3.12.36 \(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx\) [1136]

Optimal. Leaf size=446 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 (c-i d)^{5/2} f}+\frac {\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{11/2} f}+\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}} \]

[Out]

-1/8*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^3/(c-I*d)^(5/2)/f+1/16*(2*I*c^3-16*c^2*d-61*I*c*d^2+152
*d^3)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^3/(c+I*d)^(11/2)/f+1/16*d*(2*c^4+11*I*c^3*d-26*c^2*d^2+2
53*I*c*d^3+150*d^4)/a^3/(c-I*d)^2/(c+I*d)^5/f/(c+d*tan(f*x+e))^(1/2)+1/48*d*(6*c^3+33*I*c^2*d-83*c*d^2+154*I*d
^3)/a^3/(c-I*d)/(c+I*d)^4/f/(c+d*tan(f*x+e))^(3/2)-1/6/(I*c-d)/f/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2)+1
/8*(I*c-4*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2)+1/16*(2*c^2+11*I*c*d-30*d^2)/(I*c-d)^3/
f/(a^3+I*a^3*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2)

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Rubi [A]
time = 1.07, antiderivative size = 446, normalized size of antiderivative = 1.00, number of steps used = 12, 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 {2 c^2+11 i c d-30 d^2}{16 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac {\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{11/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 f (c-i d)^2 (c+i d)^5 \sqrt {c+d \tan (e+f x)}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f (c-i d)^{5/2}}+\frac {-4 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

