-62 i c^{3} d^{4}-32 i c \,d^{6}-19 c^{6} d -23 c^{4} d^{3}+11 c^{2} d^{5}+15 d^{7}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{4 i c d +2 c^{2}-2 d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (16 i c^{6} d -15 i c^{4} d^{3}-78 i c^{2} d^{5}-47 i d^{7}+2 c^{7}-57 c^{5} d^{2}-120 c^{3} d^{4}-61 c \,d^{6}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}\right )}{8 \left (i d -c \right )^{2} \left (i d +c \right )^{5} d^{3}}-\frac {-2 i c d -4 c^{2}-2 d^{2}}{\left (i d -c \right )^{2} \left (i d +c \right )^{5} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-c^{2}-d^{2}}{3 \left (i d +c \right )^{4} \left (i d -c \right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {i \left (5 i c^{4} d -10 i c^{2} d^{3}+i d^{5}+c^{5}-10 c^{3} d^{2}+5 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{5} d^{3}}\right )}{f \,a^{2}}\)	\(518\)
default	\(\frac {2 d^{3} \left (\frac {i \left (\frac {-\frac {d \left (2 i c^{6}-9 i c^{4} d^{2}-24 i c^{2} d^{4}-13 i d^{6}-15 c^{5} d -30 c^{3} d^{3}-15 c \,d^{5}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2 \left (2 i c d +c^{2}-d^{2}\right )}+\frac {d \left (2 i c^{7}-28 i c^{5} d^{2}-62 i c^{3} d^{4}-32 i c \,d^{6}-19 c^{6} d -23 c^{4} d^{3}+11 c^{2} d^{5}+15 d^{7}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{4 i c d +2 c^{2}-2 d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (16 i c^{6} d -15 i c^{4} d^{3}-78 i c^{2} d^{5}-47 i d^{7}+2 c^{7}-57 c^{5} d^{2}-120 c^{3} d^{4}-61 c \,d^{6}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}\right )}{8 \left (i d -c \right )^{2} \left (i d +c \right )^{5} d^{3}}-\frac {-2 i c d -4 c^{2}-2 d^{2}}{\left (i d -c \right )^{2} \left (i d +c \right )^{5} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-c^{2}-d^{2}}{3 \left (i d +c \right )^{4} \left (i d -c \right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {i \left (5 i c^{4} d -10 i c^{2} d^{3}+i d^{5}+c^{5}-10 c^{3} d^{2}+5 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{5} d^{3}}\right )}{f \,a^{2}}\)	\(518\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/f/a^2*d^3*(1/8*I/(I*d-c)^2/(c+I*d)^5/d^3*((-1/2*d*(2*I*c^6-9*I*c^4*d^2-24*I*c^2*d^4-13*I*d^6-15*c^5*d-30*c^3
*d^3-15*c*d^5)/(2*I*c*d+c^2-d^2)*(c+d*tan(f*x+e))^(3/2)+1/2*d*(2*I*c^7-28*I*c^5*d^2-62*I*c^3*d^4-32*I*c*d^6-19
*c^6*d-23*c^4*d^3+11*c^2*d^5+15*d^7)/(2*I*c*d+c^2-d^2)*(c+d*tan(f*x+e))^(1/2))/(-d*tan(f*x+e)+I*d)^2-1/2*(-57*
c^5*d^2-120*c^3*d^4-61*c*d^6+16*I*c^6*d-15*I*c^4*d^3-78*I*c^2*d^5-47*I*d^7+2*c^7)/(2*I*c*d+c^2-d^2)/(-I*d-c)^(
1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2)))-1/(I*d-c)^2/(c+I*d)^5*(-2*I*c*d-4*c^2-2*d^2)/(c+d*tan(f*x+
e))^(1/2)-1/3*(-c^2-d^2)/(c+I*d)^4/(I*d-c)^2/(c+d*tan(f*x+e))^(3/2)+1/8*I/(I*d-c)^(5/2)/(c+I*d)^5*(5*c*d^4-10*
I*c^2*d^3+I*d^5-10*c^3*d^2+5*I*c^4*d+c^5)/d^3*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3287 vs. \(2 (289) = 578\).
