3.9.6 \(\int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx\) [806]

Optimal. Leaf size=213 \[ -\frac {3 i a^{5/2} A c^{5/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {3 a^2 A c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f} \]

[Out]

-3/4*I*a^(5/2)*A*c^(5/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c-I*c*tan(f*x+e))^(1/2))/f+3/8*a^2*A
*c^2*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)*tan(f*x+e)/f+1/4*a*A*c*tan(f*x+e)*(a+I*a*tan(f*x+e))^(3
/2)*(c-I*c*tan(f*x+e))^(3/2)/f+1/5*B*(a+I*a*tan(f*x+e))^(5/2)*(c-I*c*tan(f*x+e))^(5/2)/f

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Rubi [A]
time = 0.18, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3669, 81, 38, 65, 223, 209} \begin {gather*} -\frac {3 i a^{5/2} A c^{5/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {3 a^2 A c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

(((-3*I)/4)*a^(5/2)*A*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])
])/f + (3*a^2*A*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) + (a*A*c*Tan[e +
f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(4*f) + (B*(a + I*a*Tan[e + f*x])^(5/2)*(c -
I*c*Tan[e + f*x])^(5/2))/(5*f)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

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 81

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 3669

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

Rubi steps

\begin {align*} \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx &=\frac {(a c) \text {Subst}\left (\int (a+i a x)^{3/2} (A+B x) (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f}+\frac {(a A c) \text {Subst}\left (\int (a+i a x)^{3/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f}+\frac {\left (3 a^2 A c^2\right ) \text {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=\frac {3 a^2 A c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f}+\frac {\left (3 a^3 A c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {3 a^2 A c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {\left (3 i a^2 A c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{4 f}\\ &=\frac {3 a^2 A c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {\left (3 i a^2 A c^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{4 f}\\ &=-\frac {3 i a^{5/2} A c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {3 a^2 A c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a A c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{5 f}\\ \end {align*}

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Mathematica [A]
time = 4.92, size = 119, normalized size = 0.56 \begin {gather*} -\frac {a^2 c^3 \sec ^4(e+f x) \left (240 A \text {ArcTan}\left (e^{i (e+f x)}\right ) \cos ^5(e+f x)+i (64 B+70 A \sin (2 (e+f x))+15 A \sin (4 (e+f x)))\right ) (i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)}}{320 f \sqrt {c-i c \tan (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

-1/320*(a^2*c^3*Sec[e + f*x]^4*(240*A*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x]^5 + I*(64*B + 70*A*Sin[2*(e + f*x)]
+ 15*A*Sin[4*(e + f*x)]))*(I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])

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Maple [A]
time = 0.43, size = 252, normalized size = 1.18