((-1/8*I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(5/2)*f) + (((2*I)*c^3 - 16*c^2*d -
(61*I)*c*d^2 + 152*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(11/2)*f) + (d*(6*c
^3 + (33*I)*c^2*d - 83*c*d^2 + (154*I)*d^3))/(48*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])^(3/2)) - 1/(
6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)) + (I*c - 4*d)/(8*a*(c + I*d)^2*f*(a + I*a*T
an[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)) + (2*c^2 + (11*I)*c*d - 30*d^2)/(16*(I*c - d)^3*f*(a^3 + I*a^3*Tan[
e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(2*c^4 + (11*I)*c^3*d - 26*c^2*d^2 + (253*I)*c*d^3 + 150*d^4))/(16*
a^3*(c - I*d)^2*(c + I*d)^5*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))^3 (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {3}{2} a (2 i c-5 d)-\frac {9}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx}{6 a^2 (i c-d)}\\ &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {-\frac {3}{2} a^2 \left (4 c^2+15 i c d-32 d^2\right )-\frac {21}{2} a^2 (c+4 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {\frac {3}{2} a^3 \left (4 i c^3-22 c^2 d-67 i c d^2+154 d^3\right )+\frac {15}{2} a^3 d \left (2 i c^2-11 c d-30 i d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}-\frac {\int \frac {\frac {3}{2} a^3 \left (4 i c^4-22 c^3 d-57 i c^2 d^2+99 c d^3-150 i d^4\right )+\frac {3}{2} a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )}\\ &=\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {\frac {3}{2} a^3 \left (4 i c^5-22 c^4 d-51 i c^3 d^2+66 c^2 d^3-233 i c d^4-154 d^5\right )-\frac {3}{2} a^3 d \left (11 c^3 d-i \left (2 c^4-26 c^2 d^2+253 i c d^3+150 d^4\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )^2}\\ &=\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 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}{16 a^3 (c-i d)^2}+\frac {\left (2 c^3+16 i c^2 d-61 c d^2-152 i d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{32 a^3 (c+i d)^5}\\ &=\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 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 )}{16 a^3 (c-i d)^2 f}-\frac {\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{32 a^3 (c+i d)^5 f}\\ &=\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 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 )}{8 a^3 (c-i d)^2 d f}+\frac {\left (i \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right )\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 )}{16 a^3 (c+i d)^5 d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 (c-i d)^{5/2} f}+\frac {\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{11/2} f}+\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 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. \(1160\) vs. \(2(446)=892\).
time = 10.93, size = 1160, normalized size = 2.60 \begin {gather*} \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac {\left (18 c^2+103 i c d-208 d^2\right ) \cos (2 f x) \left (\frac {1}{96} i \cos (e)-\frac {\sin (e)}{96}\right )}{(c+i d)^5}+\frac {(9 c+26 i d) \cos (4 f x) \left (\frac {1}{96} i \cos (e)+\frac {\sin (e)}{96}\right )}{(c+i d)^4}+\frac {\left (11 i c^5 \cos (e)-50 c^4 d \cos (e)-51 i c^3 d^2 \cos (e)-296 c^2 d^3 \cos (e)+1208 i c d^4 \cos (e)+576 d^5 \cos (e)+11 i c^4 d \sin (e)-50 c^3 d^2 \sin (e)-51 i c^2 d^3 \sin (e)-296 c d^4 \sin (e)+120 i d^5 \sin (e)\right ) \left (\frac {1}{96} \cos (3 e)+\frac {1}{96} i \sin (3 e)\right )}{(c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e))}+\frac {\cos (6 f x) \left (\frac {1}{48} i \cos (3 e)+\frac {1}{48} \sin (3 e)\right )}{(c+i d)^3}+\frac {\left (18 c^2+103 i c d-208 d^2\right ) \left (\frac {\cos (e)}{96}+\frac {1}{96} i \sin (e)\right ) \sin (2 f x)}{(c+i d)^5}+\frac {(9 c+26 i d) \left (\frac {\cos (e)}{96}-\frac {1}{96} i \sin (e)\right ) \sin (4 f x)}{(c+i d)^4}+\frac {\left (\frac {1}{48} \cos (3 e)-\frac {1}{48} i \sin (3 e)\right ) \sin (6 f x)}{(c+i d)^3}+\frac {\frac {2}{3} i d^6 \cos (3 e)-\frac {2}{3} d^6 \sin (3 e)}{(c-i d)^2 (c+i d)^5 (c \cos (e+f x)+d \sin (e+f x))^2}+\frac {2 \left (\frac {17}{2} c d^5 \cos (3 e-f x)-\frac {9}{2} i d^6 \cos (3 e-f x)-\frac {17}{2} c d^5 \cos (3 e+f x)+\frac {9}{2} i d^6 \cos (3 e+f x)+\frac {17}{2} i c d^5 \sin (3 e-f x)+\frac {9}{2} d^6 \sin (3 e-f x)-\frac {17}{2} i c d^5 \sin (3 e+f x)-\frac {9}{2} d^6 \sin (3 e+f x)\right )}{3 (c-i d)^2 (c+i d)^5 (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))^3}+\frac {\sec ^3(e+f x) (\cos (3 e)+i \sin (3 e)) (\cos (f x)+i \sin (f x))^3 \left (-\frac {i \left (4 c^5+22 i c^4 d-51 c^3 d^2-66 i c^2 d^3-233 c d^4+154 i d^5\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^4 d+11 i c^3 d^2-26 c^2 d^3+253 i c d^4+150 d^5\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 )}{32 (c-i d)^2 (c+i d)^5 f (a+i a \tan (e+f x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)),x]

