3.1154 \(\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx\)
Optimal. Leaf size=225 \[ \frac{i (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{4 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{4 \sqrt{2} a^{5/2} f}+\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(d+i c) (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}} \]
[Out]
((-I/4)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e +
f*x]])])/(Sqrt[2]*a^(5/2)*f) + ((I/4)*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x]])/(a^2*f*Sqrt[a + I*a*Tan[e + f*x]]
) + ((I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(6*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I/5)*(c + d*Tan[e + f*x])^
(5/2))/(f*(a + I*a*Tan[e + f*x])^(5/2))
________________________________________________________________________________________
Rubi [A] time = 0.477117, antiderivative size = 225, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used =
{3546, 3544, 208} \[ \frac{i (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{4 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{4 \sqrt{2} a^{5/2} f}+\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(d+i c) (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(5/2),x]
[Out]
((-I/4)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e +
f*x]])])/(Sqrt[2]*a^(5/2)*f) + ((I/4)*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x]])/(a^2*f*Sqrt[a + I*a*Tan[e + f*x]]
) + ((I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(6*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I/5)*(c + d*Tan[e + f*x])^
(5/2))/(f*(a + I*a*Tan[e + f*x])^(5/2))
Rule 3546
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)/(2*b*f*m), x] - Dist[(a*c - b*d)/(2*b^2), Int[(a + b*Tan[e
+ f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && LeQ[m, -2^(-1)]
Rule 3544
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
(-2*a*b)/f, Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx &=\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(c-i d) \int \frac{(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx}{2 a}\\ &=\frac{(i c+d) (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(c-i d)^2 \int \frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{a+i a \tan (e+f x)}} \, dx}{4 a^2}\\ &=\frac{i (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{4 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c+d) (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(c-i d)^3 \int \frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}} \, dx}{8 a^3}\\ &=\frac{i (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{4 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c+d) (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(i c+d)^3 \operatorname{Subst}\left (\int \frac{1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{a+i a \tan (e+f x)}}\right )}{4 a f}\\ &=-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{4 \sqrt{2} a^{5/2} f}+\frac{i (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{4 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{(i c+d) (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 6.05846, size = 292, normalized size = 1.3 \[ \frac{\sec ^{\frac{5}{2}}(e+f x) \left (\frac{2 \sqrt{c+d \tan (e+f x)} \left (\left (-20 c^2+52 i c d+20 d^2\right ) \sin (2 (e+f x))+\left (26 i c^2+40 c d-26 i d^2\right ) \cos (2 (e+f x))+11 i \left (c^2+d^2\right )\right )}{15 \sqrt{\sec (e+f x)}}-i \sqrt{2} (c-i d)^{5/2} e^{2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left (2 \left (\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )\right )}{8 f (a+i a \tan (e+f x))^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(5/2),x]
[Out]
(Sec[e + f*x]^(5/2)*((-I)*Sqrt[2]*(c - I*d)^(5/2)*E^((2*I)*(e + f*x))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e +
f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqr
t[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] + (2*((11*I)*(c^2 + d^2) + ((26*I)*c^2 + 4
0*c*d - (26*I)*d^2)*Cos[2*(e + f*x)] + (-20*c^2 + (52*I)*c*d + 20*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*
x]])/(15*Sqrt[Sec[e + f*x]])))/(8*f*(a + I*a*Tan[e + f*x])^(5/2))
________________________________________________________________________________________
Maple [B] time = 0.059, size = 2559, normalized size = 11.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(5/2),x)
[Out]
1/240/f*(c+d*tan(f*x+e))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a^3*(120*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*ta
n(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(t
an(f*x+e)+I))*tan(f*x+e)^3*c^2*d^2-112*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^2*d^2+220*I*tan(f*x+e)^
2*c^4*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)-112*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3