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} c^{2} \left (8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{4}\left (f x +e \right )\right )+10 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+16 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+15 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +25 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{40 f \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}\)	\(252\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} c^{2} \left (8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{4}\left (f x +e \right )\right )+10 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+16 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+15 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +25 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{40 f \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}\)	\(252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^2*c^2*(8*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^
(1/2)*tan(f*x+e)^4+10*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^3+16*B*(a*c*(1+tan(f*x+e)^2))^(1/2
)*(a*c)^(1/2)*tan(f*x+e)^2+15*A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c+
25*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)+8*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(
1/2)/(a*c*(1+tan(f*x+e)^2))^(1/2)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1535 vs. \(2 (173) = 346\).
time = 1.19, size = 1535, normalized size = 7.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(60*A*a^2*c^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 280*A*a^2*c^2*cos(7/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) + 512*I*B*a^2*c^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 280*A*a^2*
c^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 60*A*a^2*c^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) + 60*I*A*a^2*c^2*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 280*I*A*a^2*c^2*sin(7/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 512*B*a^2*c^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e))) - 280*I*A*a^2*c^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 60*I*A*a^2*c^2*sin(1/2*arctan2(s
in(2*f*x + 2*e), cos(2*f*x + 2*e))) + 30*(A*a^2*c^2*cos(10*f*x + 10*e) + 5*A*a^2*c^2*cos(8*f*x + 8*e) + 10*A*a
^2*c^2*cos(6*f*x + 6*e) + 10*A*a^2*c^2*cos(4*f*x + 4*e) + 5*A*a^2*c^2*cos(2*f*x + 2*e) + I*A*a^2*c^2*sin(10*f*
x + 10*e) + 5*I*A*a^2*c^2*sin(8*f*x + 8*e) + 10*I*A*a^2*c^2*sin(6*f*x + 6*e) + 10*I*A*a^2*c^2*sin(4*f*x + 4*e)
+ 5*I*A*a^2*c^2*sin(2*f*x + 2*e) + A*a^2*c^2)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), s
in(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 30*(A*a^2*c^2*cos(10*f*x + 10*e) + 5*A*a^2*c^2*cos(
8*f*x + 8*e) + 10*A*a^2*c^2*cos(6*f*x + 6*e) + 10*A*a^2*c^2*cos(4*f*x + 4*e) + 5*A*a^2*c^2*cos(2*f*x + 2*e) +
I*A*a^2*c^2*sin(10*f*x + 10*e) + 5*I*A*a^2*c^2*sin(8*f*x + 8*e) + 10*I*A*a^2*c^2*sin(6*f*x + 6*e) + 10*I*A*a^2
*c^2*sin(4*f*x + 4*e) + 5*I*A*a^2*c^2*sin(2*f*x + 2*e) + A*a^2*c^2)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))), -sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 15*(I*A*a^2*c^2*cos(10*f*x +
10*e) + 5*I*A*a^2*c^2*cos(8*f*x + 8*e) + 10*I*A*a^2*c^2*cos(6*f*x + 6*e) + 10*I*A*a^2*c^2*cos(4*f*x + 4*e) + 5
*I*A*a^2*c^2*cos(2*f*x + 2*e) - A*a^2*c^2*sin(10*f*x + 10*e) - 5*A*a^2*c^2*sin(8*f*x + 8*e) - 10*A*a^2*c^2*sin
(6*f*x + 6*e) - 10*A*a^2*c^2*sin(4*f*x + 4*e) - 5*A*a^2*c^2*sin(2*f*x + 2*e) + I*A*a^2*c^2)*log(cos(1/2*arctan
2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/2*
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 15*(-I*A*a^2*c^2*cos(10*f*x + 10*e) - 5*I*A*a^2*c^2*cos(8*
f*x + 8*e) - 10*I*A*a^2*c^2*cos(6*f*x + 6*e) - 10*I*A*a^2*c^2*cos(4*f*x + 4*e) - 5*I*A*a^2*c^2*cos(2*f*x + 2*e
) + A*a^2*c^2*sin(10*f*x + 10*e) + 5*A*a^2*c^2*sin(8*f*x + 8*e) + 10*A*a^2*c^2*sin(6*f*x + 6*e) + 10*A*a^2*c^2
*sin(4*f*x + 4*e) + 5*A*a^2*c^2*sin(2*f*x + 2*e) - I*A*a^2*c^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sin(1/2*arctan2(sin(2*f*x + 2*e), co
s(2*f*x + 2*e))) + 1))*sqrt(a)*sqrt(c)/(f*(-80*I*cos(10*f*x + 10*e) - 400*I*cos(8*f*x + 8*e) - 800*I*cos(6*f*x
+ 6*e) - 800*I*cos(4*f*x + 4*e) - 400*I*cos(2*f*x + 2*e) + 80*sin(10*f*x + 10*e) + 400*sin(8*f*x + 8*e) + 800
*sin(6*f*x + 6*e) + 800*sin(4*f*x + 4*e) + 400*sin(2*f*x + 2*e) - 80*I))

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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. 616 vs. \(2 (173) = 346\).
time = 1.23, size = 616, normalized size = 2.89 \begin {gather*} \frac {15 \, \sqrt {\frac {A^{2} a^{5} c^{5}}{f^{2}}} {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (A a^{2} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + A a^{2} c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - \sqrt {\frac {A^{2} a^{5} c^{5}}{f^{2}}} {\left (i \, f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, f\right )}\right )}}{A a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + A a^{2} c^{2}}\right ) - 15 \, \sqrt {\frac {A^{2} a^{5} c^{5}}{f^{2}}} {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (A a^{2} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + A a^{2} c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - \sqrt {\frac {A^{2} a^{5} c^{5}}{f^{2}}} {\left (-i \, f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, f\right )}\right )}}{A a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + A a^{2} c^{2}}\right ) + 4 \, {\left (-15 i \, A a^{2} c^{2} e^{\left (9 i \, f x + 9 i \, e\right )} - 70 i \, A a^{2} c^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 128 \, B a^{2} c^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + 70 i \, A a^{2} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + 15 i \, A a^{2} c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{80 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*(15*sqrt(A^2*a^5*c^5/f^2)*(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*
f*e^(2*I*f*x + 2*I*e) + f)*log(4*(2*(A*a^2*c^2*e^(3*I*f*x + 3*I*e) + A*a^2*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I
*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - sqrt(A^2*a^5*c^5/f^2)*(I*f*e^(2*I*f*x + 2*I*e) - I*f))
/(A*a^2*c^2*e^(2*I*f*x + 2*I*e) + A*a^2*c^2)) - 15*sqrt(A^2*a^5*c^5/f^2)*(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f
*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e) + f)*log(4*(2*(A*a^2*c^2*e^(3*I*f*x + 3*I*e) +
A*a^2*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - sqrt(A^2*a^5
*c^5/f^2)*(-I*f*e^(2*I*f*x + 2*I*e) + I*f))/(A*a^2*c^2*e^(2*I*f*x + 2*I*e) + A*a^2*c^2)) + 4*(-15*I*A*a^2*c^2*
e^(9*I*f*x + 9*I*e) - 70*I*A*a^2*c^2*e^(7*I*f*x + 7*I*e) + 128*B*a^2*c^2*e^(5*I*f*x + 5*I*e) + 70*I*A*a^2*c^2*
e^(3*I*f*x + 3*I*e) + 15*I*A*a^2*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2
*I*e) + 1)))/(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I
*e) + f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4847 deep

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Giac [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((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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