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((18*c^2 + (10
3*I)*c*d - 208*d^2)*Cos[2*f*x]*((I/96)*Cos[e] - Sin[e]/96))/(c + I*d)^5 + ((9*c + (26*I)*d)*Cos[4*f*x]*((I/96)
*Cos[e] + Sin[e]/96))/(c + I*d)^4 + (((11*I)*c^5*Cos[e] - 50*c^4*d*Cos[e] - (51*I)*c^3*d^2*Cos[e] - 296*c^2*d^
3*Cos[e] + (1208*I)*c*d^4*Cos[e] + 576*d^5*Cos[e] + (11*I)*c^4*d*Sin[e] - 50*c^3*d^2*Sin[e] - (51*I)*c^2*d^3*S
in[e] - 296*c*d^4*Sin[e] + (120*I)*d^5*Sin[e])*(Cos[3*e]/96 + (I/96)*Sin[3*e]))/((c - I*d)^2*(c + I*d)^5*(c*Co
s[e] + d*Sin[e])) + (Cos[6*f*x]*((I/48)*Cos[3*e] + Sin[3*e]/48))/(c + I*d)^3 + ((18*c^2 + (103*I)*c*d - 208*d^
2)*(Cos[e]/96 + (I/96)*Sin[e])*Sin[2*f*x])/(c + I*d)^5 + ((9*c + (26*I)*d)*(Cos[e]/96 - (I/96)*Sin[e])*Sin[4*f
*x])/(c + I*d)^4 + ((Cos[3*e]/48 - (I/48)*Sin[3*e])*Sin[6*f*x])/(c + I*d)^3 + (((2*I)/3)*d^6*Cos[3*e] - (2*d^6
*Sin[3*e])/3)/((c - I*d)^2*(c + I*d)^5*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) + (2*((17*c*d^5*Cos[3*e - f*x])/2
- ((9*I)/2)*d^6*Cos[3*e - f*x] - (17*c*d^5*Cos[3*e + f*x])/2 + ((9*I)/2)*d^6*Cos[3*e + f*x] + ((17*I)/2)*c*d^5
*Sin[3*e - f*x] + (9*d^6*Sin[3*e - f*x])/2 - ((17*I)/2)*c*d^5*Sin[3*e + f*x] - (9*d^6*Sin[3*e + f*x])/2))/(3*(
c - I*d)^2*(c + I*d)^5*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^3)
+ (Sec[e + f*x]^3*(Cos[3*e] + I*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(((-I)*(4*c^5 + (22*I)*c^4*d - 51*c^3*d^2
- (66*I)*c^2*d^3 - 233*c*d^4 + (154*I)*d^5)*(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^4*d + (11*I)*c^3*d^2 - 26*c^2*d^3 + (253*I)*c*d^4 + 1
50*d^5)*(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))))/(32*(c - I*d)^2*(c + I*d)^5*f*(a + I*a*Tan[e + f*x])^3)

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Maple [A]
time = 0.42, size = 743, normalized size = 1.67

