\(\int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx\) [1011]

Optimal result

Integrand size = 35, antiderivative size = 155 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=-\frac {2 i c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {2 i c^2 \sqrt {c-i c \tan (e+f x)}}{a f \sqrt {a+i a \tan (e+f x)}}+\frac {2 i c (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]

[Out]

-2*I*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))/a^(3/2)/f-2*I*c^2*(c-I*
c*tan(f*x+e))^(1/2)/a/f/(a+I*a*tan(f*x+e))^(1/2)+2/3*I*c*(c-I*c*tan(f*x+e))^(3/2)/f/(a+I*a*tan(f*x+e))^(3/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3604, 49, 65, 223, 209} \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=-\frac {2 i c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {2 i c^2 \sqrt {c-i c \tan (e+f x)}}{a f \sqrt {a+i a \tan (e+f x)}}+\frac {2 i c (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]

[In]

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

[Out]

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

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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 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 3604

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((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), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f,
m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.69 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.57 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {4 i c^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1+i \tan (e+f x))\right ) \sqrt {2-2 i \tan (e+f x)}}{3 f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \]

[In]

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

[Out]

(((4*I)/3)*c^3*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 + I*Tan[e + f*x])/2]*Sqrt[2 - (2*I)*Tan[e + f*x]])/(f*(a
+ I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (124 ) = 248\).

Time = 0.82 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.26

method	result	size
derivativedivides	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (9 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )-3 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )-3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -12 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )+9 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+8 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )-4 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,a^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (-\tan \left (f x +e \right )+i\right )^{3} \sqrt {a c}}\)	\(350\)
default	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (9 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )-3 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )-3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -12 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )+9 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )+8 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )-4 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,a^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (-\tan \left (f x +e \right )+i\right )^{3} \sqrt {a c}}\)	\(350\)

[In]

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

[Out]

1/3/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*c^2/a^2*(9*I*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e
)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^2-3*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)
^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^3-3*I*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(
1/2))*a*c-12*I*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)+9*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))
^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)+8*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2-4*(a*c
*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c*(1+tan(f*x+e)^2))^(1/2)/(-tan(f*x+e)+I)^3/(a*c)^(1/2)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (115) = 230\).

Time = 0.26 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.49 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {{\left (3 \, a^{2} f \sqrt {\frac {c^{5}}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (\frac {4 \, {\left (2 \, {\left (c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (i \, a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{2} f\right )} \sqrt {\frac {c^{5}}{a^{3} f^{2}}}\right )}}{c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2}}\right ) - 3 \, a^{2} f \sqrt {\frac {c^{5}}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (\frac {4 \, {\left (2 \, {\left (c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (-i \, a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, a^{2} f\right )} \sqrt {\frac {c^{5}}{a^{3} f^{2}}}\right )}}{c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2}}\right ) - 4 \, {\left (3 i \, c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, c^{2}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-3 i \, f x - 3 i \, e\right )}}{6 \, a^{2} f} \]

[In]

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

[Out]

1/6*(3*a^2*f*sqrt(c^5/(a^3*f^2))*e^(3*I*f*x + 3*I*e)*log(4*(2*(c^2*e^(3*I*f*x + 3*I*e) + 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)) - (I*a^2*f*e^(2*I*f*x + 2*I*e) - I*a^2*f)*
sqrt(c^5/(a^3*f^2)))/(c^2*e^(2*I*f*x + 2*I*e) + c^2)) - 3*a^2*f*sqrt(c^5/(a^3*f^2))*e^(3*I*f*x + 3*I*e)*log(4*
(2*(c^2*e^(3*I*f*x + 3*I*e) + 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)) - (-I*a^2*f*e^(2*I*f*x + 2*I*e) + I*a^2*f)*sqrt(c^5/(a^3*f^2)))/(c^2*e^(2*I*f*x + 2*I*e) + c^2)) - 4*
(3*I*c^2*e^(4*I*f*x + 4*I*e) + 2*I*c^2*e^(2*I*f*x + 2*I*e) - I*c^2)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(
e^(2*I*f*x + 2*I*e) + 1)))*e^(-3*I*f*x - 3*I*e)/(a^2*f)

Sympy [F]

\[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((c-I*c*tan(f*x+e))**(5/2)/(a+I*a*tan(f*x+e))**(3/2),x)

[Out]

Integral((-I*c*(tan(e + f*x) + I))**(5/2)/(I*a*(tan(e + f*x) - I))**(3/2), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (115) = 230\).

