Optimal. Leaf size=102 \[ -\frac{(-4 B+i A) (a+i a \tan (e+f x))^{3/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{5 f (c-i c \tan (e+f x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.233717, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3588, 78, 37} \[ -\frac{(-4 B+i A) (a+i a \tan (e+f x))^{3/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{5 f (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3588
Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{\sqrt{a+i a x} (A+B x)}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{3/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac{(a (A+4 i B)) \operatorname{Subst}\left (\int \frac{\sqrt{a+i a x}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{3/2}}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac{(i A-4 B) (a+i a \tan (e+f x))^{3/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 11.6563, size = 117, normalized size = 1.15 \[ \frac{a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (4 e+5 f x)+i \sin (4 e+5 f x)) ((B-4 i A) \cos (e+f x)-(A+4 i B) \sin (e+f x))}{15 c^3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.106, size = 90, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{15}}a \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( -4\,A+iA\tan \left ( fx+e \right ) -iB-4\,B\tan \left ( fx+e \right ) \right ) }{f{c}^{3} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.28958, size = 209, normalized size = 2.05 \begin{align*} -\frac{{\left ({\left (90 \, A - 90 i \, B\right )} a \cos \left (7 \, f x + 7 \, e\right ) +{\left (240 \, A + 60 i \, B\right )} a \cos \left (5 \, f x + 5 \, e\right ) +{\left (150 \, A + 150 i \, B\right )} a \cos \left (3 \, f x + 3 \, e\right ) - 90 \,{\left (-i \, A - B\right )} a \sin \left (7 \, f x + 7 \, e\right ) - 60 \,{\left (-4 i \, A + B\right )} a \sin \left (5 \, f x + 5 \, e\right ) - 150 \,{\left (-i \, A + B\right )} a \sin \left (3 \, f x + 3 \, e\right )\right )} \sqrt{a} \sqrt{c}}{{\left (-900 i \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) + 900 \, c^{3} \sin \left (2 \, f x + 2 \, e\right ) - 900 i \, c^{3}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.44042, size = 290, normalized size = 2.84 \begin{align*} \frac{{\left ({\left (-3 i \, A - 3 \, B\right )} a e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-8 i \, A + 2 \, B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-5 i \, A + 5 \, B\right )} a e^{\left (2 i \, f x + 2 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}} e^{\left (i \, f x + i \, e\right )}}{30 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (f x + e\right ) + A\right )}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]