method	result	size
derivativedivides	\(\frac {2 d^{4} \left (\frac {\left (i c^{6}-15 i c^{4} d^{2}+15 i c^{2} d^{4}-i d^{6}-6 c^{5} d +20 c^{3} d^{3}-6 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{6} d^{4}}+\frac {i \left (\frac {\frac {d \left (2 i c^{8}-82 i c^{6} d^{2}-116 i c^{4} d^{4}+22 i c^{2} d^{6}+54 i d^{8}-19 c^{7} d +85 c^{5} d^{3}+227 c^{3} d^{5}+123 c \,d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{9}-166 i c^{7} d^{2}-44 i c^{5} d^{4}+422 i c^{3} d^{6}+297 i c \,d^{8}-33 c^{8} d +282 c^{6} d^{3}+572 c^{4} d^{5}+166 c^{2} d^{7}-91 d^{9}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{10}-146 i c^{8} d^{2}+192 i c^{6} d^{4}+760 i c^{4} d^{6}+350 i c^{2} d^{8}-70 i d^{10}-25 c^{9} d +340 c^{7} d^{3}+458 c^{5} d^{5}-204 c^{3} d^{7}-297 c \,d^{9}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (20 i c^{8} d -250 i c^{6} d^{3}-408 i c^{4} d^{5}+14 i c^{2} d^{7}+152 i d^{9}+2 c^{9}-91 c^{7} d^{2}+177 c^{5} d^{4}+635 c^{3} d^{6}+365 c \,d^{8}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 \left (i d -c \right )^{2} \left (i d +c \right )^{6} d^{4}}-\frac {-5 i c^{2}-3 i d^{2}+2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{6} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-i c^{3}-i c \,d^{2}+c^{2} d +d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{6} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f \,a^{3}}\)	\(743\)
default	\(\frac {2 d^{4} \left (\frac {\left (i c^{6}-15 i c^{4} d^{2}+15 i c^{2} d^{4}-i d^{6}-6 c^{5} d +20 c^{3} d^{3}-6 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{6} d^{4}}+\frac {i \left (\frac {\frac {d \left (2 i c^{8}-82 i c^{6} d^{2}-116 i c^{4} d^{4}+22 i c^{2} d^{6}+54 i d^{8}-19 c^{7} d +85 c^{5} d^{3}+227 c^{3} d^{5}+123 c \,d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{9}-166 i c^{7} d^{2}-44 i c^{5} d^{4}+422 i c^{3} d^{6}+297 i c \,d^{8}-33 c^{8} d +282 c^{6} d^{3}+572 c^{4} d^{5}+166 c^{2} d^{7}-91 d^{9}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{10}-146 i c^{8} d^{2}+192 i c^{6} d^{4}+760 i c^{4} d^{6}+350 i c^{2} d^{8}-70 i d^{10}-25 c^{9} d +340 c^{7} d^{3}+458 c^{5} d^{5}-204 c^{3} d^{7}-297 c \,d^{9}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (20 i c^{8} d -250 i c^{6} d^{3}-408 i c^{4} d^{5}+14 i c^{2} d^{7}+152 i d^{9}+2 c^{9}-91 c^{7} d^{2}+177 c^{5} d^{4}+635 c^{3} d^{6}+365 c \,d^{8}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 \left (i d -c \right )^{2} \left (i d +c \right )^{6} d^{4}}-\frac {-5 i c^{2}-3 i d^{2}+2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{6} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-i c^{3}-i c \,d^{2}+c^{2} d +d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{6} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f \,a^{3}}\)	\(743\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/f/a^3*d^4*(1/16/(I*d-c)^(5/2)/(c+I*d)^6*(-15*I*c^4*d^2+15*I*c^2*d^4+20*c^3*d^3-6*c*d^5+I*c^6-I*d^6-6*c^5*d)/
d^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))+1/16*I/(I*d-c)^2/(c+I*d)^6/d^4*((1/2*d*(2*I*c^8-82*I*c^6*d^2-
116*I*c^4*d^4+22*I*c^2*d^6+54*I*d^8-19*c^7*d+85*c^5*d^3+227*c^3*d^5+123*c*d^7)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(
c+d*tan(f*x+e))^(5/2)-2/3*d*(-33*c^8*d+282*c^6*d^3+572*c^4*d^5+166*c^2*d^7-91*d^9+3*I*c^9-166*I*c^7*d^2-44*I*c
^5*d^4+422*I*c^3*d^6+297*I*c*d^8)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2)+1/2*d*(2*I*c^10-146*I*c
^8*d^2+192*I*c^6*d^4+760*I*c^4*d^6+350*I*c^2*d^8-70*I*d^10-25*c^9*d+340*c^7*d^3+458*c^5*d^5-204*c^3*d^7-297*c*
d^9)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2))/(-d*tan(f*x+e)+I*d)^3-1/2*(-91*c^7*d^2+177*c^5*d^4+
635*c^3*d^6+365*c*d^8+20*I*c^8*d-250*I*c^6*d^3-408*I*c^4*d^5+14*I*c^2*d^7+152*I*d^9+2*c^9)/(3*I*c^2*d-I*d^3+c^
3-3*c*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)^6*(-5*I*c^2-3*I*d
^2+2*c*d)/(c+d*tan(f*x+e))^(1/2)-1/3/(I*d-c)^2/(c+I*d)^6*(-I*c^3-I*c*d^2+c^2*d+d^3)/(c+d*tan(f*x+e))^(3/2))