time = 10.19, size = 3287, normalized size = 9.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-1/96*(24*((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(8*I*f*x + 8*I*e) + 2*(a^2*
c^8 + 2*I*a^2*c^7*d + 2*a^2*c^6*d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d^6 + 2*I*a^2*c*d^7 - a^2*
d^8)*f*e^(6*I*f*x + 6*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2*c^5*d^3 - 10*a^2*c^4*d^4 - 4*I
*a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4*I*a^2*c*d^7 + a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt(1/16*I/((-I*a^4*c^5 - 5*a^
4*c^4*d + 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 - 5*I*a^4*c*d^4 - a^4*d^5)*f^2))*log(-2*(4*((I*a^2*c^3 + 3*a^2*c^2
*d - 3*I*a^2*c*d^2 - a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c^3 + 3*a^2*c^2*d - 3*I*a^2*c*d^2 - a^2*d^3)*f)*s
qrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/16*I/((-I*a^4*c^5 - 5*a^4*c^4*
d + 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 - 5*I*a^4*c*d^4 - a^4*d^5)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^
(-2*I*f*x - 2*I*e)) - 24*((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(8*I*f*x + 8
*I*e) + 2*(a^2*c^8 + 2*I*a^2*c^7*d + 2*a^2*c^6*d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d^6 + 2*I*a
^2*c*d^7 - a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2*c^5*d^3 - 10*a^
2*c^4*d^4 - 4*I*a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4*I*a^2*c*d^7 + a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt(1/16*I/((-I
*a^4*c^5 - 5*a^4*c^4*d + 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 - 5*I*a^4*c*d^4 - a^4*d^5)*f^2))*log(-2*(4*((-I*a^2
*c^3 - 3*a^2*c^2*d + 3*I*a^2*c*d^2 + a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (-I*a^2*c^3 - 3*a^2*c^2*d + 3*I*a^2*c*d^
2 + a^2*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/16*I/((-I*a^4
*c^5 - 5*a^4*c^4*d + 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 - 5*I*a^4*c*d^4 - a^4*d^5)*f^2)) - (c - I*d)*e^(2*I*f*x
+ 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) - 3*((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*
f*e^(8*I*f*x + 8*I*e) + 2*(a^2*c^8 + 2*I*a^2*c^7*d + 2*a^2*c^6*d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2
*c^2*d^6 + 2*I*a^2*c*d^7 - a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2
*c^5*d^3 - 10*a^2*c^4*d^4 - 4*I*a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4*I*a^2*c*d^7 + a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*
sqrt(-(4*I*c^4 - 56*c^3*d - 384*I*c^2*d^2 + 1316*c*d^3 + 2209*I*d^4)/((I*a^4*c^9 - 9*a^4*c^8*d - 36*I*a^4*c^7*
d^2 + 84*a^4*c^6*d^3 + 126*I*a^4*c^5*d^4 - 126*a^4*c^4*d^5 - 84*I*a^4*c^3*d^6 + 36*a^4*c^2*d^7 + 9*I*a^4*c*d^8
