3.1134 \(\int \frac{1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=267 \[ \frac{d (7 d+3 i c)}{6 a f (-d+i c) \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (c^2-14 i c d-5 d^2\right )}{2 a f (c-i d)^2 (c+i d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{2 a f (c-i d)^{5/2}}+\frac{(-6 d+i c) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{2 a f (c+i d)^{7/2}} \]

[Out]

((-I/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*(c - I*d)^(5/2)*f) + ((I*c - 6*d)*ArcTanh[Sqrt[c +
d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*(c + I*d)^(7/2)*f) + (d*((3*I)*c + 7*d))/(6*a*(I*c - d)*(c^2 + d^2)*f*(c
+ d*Tan[e + f*x])^(3/2)) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(c^2 - (1
4*I)*c*d - 5*d^2))/(2*a*(c - I*d)^2*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]])

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Rubi [A]  time = 0.613968, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3552, 3529, 3539, 3537, 63, 208} \[ \frac{d (7 d+3 i c)}{6 a f (-d+i c) \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (c^2-14 i c d-5 d^2\right )}{2 a f (c-i d)^2 (c+i d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{2 a f (c-i d)^{5/2}}+\frac{(-6 d+i c) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{2 a f (c+i d)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*(c - I*d)^(5/2)*f) + ((I*c - 6*d)*ArcTanh[Sqrt[c +
d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*(c + I*d)^(7/2)*f) + (d*((3*I)*c + 7*d))/(6*a*(I*c - d)*(c^2 + d^2)*f*(c
+ d*Tan[e + f*x])^(3/2)) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(c^2 - (1
4*I)*c*d - 5*d^2))/(2*a*(c - I*d)^2*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]])

Rule 3552

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(a
*(c + d*Tan[e + f*x])^(n + 1))/(2*f*(b*c - a*d)*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c +
d*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*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] &&  !GtQ[n, 0]

Rule 3529

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 3539

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 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 63

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 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{1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac{1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac{\int \frac{\frac{1}{2} a (2 i c-7 d)+\frac{5}{2} i a d \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx}{2 a^2 (i c-d)}\\ &=\frac{(3 c-7 i d) d}{6 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac{\int \frac{-\frac{1}{2} a \left (7 c d-i \left (2 c^2+5 d^2\right )\right )+\frac{1}{2} a d (3 i c+7 d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{2 a^2 (i c-d) \left (c^2+d^2\right )}\\ &=\frac{(3 c-7 i d) d}{6 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (c^2-14 i c d-5 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 f \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{\frac{1}{2} a \left (2 i c^3-7 c^2 d+8 i c d^2+7 d^3\right )+\frac{1}{2} a d \left (i c^2+14 c d-5 i d^2\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 a^2 (i c-d)^3 (c-i d)^2}\\ &=\frac{(3 c-7 i d) d}{6 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (c^2-14 i c d-5 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 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}{4 a (c-i d)^2}+\frac{(c+6 i d) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{4 a (c+i d)^3}\\ &=\frac{(3 c-7 i d) d}{6 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (c^2-14 i c d-5 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 f \sqrt{c+d \tan (e+f x)}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{4 a (c-i d)^2 f}-\frac{(c+6 i d) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{4 a (i c-d)^3 f}\\ &=\frac{(3 c-7 i d) d}{6 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (c^2-14 i c d-5 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 f \sqrt{c+d \tan (e+f x)}}-\frac{\operatorname{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 )}{2 a (c-i d)^2 d f}-\frac{(c+6 i d) \operatorname{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 )}{2 a (c+i d)^3 d f}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{2 a (c-i d)^{5/2} f}+\frac{(i c-6 d) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{2 a (c+i d)^{7/2} f}+\frac{(3 c-7 i d) d}{6 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^{3/2}}-\frac{1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac{d \left (c^2-14 i c d-5 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.00469, size = 371, normalized size = 1.39 \[ \frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left (\frac{(\cos (f x)-i \sin (f x)) \sqrt{c+d \tan (e+f x)} \left (3 \left (-42 i c^2 d^2+6 c^3 d+3 i c^4+2 c d^3-9 i d^4\right ) \cos (e+f x)+(d+i c) \left (\left (-3 i c^2 d+3 c^3-43 c d^2+11 i d^3\right ) \cos (3 (e+f x))-8 d \left (-3 c^2+23 i c d+4 d^2\right ) \sin (e+f x) \cos ^2(e+f x)\right )\right )}{6 (c-i d)^2 (c+i d)^3 (c \cos (e+f x)+d \sin (e+f x))^2}-\frac{2 (\cos (e)+i \sin (e)) \left ((-c+i d)^{5/2} (6 d-i c) \tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right )-i (-c-i d)^{7/2} \tan ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right )\right )}{(-c-i d)^{7/2} (-c+i d)^{5/2}}\right )}{4 f (a+i a \tan (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-2*((-c + I*d)^(5/2)*((-I)*c + 6*d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sq
rt[-c - I*d]] - I*(-c - I*d)^(7/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[e] + I*Sin[e]))/((-c
- I*d)^(7/2)*(-c + I*d)^(5/2)) + ((Cos[f*x] - I*Sin[f*x])*(3*((3*I)*c^4 + 6*c^3*d - (42*I)*c^2*d^2 + 2*c*d^3 -
(9*I)*d^4)*Cos[e + f*x] + (I*c + d)*((3*c^3 - (3*I)*c^2*d - 43*c*d^2 + (11*I)*d^3)*Cos[3*(e + f*x)] - 8*d*(-3
*c^2 + (23*I)*c*d + 4*d^2)*Cos[e + f*x]^2*Sin[e + f*x]))*Sqrt[c + d*Tan[e + f*x]])/(6*(c - I*d)^2*(c + I*d)^3*
(c*Cos[e + f*x] + d*Sin[e + f*x])^2)))/(4*f*(a + I*a*Tan[e + f*x]))