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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))^3/(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. 3845 vs. \(2 (374) = 748\).
time = 20.58, size = 3845, normalized size = 8.62 \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))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/192*(48*((-I*a^3*c^9 + a^3*c^8*d - 4*I*a^3*c^7*d^2 + 4*a^3*c^6*d^3 - 6*I*a^3*c^5*d^4 + 6*a^3*c^4*d^5 - 4*I*a
^3*c^3*d^6 + 4*a^3*c^2*d^7 - I*a^3*c*d^8 + a^3*d^9)*f*e^(10*I*f*x + 10*I*e) + 2*(-I*a^3*c^9 + 3*a^3*c^8*d + 8*
a^3*c^6*d^3 + 6*I*a^3*c^5*d^4 + 6*a^3*c^4*d^5 + 8*I*a^3*c^3*d^6 + 3*I*a^3*c*d^8 - a^3*d^9)*f*e^(8*I*f*x + 8*I*
e) + (-I*a^3*c^9 + 5*a^3*c^8*d + 8*I*a^3*c^7*d^2 + 14*I*a^3*c^5*d^4 - 14*a^3*c^4*d^5 - 8*a^3*c^2*d^7 - 5*I*a^3
*c*d^8 + a^3*d^9)*f*e^(6*I*f*x + 6*I*e))*sqrt(1/64*I/((-I*a^6*c^5 - 5*a^6*c^4*d + 10*I*a^6*c^3*d^2 + 10*a^6*c^
2*d^3 - 5*I*a^6*c*d^4 - a^6*d^5)*f^2))*log(-2*(8*((I*a^3*c^3 + 3*a^3*c^2*d - 3*I*a^3*c*d^2 - a^3*d^3)*f*e^(2*I
*f*x + 2*I*e) + (I*a^3*c^3 + 3*a^3*c^2*d - 3*I*a^3*c*d^2 - a^3*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(1/64*I/((-I*a^6*c^5 - 5*a^6*c^4*d + 10*I*a^6*c^3*d^2 + 10*a^6*c^2*d^3
- 5*I*a^6*c*d^4 - a^6*d^5)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + 48*((I*a^3*c^9 -
a^3*c^8*d + 4*I*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 6*I*a^3*c^5*d^4 - 6*a^3*c^4*d^5 + 4*I*a^3*c^3*d^6 - 4*a^3*c^2*d
^7 + I*a^3*c*d^8 - a^3*d^9)*f*e^(10*I*f*x + 10*I*e) + 2*(I*a^3*c^9 - 3*a^3*c^8*d - 8*a^3*c^6*d^3 - 6*I*a^3*c^5
*d^4 - 6*a^3*c^4*d^5 - 8*I*a^3*c^3*d^6 - 3*I*a^3*c*d^8 + a^3*d^9)*f*e^(8*I*f*x + 8*I*e) + (I*a^3*c^9 - 5*a^3*c
^8*d - 8*I*a^3*c^7*d^2 - 14*I*a^3*c^5*d^4 + 14*a^3*c^4*d^5 + 8*a^3*c^2*d^7 + 5*I*a^3*c*d^8 - a^3*d^9)*f*e^(6*I
*f*x + 6*I*e))*sqrt(1/64*I/((-I*a^6*c^5 - 5*a^6*c^4*d + 10*I*a^6*c^3*d^2 + 10*a^6*c^2*d^3 - 5*I*a^6*c*d^4 - a^
6*d^5)*f^2))*log(-2*(8*((-I*a^3*c^3 - 3*a^3*c^2*d + 3*I*a^3*c*d^2 + a^3*d^3)*f*e^(2*I*f*x + 2*I*e) + (-I*a^3*c
^3 - 3*a^3*c^2*d + 3*I*a^3*c*d^2 + a^3*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(1/64*I/((-I*a^6*c^5 - 5*a^6*c^4*d + 10*I*a^6*c^3*d^2 + 10*a^6*c^2*d^3 - 5*I*a^6*c*d^4 - a^6*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*((-I*a^3*c^9 + a^3*c^8*d - 4*I*a^3*c^7
*d^2 + 4*a^3*c^6*d^3 - 6*I*a^3*c^5*d^4 + 6*a^3*c^4*d^5 - 4*I*a^3*c^3*d^6 + 4*a^3*c^2*d^7 - I*a^3*c*d^8 + a^3*d
^9)*f*e^(10*I*f*x + 10*I*e) + 2*(-I*a^3*c^9 + 3*a^3*c^8*d + 8*a^3*c^6*d^3 + 6*I*a^3*c^5*d^4 + 6*a^3*c^4*d^5 +
8*I*a^3*c^3*d^6 + 3*I*a^3*c*d^8 - a^3*d^9)*f*e^(8*I*f*x + 8*I*e) + (-I*a^3*c^9 + 5*a^3*c^8*d + 8*I*a^3*c^7*d^2