- a^4*d^9)*f^2))*log(-1/8*(2*c^3 + 16*I*c^2*d - 61*c*d^2 - 47*I*d^3 - ((I*a^2*c^5 - 5*a^2*c^4*d - 10*I*a^2*c^
3*d^2 + 10*a^2*c^2*d^3 + 5*I*a^2*c*d^4 - a^2*d^5)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c^5 - 5*a^2*c^4*d - 10*I*a^2*
c^3*d^2 + 10*a^2*c^2*d^3 + 5*I*a^2*c*d^4 - a^2*d^5)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*
f*x + 2*I*e) + 1))*sqrt(-(4*I*c^4 - 56*c^3*d - 384*I*c^2*d^2 + 1316*c*d^3 + 2209*I*d^4)/((I*a^4*c^9 - 9*a^4*c^
8*d - 36*I*a^4*c^7*d^2 + 84*a^4*c^6*d^3 + 126*I*a^4*c^5*d^4 - 126*a^4*c^4*d^5 - 84*I*a^4*c^3*d^6 + 36*a^4*c^2*
d^7 + 9*I*a^4*c*d^8 - a^4*d^9)*f^2)) + (2*c^3 + 14*I*c^2*d - 47*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*
e)/((I*a^2*c^5 - 5*a^2*c^4*d - 10*I*a^2*c^3*d^2 + 10*a^2*c^2*d^3 + 5*I*a^2*c*d^4 - a^2*d^5)*f)) + 3*((a^2*c^8
+ 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(8*I*f*x + 8*I*e) + 2*(a^2*c^8 + 2*I*a^2*c^7*d
+ 2*a^2*c^6*d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d^6 + 2*I*a^2*c*d^7 - a^2*d^8)*f*e^(6*I*f*x +
6*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2*c^5*d^3 - 10*a^2*c^4*d^4 - 4*I*a^2*c^3*d^5 - 4*a^2
*c^2*d^6 - 4*I*a^2*c*d^7 + a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt(-(4*I*c^4 - 56*c^3*d - 384*I*c^2*d^2 + 1316*c*
d^3 + 2209*I*d^4)/((I*a^4*c^9 - 9*a^4*c^8*d - 36*I*a^4*c^7*d^2 + 84*a^4*c^6*d^3 + 126*I*a^4*c^5*d^4 - 126*a^4*
c^4*d^5 - 84*I*a^4*c^3*d^6 + 36*a^4*c^2*d^7 + 9*I*a^4*c*d^8 - a^4*d^9)*f^2))*log(-1/8*(2*c^3 + 16*I*c^2*d - 61
*c*d^2 - 47*I*d^3 - ((-I*a^2*c^5 + 5*a^2*c^4*d + 10*I*a^2*c^3*d^2 - 10*a^2*c^2*d^3 - 5*I*a^2*c*d^4 + a^2*d^5)*
f*e^(2*I*f*x + 2*I*e) + (-I*a^2*c^5 + 5*a^2*c^4*d + 10*I*a^2*c^3*d^2 - 10*a^2*c^2*d^3 - 5*I*a^2*c*d^4 + a^2*d^
5)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(4*I*c^4 - 56*c^3*d - 38
4*I*c^2*d^2 + 1316*c*d^3 + 2209*I*d^4)/((I*a^4*c^9 - 9*a^4*c^8*d - 36*I*a^4*c^7*d^2 + 84*a^4*c^6*d^3 + 126*I*a
^4*c^5*d^4 - 126*a^4*c^4*d^5 - 84*I*a^4*c^3*d^6 + 36*a^4*c^2*d^7 + 9*I*a^4*c*d^8 - a^4*d^9)*f^2)) + (2*c^3 + 1
4*I*c^2*d - 47*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((I*a^2*c^5 - 5*a^2*c^4*d - 10*I*a^2*c^3*d^2 +
10*a^2*c^2*d^3 + 5*I*a^2*c*d^4 - a^2*d^5)*f)) - 2*(3*I*c^5 - 3*c^4*d + 6*I*c^3*d^2 - 6*c^2*d^3 + 3*I*c*d^4 -
3*d^5 + (9*I*c^5 - 6*c^4*d + 114*I*c^3*d^2 + 632*c^2*d^3 - 735*I*c*d^4 - 202*d^5)*e^(8*I*f*x + 8*I*e) + (30*I*
c^5 - 45*c^4*d + 276*I*c^3*d^2 + 1090*c^2*d^3 - 402*I*c*d^4 + 103*d^5)*e^(6*I*f*x + 6*I*e) + (36*I*c^5 - 75*c^
4*d + 192*I*c^3*d^2 + 386*c^2*d^3 + 348*I*c*d^4 + 269*d^5)*e^(4*I*f*x + 4*I*e) - 3*(-6*I*c^5 + 13*c^4*d - 12*I
*c^3*d^2 + 26*c^2*d^3 - 6*I*c*d^4 + 13*d^5)*e^(...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))**2/(c+d*tan(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (289) = 578\).