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Maple [B]  time = 0.077, size = 1137, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I/f/a*d^4/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e))^(1/2)-6*I/f/a*d^2/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e))^(1/2)
*c^2-I/f/a*d^2/(I*d-c)^2/(c+I*d)^5/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^3-2/f/a*d/(I
*d-c)^(5/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^3+2/f/a*d^3/(I*d-c)^(5/2)/(c+I*d)^4*arcta
n((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c+1/2/f/a*d/(I*d-c)^2/(c+I*d)^5*(c+d*tan(f*x+e))^(1/2)/(-I*d+d*tan(f*x
+e))*c^4+1/f/a*d^3/(I*d-c)^2/(c+I*d)^5*(c+d*tan(f*x+e))^(1/2)/(-I*d+d*tan(f*x+e))*c^2+1/2/f/a*d^5/(I*d-c)^2/(c
+I*d)^5*(c+d*tan(f*x+e))^(1/2)/(-I*d+d*tan(f*x+e))-1/2*I/f/a/(I*d-c)^2/(c+I*d)^5/(-I*d-c)^(1/2)*arctan((c+d*ta
n(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^5-2/3*I/f/a*d^2/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e))^(3/2)*c^3-2/3*I/f/a*d^4
/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e))^(3/2)*c+3/f/a*d/(I*d-c)^2/(c+I*d)^5/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e
))^(1/2)/(-I*d-c)^(1/2))*c^4+6/f/a*d^3/(I*d-c)^2/(c+I*d)^5/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-
c)^(1/2))*c^2+3/f/a*d^5/(I*d-c)^2/(c+I*d)^5/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))+1/2*I
/f/a*d^4/(I*d-c)^(5/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))-1/2*I/f/a*d^4/(I*d-c)^2/(c+I*d)^
5/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c+4/f/a*d^3/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e)
)^(1/2)*c+1/2*I/f/a/(I*d-c)^(5/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^4-3*I/f/a*d^2/(I*d-
c)^(5/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^2+2/3/f/a*d^3/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f
*x+e))^(3/2)*c^2+2/3/f/a*d^5/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e))^(3/2)