+ 14*I*a^3*c^5*d^4 - 14*a^3*c^4*d^5 - 8*a^3*c^2*d^7 - 5*I*a^3*c*d^8 + a^3*d^9)*f*e^(6*I*f*x + 6*I*e))*sqrt(-(
-4*I*c^6 + 64*c^5*d + 500*I*c^4*d^2 - 2560*c^3*d^3 - 8585*I*c^2*d^4 + 18544*c*d^5 + 23104*I*d^6)/((-I*a^6*c^11
+ 11*a^6*c^10*d + 55*I*a^6*c^9*d^2 - 165*a^6*c^8*d^3 - 330*I*a^6*c^7*d^4 + 462*a^6*c^6*d^5 + 462*I*a^6*c^5*d^
6 - 330*a^6*c^4*d^7 - 165*I*a^6*c^3*d^8 + 55*a^6*c^2*d^9 + 11*I*a^6*c*d^10 - a^6*d^11)*f^2))*log(-1/16*(-2*I*c
^4 + 18*c^3*d + 77*I*c^2*d^2 - 213*c*d^3 - 152*I*d^4 + ((a^3*c^6 + 6*I*a^3*c^5*d - 15*a^3*c^4*d^2 - 20*I*a^3*c
^3*d^3 + 15*a^3*c^2*d^4 + 6*I*a^3*c*d^5 - a^3*d^6)*f*e^(2*I*f*x + 2*I*e) + (a^3*c^6 + 6*I*a^3*c^5*d - 15*a^3*c
^4*d^2 - 20*I*a^3*c^3*d^3 + 15*a^3*c^2*d^4 + 6*I*a^3*c*d^5 - a^3*d^6)*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^6 + 64*c^5*d + 500*I*c^4*d^2 - 2560*c^3*d^3 - 8585*I*c^2*d^
4 + 18544*c*d^5 + 23104*I*d^6)/((-I*a^6*c^11 + 11*a^6*c^10*d + 55*I*a^6*c^9*d^2 - 165*a^6*c^8*d^3 - 330*I*a^6*
c^7*d^4 + 462*a^6*c^6*d^5 + 462*I*a^6*c^5*d^6 - 330*a^6*c^4*d^7 - 165*I*a^6*c^3*d^8 + 55*a^6*c^2*d^9 + 11*I*a^
6*c*d^10 - a^6*d^11)*f^2)) + (-2*I*c^4 + 16*c^3*d + 61*I*c^2*d^2 - 152*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x
- 2*I*e)/((a^3*c^6 + 6*I*a^3*c^5*d - 15*a^3*c^4*d^2 - 20*I*a^3*c^3*d^3 + 15*a^3*c^2*d^4 + 6*I*a^3*c*d^5 - a^3
*d^6)*f)) + 3*((I*a^3*c^9 - a^3*c^8*d + 4*I*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 6*I*a^3*c^5*d^4 - 6*a^3*c^4*d^5 + 4*
I*a^3*c^3*d^6 - 4*a^3*c^2*d^7 + I*a^3*c*d^8 - a^3*d^9)*f*e^(10*I*f*x + 10*I*e) + 2*(I*a^3*c^9 - 3*a^3*c^8*d -
8*a^3*c^6*d^3 - 6*I*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 8*I*a^3*c^3*d^6 - 3*I*a^3*c*d^8 + a^3*d^9)*f*e^(8*I*f*x + 8*
I*e) + (I*a^3*c^9 - 5*a^3*c^8*d - 8*I*a^3*c^7*d^2 - 14*I*a^3*c^5*d^4 + 14*a^3*c^4*d^5 + 8*a^3*c^2*d^7 + 5*I*a^
3*c*d^8 - a^3*d^9)*f*e^(6*I*f*x + 6*I*e))*sqrt(-(-4*I*c^6 + 64*c^5*d + 500*I*c^4*d^2 - 2560*c^3*d^3 - 8585*I*c
^2*d^4 + 18544*c*d^5 + 23104*I*d^6)/((-I*a^6*c^11 + 11*a^6*c^10*d + 55*I*a^6*c^9*d^2 - 165*a^6*c^8*d^3 - 330*I
*a^6*c^7*d^4 + 462*a^6*c^6*d^5 + 462*I*a^6*c^5*d^6 - 330*a^6*c^4*d^7 - 165*I*a^6*c^3*d^8 + 55*a^6*c^2*d^9 + 11
*I*a^6*c*d^10 - a^6*d^11)*f^2))*log(-1/16*(-2*I*c^4 + 18*c^3*d + 77*I*c^2*d^2 - 213*c*d^3 - 152*I*d^4 - ((a^3*
c^6 + 6*I*a^3*c^5*d - 15*a^3*c^4*d^2 - 20*I*a^3*c^3*d^3 + 15*a^3*c^2*d^4 + 6*I*a^3*c*d^5 - a^3*d^6)*f*e^(2*I*f
*x + 2*I*e) + (a^3*c^6 + 6*I*a^3*c^5*d - 15*a^3*c^4*d^2 - 20*I*a^3*c^3*d^3 + 15*a^3*c^2*d^4 + 6*I*a^3*c*d^5 -
a^3*d^6)*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^6 + 64*c^5
*d + 500*I*c^4*d^2 - 2560*c^3*d^3 - 8585*I*c^2*d^4 + 18544*c*d^5 + 23104*I*d^6)/((-I*a^6*c^11 + 11*a^6*c^10*d
+ 55*I*a^6*c^9*d^2 - 165*a^6*c^8*d^3 - 330*I*a^...