Time = 0.50 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.65 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {{\left (-6 i \, c^{2} \arctan \left (\cos \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right ), \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right ) + 1\right ) - 6 i \, c^{2} \arctan \left (\cos \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right ), -\sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right ) + 1\right ) + 4 i \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, c^{2} \log \left (\cos \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right ) + 1\right ) - 3 \, c^{2} \log \left (\cos \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right ) + 1\right ) + 4 \, c^{2} \sin \left (3 \, f x + 3 \, e\right ) - 12 \, {\left (i \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) + c^{2} \sin \left (3 \, f x + 3 \, e\right )\right )} \cos \left (\frac {2}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right ) + 12 \, {\left (c^{2} \cos \left (3 \, f x + 3 \, e\right ) - i \, c^{2} \sin \left (3 \, f x + 3 \, e\right )\right )} \sin \left (\frac {2}{3} \, \arctan \left (\sin \left (3 \, f x + 3 \, e\right ), \cos \left (3 \, f x + 3 \, e\right )\right )\right )\right )} \sqrt {c}}{6 \, a^{\frac {3}{2}} f} \]

[In]

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

[Out]

1/6*(-6*I*c^2*arctan2(cos(1/3*arctan2(sin(3*f*x + 3*e), cos(3*f*x + 3*e))), sin(1/3*arctan2(sin(3*f*x + 3*e),
cos(3*f*x + 3*e))) + 1) - 6*I*c^2*arctan2(cos(1/3*arctan2(sin(3*f*x + 3*e), cos(3*f*x + 3*e))), -sin(1/3*arcta
n2(sin(3*f*x + 3*e), cos(3*f*x + 3*e))) + 1) + 4*I*c^2*cos(3*f*x + 3*e) + 3*c^2*log(cos(1/3*arctan2(sin(3*f*x
+ 3*e), cos(3*f*x + 3*e)))^2 + sin(1/3*arctan2(sin(3*f*x + 3*e), cos(3*f*x + 3*e)))^2 + 2*sin(1/3*arctan2(sin(
3*f*x + 3*e), cos(3*f*x + 3*e))) + 1) - 3*c^2*log(cos(1/3*arctan2(sin(3*f*x + 3*e), cos(3*f*x + 3*e)))^2 + sin
(1/3*arctan2(sin(3*f*x + 3*e), cos(3*f*x + 3*e)))^2 - 2*sin(1/3*arctan2(sin(3*f*x + 3*e), cos(3*f*x + 3*e))) +
1) + 4*c^2*sin(3*f*x + 3*e) - 12*(I*c^2*cos(3*f*x + 3*e) + c^2*sin(3*f*x + 3*e))*cos(2/3*arctan2(sin(3*f*x +
3*e), cos(3*f*x + 3*e))) + 12*(c^2*cos(3*f*x + 3*e) - I*c^2*sin(3*f*x + 3*e))*sin(2/3*arctan2(sin(3*f*x + 3*e)
, cos(3*f*x + 3*e))))*sqrt(c)/(a^(3/2)*f)

Giac [F]

\[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((-I*c*tan(f*x + e) + c)^(5/2)/(I*a*tan(f*x + e) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]

[In]

int((c - c*tan(e + f*x)*1i)^(5/2)/(a + a*tan(e + f*x)*1i)^(3/2),x)

[Out]

int((c - c*tan(e + f*x)*1i)^(5/2)/(a + a*tan(e + f*x)*1i)^(3/2), x)