time = 1.33, size = 701, normalized size = 2.00 \begin {gather*} -\frac {{\left (2 i \, c^{2} - 14 \, c d - 47 i \, d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{4 \, {\left (a^{2} c^{4} f + 4 i \, a^{2} c^{3} d f - 6 \, a^{2} c^{2} d^{2} f - 4 i \, a^{2} c d^{3} f + a^{2} d^{4} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (12 \, {\left (d \tan \left (f x + e\right ) + c\right )} c d^{3} + c^{2} d^{3} - 6 \, {\left (i \, d \tan \left (f x + e\right ) + i \, c\right )} d^{4} + d^{5}\right )}}{3 \, {\left (a^{2} c^{6} f + 2 i \, a^{2} c^{5} d f + a^{2} c^{4} d^{2} f + 4 i \, a^{2} c^{3} d^{3} f - a^{2} c^{2} d^{4} f + 2 i \, a^{2} c d^{5} f - a^{2} d^{6} f\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} - \frac {2 \, \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{-4 \, {\left (-i \, a^{2} c^{2} f - 2 \, a^{2} c d f + i \, a^{2} d^{2} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d - 2 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d + 13 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{2} - 17 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{2} + 15 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{3}}{8 \, {\left (a^{2} c^{4} f + 4 i \, a^{2} c^{3} d f - 6 \, a^{2} c^{2} d^{2} f - 4 i \, a^{2} c d^{3} f + a^{2} d^{4} f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

-1/4*(2*I*c^2 - 14*c*d - 47*I*d^2)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e)
+ c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) + I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*s
qrt(c^2 + d^2))))/((a^2*c^4*f + 4*I*a^2*c^3*d*f - 6*a^2*c^2*d^2*f - 4*I*a^2*c*d^3*f + a^2*d^4*f)*sqrt(-2*c + 2
*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 2/3*(12*(d*tan(f*x + e) + c)*c*d^3 + c^2*d^3 - 6*(I*d*tan
(f*x + e) + I*c)*d^4 + d^5)/((a^2*c^6*f + 2*I*a^2*c^5*d*f + a^2*c^4*d^2*f + 4*I*a^2*c^3*d^3*f - a^2*c^2*d^4*f
+ 2*I*a^2*c*d^5*f - a^2*d^6*f)*(d*tan(f*x + e) + c)^(3/2)) - 2*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2
+ d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) - I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqr
t(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/((4*I*a^2*c^2*f + 8*a^2*c*d*f - 4*I*a^2*d^2*f)*sqrt(-2*c + 2*sqr
t(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) + 1/8*(2*(d*tan(f*x + e) + c)^(3/2)*c*d - 2*sqrt(d*tan(f*x + e
) + c)*c^2*d + 13*I*(d*tan(f*x + e) + c)^(3/2)*d^2 - 17*I*sqrt(d*tan(f*x + e) + c)*c*d^2 + 15*sqrt(d*tan(f*x +
e) + c)*d^3)/((a^2*c^4*f + 4*I*a^2*c^3*d*f - 6*a^2*c^2*d^2*f - 4*I*a^2*c*d^3*f + a^2*d^4*f)*(d*tan(f*x + e) -
I*d)^2)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*tan(e + f*x)*1i)^2*(c + d*tan(e + f*x))^(5/2)),x)

[Out]

\text{Hanged}

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