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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))**3/(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. 1061 vs. \(2 (374) = 748\).
time = 1.29, size = 1061, normalized size = 2.38 \begin {gather*} -\frac {{\left (-2 i \, c^{3} + 16 \, c^{2} d + 61 i \, c d^{2} - 152 \, d^{3}\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 )}{8 \, {\left (a^{3} c^{5} f + 5 i \, a^{3} c^{4} d f - 10 \, a^{3} c^{3} d^{2} f - 10 i \, a^{3} c^{2} d^{3} f + 5 \, a^{3} c d^{4} f + i \, a^{3} d^{5} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {i \, \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^{3} c^{2} f - 2 i \, a^{3} c d f - a^{3} 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 {6 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{4} c^{4} d - 12 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{3} c^{5} d + 6 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} c^{6} d - 33 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{4} c^{3} d^{2} + 84 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{3} c^{4} d^{2} - 51 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} c^{5} d^{2} - 78 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{4} c^{2} d^{3} + 256 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{3} c^{3} d^{3} - 198 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} c^{4} d^{3} - 759 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{4} c d^{4} + 1856 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{3} c^{2} d^{4} - 1446 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} c^{3} d^{4} + 384 \, {\left (d \tan \left (f x + e\right ) + c\right )} c^{4} d^{4} + 32 \, c^{5} d^{4} + 450 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{4} d^{5} + 844 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{3} c d^{5} - 2334 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} c^{2} d^{5} - 960 \, {\left (-i \, d \tan \left (f x + e\right ) - i \, c\right )} c^{3} d^{5} + 96 i \, c^{4} d^{5} + 1196 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{3} d^{6} - 243 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} c d^{6} - 576 \, {\left (d \tan \left (f x + e\right ) + c\right )} c^{2} d^{6} - 64 \, c^{3} d^{6} - 978 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} d^{7} - 192 \, {\left (-i \, d \tan \left (f x + e\right ) - i \, c\right )} c d^{7} + 64 i \, c^{2} d^{7} - 192 \, {\left (d \tan \left (f x + e\right ) + c\right )} d^{8} - 96 \, c d^{8} - 32 i \, d^{9}}{48 \, {\left (a^{3} c^{7} f + 3 i \, a^{3} c^{6} d f - a^{3} c^{5} d^{2} f + 5 i \, a^{3} c^{4} d^{3} f - 5 \, a^{3} c^{3} d^{4} f + i \, a^{3} c^{2} d^{5} f - 3 \, a^{3} c d^{6} f - i \, a^{3} d^{7} f\right )} {\left (-i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} + i \, \sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {d \tan \left (f x + e\right ) + c} d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