*c^2*d^2+136*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*c^3*d+40*(a*(c+d*tan(f*x+e))*(1+I*tan(f*
x+e)))^(1/2)*tan(f*x+e)^2*c*d^3+624*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^2*d^2+308*I*(a*(c
+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2*d^4+40*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d^3-1
48*I*c^4*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)-60*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d^4+308*
tan(f*x+e)*c^4*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)-60*tan(f*x+e)^3*c^4*(a*(c+d*tan(f*x+e))*(1+I*tan(f*
x+e)))^(1/2)-148*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*d^4+220*(a*(c+d*tan(f*x+e))*(1+I*tan
(f*x+e)))^(1/2)*tan(f*x+e)*d^4+136*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c^3*d+136*I*(a*(c+d*tan(f*x+e))
*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c*d^3+60*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*t
an(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+
e)^3*c^4+60*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c)
)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^3*d^4-60*2^(1/2)*(-a*(I*d-c))^
(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*
tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)*c^4-60*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*
a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))
*tan(f*x+e)*d^4+40*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*c^3*d+136*I*(a*(c+d*tan(f*x+e))*
(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^3*c*d^3+15*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*
a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^4+1
5*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*
(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*d^4+624*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(
1/2)*tan(f*x+e)^2*c^2*d^2+40*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)*c^3*d-120*2^(1/2)*(-a*(I
*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e)
)*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)*c^2*d^2+15*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*ta
n(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(t
an(f*x+e)+I))*tan(f*x+e)^4*c^4+15*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)
*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^4*d^4-
90*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)
*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^2*c^4-90*I*2^(1/2)*(-a*(I*d-c))^(1/2)
*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f
*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^2*d^4+30*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a
*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*
c^2*d^2+30*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c
))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^4*c^2*d^2-180*I*2^(1/2)*(-a*(
I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*tan(f*x+e)*d+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e
))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*tan(f*x+e)^2*c^2*d^2)/(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)/
(I*c-d)^2/(-tan(f*x+e)+I)^4
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
________________________________________________________________________________________
Fricas [B] time = 1.98143, size = 1805, normalized size = 8.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="fricas")
[Out]
-1/120*(15*sqrt(1/2)*a^3*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^5*f^2))*e^
(6*I*f*x + 6*I*e)*log(-(2*I*sqrt(1/2)*a^3*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d
^5)/(a^5*f^2))*e^(2*I*f*x + 2*I*e) - sqrt(2)*(c^2 - 2*I*c*d - d^2 + (c^2 - 2*I*c*d - d^2)*e^(2*I*f*x + 2*I*e))
*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(a/(e^(2*I*f*x + 2*I*e) + 1))*e
^(I*f*x + I*e))*e^(-I*f*x - I*e)/(c^2 - 2*I*c*d - d^2)) - 15*sqrt(1/2)*a^3*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d
^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^5*f^2))*e^(6*I*f*x + 6*I*e)*log(-(-2*I*sqrt(1/2)*a^3*f*sqrt(-(c^5 - 5*
I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^5*f^2))*e^(2*I*f*x + 2*I*e) - sqrt(2)*(c^2 - 2*I*c*d
- d^2 + (c^2 - 2*I*c*d - d^2)*e^(2*I*f*x + 2*I*e))*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(a/(e^(2*I*f*x + 2*I*e) + 1))*e^(I*f*x + I*e))*e^(-I*f*x - I*e)/(c^2 - 2*I*c*d - d^2)) - s
qrt(2)*(3*I*c^2 - 6*c*d - 3*I*d^2 + (23*I*c^2 + 46*c*d - 23*I*d^2)*e^(6*I*f*x + 6*I*e) + (34*I*c^2 + 46*c*d -
12*I*d^2)*e^(4*I*f*x + 4*I*e) + (14*I*c^2 - 6*c*d + 8*I*d^2)*e^(2*I*f*x + 2*I*e))*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(a/(e^(2*I*f*x + 2*I*e) + 1))*e^(I*f*x + I*e))*e^(-6*I*f*x -
6*I*e)/(a^3*f)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))**(5/2)/(a+I*a*tan(f*x+e))**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError