Optimal. Leaf size=91 \[ -\frac{a^2 (A-3 i B)}{5 c^6 f (\tan (e+f x)+i)^5}+\frac{a^2 (B+i A)}{3 c^6 f (\tan (e+f x)+i)^6}-\frac{a^2 B}{4 c^6 f (\tan (e+f x)+i)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.149589, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {3588, 77} \[ -\frac{a^2 (A-3 i B)}{5 c^6 f (\tan (e+f x)+i)^5}+\frac{a^2 (B+i A)}{3 c^6 f (\tan (e+f x)+i)^6}-\frac{a^2 B}{4 c^6 f (\tan (e+f x)+i)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3588
Rule 77
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(a+i a x) (A+B x)}{(c-i c x)^7} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (-\frac{2 i a (A-i B)}{c^7 (i+x)^7}+\frac{a (A-3 i B)}{c^7 (i+x)^6}+\frac{a B}{c^7 (i+x)^5}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a^2 (i A+B)}{3 c^6 f (i+\tan (e+f x))^6}-\frac{a^2 (A-3 i B)}{5 c^6 f (i+\tan (e+f x))^5}-\frac{a^2 B}{4 c^6 f (i+\tan (e+f x))^4}\\ \end{align*}
Mathematica [A] time = 4.82706, size = 143, normalized size = 1.57 \[ -\frac{i a^2 (\cos (8 e+10 f x)+i \sin (8 e+10 f x)) (8 (8 A+i B) \cos (2 (e+f x))+10 (2 A+i B) \cos (4 (e+f x))-16 i A \sin (2 (e+f x))-10 i A \sin (4 (e+f x))+45 A+32 B \sin (2 (e+f x))+20 B \sin (4 (e+f x)))}{960 c^6 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 66, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}}{f{c}^{6}} \left ( -{\frac{A-3\,iB}{5\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}-{\frac{B}{4\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-{\frac{-2\,B-2\,iA}{6\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{6}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.32842, size = 300, normalized size = 3.3 \begin{align*} \frac{{\left (-5 i \, A - 5 \, B\right )} a^{2} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-24 i \, A - 12 \, B\right )} a^{2} e^{\left (10 i \, f x + 10 i \, e\right )} - 45 i \, A a^{2} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-40 i \, A + 20 \, B\right )} a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-15 i \, A + 15 \, B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{960 \, c^{6} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.27928, size = 381, normalized size = 4.19 \begin{align*} \begin{cases} \frac{- 141557760 i A a^{2} c^{24} f^{4} e^{8 i e} e^{8 i f x} + \left (- 47185920 i A a^{2} c^{24} f^{4} e^{4 i e} + 47185920 B a^{2} c^{24} f^{4} e^{4 i e}\right ) e^{4 i f x} + \left (- 125829120 i A a^{2} c^{24} f^{4} e^{6 i e} + 62914560 B a^{2} c^{24} f^{4} e^{6 i e}\right ) e^{6 i f x} + \left (- 75497472 i A a^{2} c^{24} f^{4} e^{10 i e} - 37748736 B a^{2} c^{24} f^{4} e^{10 i e}\right ) e^{10 i f x} + \left (- 15728640 i A a^{2} c^{24} f^{4} e^{12 i e} - 15728640 B a^{2} c^{24} f^{4} e^{12 i e}\right ) e^{12 i f x}}{3019898880 c^{30} f^{5}} & \text{for}\: 3019898880 c^{30} f^{5} \neq 0 \\\frac{x \left (A a^{2} e^{12 i e} + 4 A a^{2} e^{10 i e} + 6 A a^{2} e^{8 i e} + 4 A a^{2} e^{6 i e} + A a^{2} e^{4 i e} - i B a^{2} e^{12 i e} - 2 i B a^{2} e^{10 i e} + 2 i B a^{2} e^{6 i e} + i B a^{2} e^{4 i e}\right )}{16 c^{6}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.62622, size = 514, normalized size = 5.65 \begin{align*} -\frac{2 \,{\left (15 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} + 60 i \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 15 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 235 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} - 20 i \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} - 480 i \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 90 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 822 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 84 i \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 904 i \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 158 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 822 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 84 i \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 480 i \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 90 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 235 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 20 i \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 60 i \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 15 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 15 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{15 \, c^{6} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]