-1/8*(-2*I*c^3 + 16*c^2*d + 61*I*c*d^2 - 152*d^3)*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*sqrt(c^2 + d^2))))/((a^3*c^5*f + 5*I*a^3*c^4*d*f - 10*a^3*c^3*d^2*f - 10*I*a^3*c^2*d^3*f + 5*a^
3*c*d^4*f + I*a^3*d^5*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 1/4*I*arctan(2*(sqr
t(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*sqrt(c^2 + d^2))))/((a^3*c^2*f - 2*I*a^3*c*d*f -
a^3*d^2*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) + 1/48*(6*I*(d*tan(f*x + e) + c)^4
*c^4*d - 12*I*(d*tan(f*x + e) + c)^3*c^5*d + 6*I*(d*tan(f*x + e) + c)^2*c^6*d - 33*(d*tan(f*x + e) + c)^4*c^3*
d^2 + 84*(d*tan(f*x + e) + c)^3*c^4*d^2 - 51*(d*tan(f*x + e) + c)^2*c^5*d^2 - 78*I*(d*tan(f*x + e) + c)^4*c^2*
d^3 + 256*I*(d*tan(f*x + e) + c)^3*c^3*d^3 - 198*I*(d*tan(f*x + e) + c)^2*c^4*d^3 - 759*(d*tan(f*x + e) + c)^4
*c*d^4 + 1856*(d*tan(f*x + e) + c)^3*c^2*d^4 - 1446*(d*tan(f*x + e) + c)^2*c^3*d^4 + 384*(d*tan(f*x + e) + c)*
c^4*d^4 + 32*c^5*d^4 + 450*I*(d*tan(f*x + e) + c)^4*d^5 + 844*I*(d*tan(f*x + e) + c)^3*c*d^5 - 2334*I*(d*tan(f
*x + e) + c)^2*c^2*d^5 - 960*(-I*d*tan(f*x + e) - I*c)*c^3*d^5 + 96*I*c^4*d^5 + 1196*(d*tan(f*x + e) + c)^3*d^
6 - 243*(d*tan(f*x + e) + c)^2*c*d^6 - 576*(d*tan(f*x + e) + c)*c^2*d^6 - 64*c^3*d^6 - 978*I*(d*tan(f*x + e) +
c)^2*d^7 - 192*(-I*d*tan(f*x + e) - I*c)*c*d^7 + 64*I*c^2*d^7 - 192*(d*tan(f*x + e) + c)*d^8 - 96*c*d^8 - 32*
I*d^9)/((a^3*c^7*f + 3*I*a^3*c^6*d*f - a^3*c^5*d^2*f + 5*I*a^3*c^4*d^3*f - 5*a^3*c^3*d^4*f + I*a^3*c^2*d^5*f -
3*a^3*c*d^6*f - I*a^3*d^7*f)*(-I*(d*tan(f*x + e) + c)^(3/2) + I*sqrt(d*tan(f*x + e) + c)*c - sqrt(d*tan(f*x +
e) + c)*d)^3)

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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)^3*(c + d*tan(e + f*x))^(5/2)),x)

[Out]

\text{Hanged}

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