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 23.4045, size = 6516, normalized size = 24.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((3*I*a*c^7 + 3*a*c^6*d + 9*I*a*c^5*d^2 + 9*a*c^4*d^3 + 9*I*a*c^3*d^4 + 9*a*c^2*d^5 + 3*I*a*c*d^6 + 3*a*d^7)*
f*e^(6*I*f*x + 6*I*e) + (6*I*a*c^7 - 6*a*c^6*d + 18*I*a*c^5*d^2 - 18*a*c^4*d^3 + 18*I*a*c^3*d^4 - 18*a*c^2*d^5
+ 6*I*a*c*d^6 - 6*a*d^7)*f*e^(4*I*f*x + 4*I*e) + (3*I*a*c^7 - 9*a*c^6*d - 3*I*a*c^5*d^2 - 15*a*c^4*d^3 - 15*I
*a*c^3*d^4 - 3*a*c^2*d^5 - 9*I*a*c*d^6 + 3*a*d^7)*f*e^(2*I*f*x + 2*I*e))*sqrt(-I/((4*I*a^2*c^5 + 20*a^2*c^4*d
- 40*I*a^2*c^3*d^2 - 40*a^2*c^2*d^3 + 20*I*a^2*c*d^4 + 4*a^2*d^5)*f^2))*log((((4*I*a*c^3 + 12*a*c^2*d - 12*I*a
*c*d^2 - 4*a*d^3)*f*e^(2*I*f*x + 2*I*e) + (4*I*a*c^3 + 12*a*c^2*d - 12*I*a*c*d^2 - 4*a*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(-I/((4*I*a^2*c^5 + 20*a^2*c^4*d - 40*I*a^2*c^3
*d^2 - 40*a^2*c^2*d^3 + 20*I*a^2*c*d^4 + 4*a^2*d^5)*f^2)) + 2*(c - I*d)*e^(2*I*f*x + 2*I*e) + 2*c)*e^(-2*I*f*x
- 2*I*e)) + ((-3*I*a*c^7 - 3*a*c^6*d - 9*I*a*c^5*d^2 - 9*a*c^4*d^3 - 9*I*a*c^3*d^4 - 9*a*c^2*d^5 - 3*I*a*c*d^
6 - 3*a*d^7)*f*e^(6*I*f*x + 6*I*e) + (-6*I*a*c^7 + 6*a*c^6*d - 18*I*a*c^5*d^2 + 18*a*c^4*d^3 - 18*I*a*c^3*d^4
+ 18*a*c^2*d^5 - 6*I*a*c*d^6 + 6*a*d^7)*f*e^(4*I*f*x + 4*I*e) + (-3*I*a*c^7 + 9*a*c^6*d + 3*I*a*c^5*d^2 + 15*a
*c^4*d^3 + 15*I*a*c^3*d^4 + 3*a*c^2*d^5 + 9*I*a*c*d^6 - 3*a*d^7)*f*e^(2*I*f*x + 2*I*e))*sqrt(-I/((4*I*a^2*c^5
+ 20*a^2*c^4*d - 40*I*a^2*c^3*d^2 - 40*a^2*c^2*d^3 + 20*I*a^2*c*d^4 + 4*a^2*d^5)*f^2))*log((((-4*I*a*c^3 - 12*
a*c^2*d + 12*I*a*c*d^2 + 4*a*d^3)*f*e^(2*I*f*x + 2*I*e) + (-4*I*a*c^3 - 12*a*c^2*d + 12*I*a*c*d^2 + 4*a*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(-I/((4*I*a^2*c^5 + 20*a^2*c^4
*d - 40*I*a^2*c^3*d^2 - 40*a^2*c^2*d^3 + 20*I*a^2*c*d^4 + 4*a^2*d^5)*f^2)) + 2*(c - I*d)*e^(2*I*f*x + 2*I*e) +
2*c)*e^(-2*I*f*x - 2*I*e)) + ((3*I*a*c^7 + 3*a*c^6*d + 9*I*a*c^5*d^2 + 9*a*c^4*d^3 + 9*I*a*c^3*d^4 + 9*a*c^2*
d^5 + 3*I*a*c*d^6 + 3*a*d^7)*f*e^(6*I*f*x + 6*I*e) + (6*I*a*c^7 - 6*a*c^6*d + 18*I*a*c^5*d^2 - 18*a*c^4*d^3 +
18*I*a*c^3*d^4 - 18*a*c^2*d^5 + 6*I*a*c*d^6 - 6*a*d^7)*f*e^(4*I*f*x + 4*I*e) + (3*I*a*c^7 - 9*a*c^6*d - 3*I*a*
c^5*d^2 - 15*a*c^4*d^3 - 15*I*a*c^3*d^4 - 3*a*c^2*d^5 - 9*I*a*c*d^6 + 3*a*d^7)*f*e^(2*I*f*x + 2*I*e))*sqrt((-I
*c^2 + 12*c*d + 36*I*d^2)/((4*I*a^2*c^7 - 28*a^2*c^6*d - 84*I*a^2*c^5*d^2 + 140*a^2*c^4*d^3 + 140*I*a^2*c^3*d^
4 - 84*a^2*c^2*d^5 - 28*I*a^2*c*d^6 + 4*a^2*d^7)*f^2))*log((I*c^2 - 7*c*d - 6*I*d^2 + ((2*a*c^4 + 8*I*a*c^3*d
