3.1125 \(\int \frac{(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ \frac{4 a^3 c \sqrt{c+d \tan (e+f x)}}{d^2 f (d+i c)}+\frac{2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{d f (c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{8 i a^3 \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (c-i d)^{3/2}} \]

[Out]

((-8*I)*a^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f) + (2*(c + I*d)*(a^3 + I*a^3*T
an[e + f*x]))/((c - I*d)*d*f*Sqrt[c + d*Tan[e + f*x]]) + (4*a^3*c*Sqrt[c + d*Tan[e + f*x]])/(d^2*(I*c + d)*f)

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Rubi [A]  time = 0.314943, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3553, 3592, 3537, 63, 208} \[ \frac{4 a^3 c \sqrt{c+d \tan (e+f x)}}{d^2 f (d+i c)}+\frac{2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{d f (c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{8 i a^3 \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (c-i d)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*I)*a^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f) + (2*(c + I*d)*(a^3 + I*a^3*T
an[e + f*x]))/((c - I*d)*d*f*Sqrt[c + d*Tan[e + f*x]]) + (4*a^3*c*Sqrt[c + d*Tan[e + f*x]])/(d^2*(I*c + d)*f)

Rule 3553

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(a^2*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] +
Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(b*c*(m
- 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; Fr
eeQ[{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] && GtQ[m, 1] && LtQ[
n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3592

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

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{(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{3/2}} \, dx &=\frac{2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt{c+d \tan (e+f x)}}-\frac{2 \int \frac{(a+i a \tan (e+f x)) \left (-a^2 (c+2 i d)+i a^2 c \tan (e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{d (i c+d)}\\ &=\frac{2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt{c+d \tan (e+f x)}}+\frac{4 a^3 c \sqrt{c+d \tan (e+f x)}}{d^2 (i c+d) f}-\frac{2 \int \frac{-2 i a^3 d+2 a^3 d \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{d (i c+d)}\\ &=\frac{2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt{c+d \tan (e+f x)}}+\frac{4 a^3 c \sqrt{c+d \tan (e+f x)}}{d^2 (i c+d) f}+\frac{\left (8 a^6 d\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{x}{2 a^3}} \left (4 a^6 d^2-2 i a^3 d x\right )} \, dx,x,2 a^3 d \tan (e+f x)\right )}{(c-i d) f}\\ &=\frac{2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt{c+d \tan (e+f x)}}+\frac{4 a^3 c \sqrt{c+d \tan (e+f x)}}{d^2 (i c+d) f}+\frac{\left (32 a^9 d\right ) \operatorname{Subst}\left (\int \frac{1}{4 i a^6 c d+4 a^6 d^2-4 i a^6 d x^2} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{(c-i d) f}\\ &=-\frac{8 i a^3 \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{(c-i d)^{3/2} f}+\frac{2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt{c+d \tan (e+f x)}}+\frac{4 a^3 c \sqrt{c+d \tan (e+f x)}}{d^2 (i c+d) f}\\ \end{align*}

Mathematica [A]  time = 5.05283, size = 219, normalized size = 1.58 \[ \frac{a^3 (\cos (e+f x)+i \sin (e+f x))^3 \left (\frac{2 (\cos (3 e)-i \sin (3 e)) \sqrt{c+d \tan (e+f x)} \left (\left (-2 i c^2+c d+i d^2\right ) \cos (e+f x)+d (-d-i c) \sin (e+f x)\right )}{d^2 (c-i d) (c \cos (e+f x)+d \sin (e+f x))}-\frac{8 i e^{-3 i e} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right )}{(c-i d)^{3/2}}\right )}{f (\cos (f x)+i \sin (f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(Cos[e + f*x] + I*Sin[e + f*x])^3*(((-8*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I
)*(e + f*x)))]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*E^((3*I)*e)) + (2*(Cos[3*e] - I*Sin[3*e])*(((-2*I)*c^2 + c*d +
I*d^2)*Cos[e + f*x] + ((-I)*c - d)*d*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/((c - I*d)*d^2*(c*Cos[e + f*x] +
d*Sin[e + f*x]))))/(f*(Cos[f*x] + I*Sin[f*x])^3)

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Maple [B]  time = 0.052, size = 2594, normalized size = 18.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2),x)

[Out]