- 12*a*c^2*d^2 - 8*I*a*c*d^3 + 2*a*d^4)*f*e^(2*I*f*x + 2*I*e) + (2*a*c^4 + 8*I*a*c^3*d - 12*a*c^2*d^2 - 8*I*a*
c*d^3 + 2*a*d^4)*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((-I*c^2 + 1
2*c*d + 36*I*d^2)/((4*I*a^2*c^7 - 28*a^2*c^6*d - 84*I*a^2*c^5*d^2 + 140*a^2*c^4*d^3 + 140*I*a^2*c^3*d^4 - 84*a
^2*c^2*d^5 - 28*I*a^2*c*d^6 + 4*a^2*d^7)*f^2)) + (I*c^2 - 6*c*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((2
*a*c^4 + 8*I*a*c^3*d - 12*a*c^2*d^2 - 8*I*a*c*d^3 + 2*a*d^4)*f)) + ((-3*I*a*c^7 - 3*a*c^6*d - 9*I*a*c^5*d^2 -
9*a*c^4*d^3 - 9*I*a*c^3*d^4 - 9*a*c^2*d^5 - 3*I*a*c*d^6 - 3*a*d^7)*f*e^(6*I*f*x + 6*I*e) + (-6*I*a*c^7 + 6*a*c
^6*d - 18*I*a*c^5*d^2 + 18*a*c^4*d^3 - 18*I*a*c^3*d^4 + 18*a*c^2*d^5 - 6*I*a*c*d^6 + 6*a*d^7)*f*e^(4*I*f*x + 4
*I*e) + (-3*I*a*c^7 + 9*a*c^6*d + 3*I*a*c^5*d^2 + 15*a*c^4*d^3 + 15*I*a*c^3*d^4 + 3*a*c^2*d^5 + 9*I*a*c*d^6 -
3*a*d^7)*f*e^(2*I*f*x + 2*I*e))*sqrt((-I*c^2 + 12*c*d + 36*I*d^2)/((4*I*a^2*c^7 - 28*a^2*c^6*d - 84*I*a^2*c^5*
d^2 + 140*a^2*c^4*d^3 + 140*I*a^2*c^3*d^4 - 84*a^2*c^2*d^5 - 28*I*a^2*c*d^6 + 4*a^2*d^7)*f^2))*log((I*c^2 - 7*
c*d - 6*I*d^2 - ((2*a*c^4 + 8*I*a*c^3*d - 12*a*c^2*d^2 - 8*I*a*c*d^3 + 2*a*d^4)*f*e^(2*I*f*x + 2*I*e) + (2*a*c
^4 + 8*I*a*c^3*d - 12*a*c^2*d^2 - 8*I*a*c*d^3 + 2*a*d^4)*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((-I*c^2 + 12*c*d + 36*I*d^2)/((4*I*a^2*c^7 - 28*a^2*c^6*d - 84*I*a^2*c^5*d^2 + 14
0*a^2*c^4*d^3 + 140*I*a^2*c^3*d^4 - 84*a^2*c^2*d^5 - 28*I*a^2*c*d^6 + 4*a^2*d^7)*f^2)) + (I*c^2 - 6*c*d)*e^(2*
I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((2*a*c^4 + 8*I*a*c^3*d - 12*a*c^2*d^2 - 8*I*a*c*d^3 + 2*a*d^4)*f)) - (3*
c^4 + 6*c^2*d^2 + 3*d^4 + (3*c^4 - 12*I*c^3*d - 98*c^2*d^2 + 108*I*c*d^3 + 19*d^4)*e^(6*I*f*x + 6*I*e) + (9*c^
4 - 24*I*c^3*d - 178*c^2*d^2 + 48*I*c*d^3 - 19*d^4)*e^(4*I*f*x + 4*I*e) + (9*c^4 - 12*I*c^3*d - 74*c^2*d^2 - 6
0*I*c*d^3 - 35*d^4)*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)))/((12*I*a*c^7 + 12*a*c^6*d + 36*I*a*c^5*d^2 + 36*a*c^4*d^3 + 36*I*a*c^3*d^4 + 36*a*c^2*d^5 + 12*I*a*c*d^6
+ 12*a*d^7)*f*e^(6*I*f*x + 6*I*e) + (24*I*a*c^7 - 24*a*c^6*d + 72*I*a*c^5*d^2 - 72*a*c^4*d^3 + 72*I*a*c^3*d^4
- 72*a*c^2*d^5 + 24*I*a*c*d^6 - 24*a*d^7)*f*e^(4*I*f*x + 4*I*e) + (12*I*a*c^7 - 36*a*c^6*d - 12*I*a*c^5*d^2 -