-2*I/f*a^3/d^2*(c+d*tan(f*x+e))^(1/2)+4/f*a^3*d/(c^2+d^2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*ar
ctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c-4/f*a^3*d/(c^2+
d^2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e)
)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c+2/f*a^3*d/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)*ln(d*tan(f*x+e)+c+(c+d
*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-4*I/f*a^3*d^
2/(c^2+d^2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan
(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))-I/f*a^3*d^2/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)*ln((c+d*tan(f*x
+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+4*I/f*a
^3/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(
c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c^3-I/f*a^3/(c^2+d^2)/((c^2+d^2)^(1/2)+c)*ln(d*tan(f*x+e
)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-I/f*
a^3/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+8/f*a^3*d/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)
^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2
))*c^2-2/f*a^3*d/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d
*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-8/f*a^3*d/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)/(
2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2
)-2*c)^(1/2))*c^2+I/f*a^3*d^2/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+4*I/f*a^3/((c^2+d^2)^(1/2)+c)/(c^2+d
^2)^(1/2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^
2+d^2)^(1/2)-2*c)^(1/2))*c+4*I/f*a^3*d^2/(c^2+d^2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2
*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))+I/f*a^3/(c^2+d^2)^(3/2)/
((c^2+d^2)^(1/2)+c)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+I/f*a^3/(c^2+d^2)/((c^2+d^2)^(1/2)+c)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-4*I/f*a^3/(c^2+d^2)^(3/2)/((c^
2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/
(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c^3-4*I/f*a^3/((c^2+d^2)^(1/2)+c)/(c^2+d^2)^(1/2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)
*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c+4*I/f*a^3*d^
2/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d
^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c-2/f*a^3*d/(c^2+d^2)/(c+d*tan(f*x+e))^(1/2)-4*I/f*a^3*d^
2/(c^2+d^2)^(3/2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c
+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*c+4/f*a^3*d/(c^2+d^2)^(1/2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d
^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(
1/2))-4/f*a^3*d/(c^2+d^2)^(1/2)/((c^2+d^2)^(1/2)+c)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(
1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))-1/f*a^3*d/(c^2+d^2)/((c^2+d^2)^(1/2)+c)*ln(
(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(
1/2)+1/f*a^3*d/(c^2+d^2)/((c^2+d^2)^(1/2)+c)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*I/f*a^3/d^2/(c^2+d^2)/(c+d*tan(f*x+e))^(1/2)*c^3+6/f*a^
3/d/(c^2+d^2)/(c+d*tan(f*x+e))^(1/2)*c^2+6*I/f*a^3/(c^2+d^2)/(c+d*tan(f*x+e))^(1/2)*c

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Maxima [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((a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 2.20237, size = 1478, normalized size = 10.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(64*I*a^6/((-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*f^2))*((c^2*d^2 - 2*I*c*d^3 - d^4)*f*e^(2*I*f*x + 2*I
*e) + (c^2*d^2 + d^4)*f)*log(1/4*(8*a^3*c + sqrt(64*I*a^6/((-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*f^2))*((I*c^2
+ 2*c*d - I*d^2)*f*e^(2*I*f*x + 2*I*e) + (I*c^2 + 2*c*d - I*d^2)*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)) + (8*a^3*c - 8*I*a^3*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/a^3) - sqrt(
64*I*a^6/((-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*f^2))*((c^2*d^2 - 2*I*c*d^3 - d^4)*f*e^(2*I*f*x + 2*I*e) + (c^2
*d^2 + d^4)*f)*log(1/4*(8*a^3*c + sqrt(64*I*a^6/((-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*f^2))*((-I*c^2 - 2*c*d +
I*d^2)*f*e^(2*I*f*x + 2*I*e) + (-I*c^2 - 2*c*d + I*d^2)*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)) + (8*a^3*c - 8*I*a^3*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/a^3) - (16*I*a^3*c^2
- 16*a^3*c*d + (16*I*a^3*c^2 - 16*I*a^3*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)))/((c^2*d^2 - 2*I*c*d^3 - d^4)*f*e^(2*I*f*x + 2*I*e) + (c^2*d^2 + d^4)*f)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int - \frac{3 \tan ^{2}{\left (e + f x \right )}}{c \sqrt{c + d \tan{\left (e + f x \right )}} + d \sqrt{c + d \tan{\left (e + f x \right )}} \tan{\left (e + f x \right )}}\, dx + \int \frac{3 i \tan{\left (e + f x \right )}}{c \sqrt{c + d \tan{\left (e + f x \right )}} + d \sqrt{c + d \tan{\left (e + f x \right )}} \tan{\left (e + f x \right )}}\, dx + \int - \frac{i \tan ^{3}{\left (e + f x \right )}}{c \sqrt{c + d \tan{\left (e + f x \right )}} + d \sqrt{c + d \tan{\left (e + f x \right )}} \tan{\left (e + f x \right )}}\, dx + \int \frac{1}{c \sqrt{c + d \tan{\left (e + f x \right )}} + d \sqrt{c + d \tan{\left (e + f x \right )}} \tan{\left (e + f x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**3/(c+d*tan(f*x+e))**(3/2),x)

[Out]

a**3*(Integral(-3*tan(e + f*x)**2/(c*sqrt(c + d*tan(e + f*x)) + d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)), x) +
Integral(3*I*tan(e + f*x)/(c*sqrt(c + d*tan(e + f*x)) + d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)), x) + Integr
al(-I*tan(e + f*x)**3/(c*sqrt(c + d*tan(e + f*x)) + d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)), x) + Integral(1/
(c*sqrt(c + d*tan(e + f*x)) + d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)), x))

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Giac [B]  time = 1.69223, size = 333, normalized size = 2.4 \begin{align*} \frac{32 \, a^{3} \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 (-i \, c f - d f\right )} \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}}{\left (-\frac{i \, d}{c - \sqrt{c^{2} + d^{2}}} + 1\right )}} - \frac{2 i \, \sqrt{d \tan \left (f x + e\right ) + c} a^{3}}{d^{2} f} - \frac{2 i \, a^{3} c^{2} - 4 \, a^{3} c d - 2 i \, a^{3} d^{2}}{{\left (c d^{2} f - i \, d^{3} f\right )} \sqrt{d \tan \left (f x + e\right ) + c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32*a^3*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))))/((-I*c*f -
d*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - 2*I*sqrt(d*tan(f*x + e) + c)*a^3/(d^2
*f) - (2*I*a^3*c^2 - 4*a^3*c*d - 2*I*a^3*d^2)/((c*d^2*f - I*d^3*f)*sqrt(d*tan(f*x + e) + c))