60*a*c^4*d^3 - 60*I*a*c^3*d^4 - 12*a*c^2*d^5 - 36*I*a*c*d^6 + 12*a*d^7)*f*e^(2*I*f*x + 2*I*e))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.62189, size = 749, normalized size = 2.81 \begin{align*} 2 \, d^{2}{\left (\frac{2 \,{\left (-i \, c + 6 \, d\right )} \arctan \left (\frac{4 \,{\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{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}} + i \, \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}} d - \sqrt{c^{2} + d^{2}} \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}}}\right )}{{\left (2 \, a c^{3} d^{2} f + 6 i \, a c^{2} d^{3} f - 6 \, a c d^{4} f - 2 i \, a d^{5} f\right )} \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}}{\left (\frac{i \, d}{c - \sqrt{c^{2} + d^{2}}} + 1\right )}} + \frac{9 \,{\left (d \tan \left (f x + e\right ) + c\right )} c + c^{2} +{\left (-3 i \, d \tan \left (f x + e\right ) - 3 i \, c\right )} d + d^{2}}{{\left (3 i \, a c^{5} f - 3 \, a c^{4} d f + 6 i \, a c^{3} d^{2} f - 6 \, a c^{2} d^{3} f + 3 i \, a c d^{4} f - 3 \, a d^{5} f\right )}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}} + \frac{\sqrt{d \tan \left (f x + e\right ) + c}}{{\left (-4 i \, a c^{3} d f + 12 \, a c^{2} d^{2} f + 12 i \, a c d^{3} f - 4 \, a d^{4} f\right )}{\left (i \, d \tan \left (f x + e\right ) + d\right )}} + \frac{2 i \, \arctan \left (\frac{4 \,{\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{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}} - i \, \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}} d - \sqrt{c^{2} + d^{2}} \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}}}\right )}{{\left (2 \, a c^{2} d^{2} f - 4 i \, a c d^{3} f - 2 \, a d^{4} f\right )} \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}}{\left (-\frac{i \, d}{c - \sqrt{c^{2} + d^{2}}} + 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^2*(2*(-I*c + 6*d)*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt
(-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2
))))/((2*a*c^3*d^2*f + 6*I*a*c^2*d^3*f - 6*a*c*d^4*f - 2*I*a*d^5*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - s
qrt(c^2 + d^2)) + 1)) + (9*(d*tan(f*x + e) + c)*c + c^2 + (-3*I*d*tan(f*x + e) - 3*I*c)*d + d^2)/((3*I*a*c^5*f
- 3*a*c^4*d*f + 6*I*a*c^3*d^2*f - 6*a*c^2*d^3*f + 3*I*a*c*d^4*f - 3*a*d^5*f)*(d*tan(f*x + e) + c)^(3/2)) + sq
rt(d*tan(f*x + e) + c)/((-4*I*a*c^3*d*f + 12*a*c^2*d^2*f + 12*I*a*c*d^3*f - 4*a*d^4*f)*(I*d*tan(f*x + e) + d))
+ 2*I*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(
c^2 + d^2)) - I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/((2*a*c^2*
d^2*f - 4*I*a*c*d^3*f - 2*a*d^4*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)))