Optimal. Leaf size=82 \[ \frac{a^4 c^3 \tan ^5(e+f x)}{5 f}+\frac{2 a^4 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^4 c^3 \tan (e+f x)}{f}+\frac{i a^4 c^3 \sec ^6(e+f x)}{6 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0963688, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3486, 3767} \[ \frac{a^4 c^3 \tan ^5(e+f x)}{5 f}+\frac{2 a^4 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^4 c^3 \tan (e+f x)}{f}+\frac{i a^4 c^3 \sec ^6(e+f x)}{6 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3522
Rule 3486
Rule 3767
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx &=\left (a^3 c^3\right ) \int \sec ^6(e+f x) (a+i a \tan (e+f x)) \, dx\\ &=\frac{i a^4 c^3 \sec ^6(e+f x)}{6 f}+\left (a^4 c^3\right ) \int \sec ^6(e+f x) \, dx\\ &=\frac{i a^4 c^3 \sec ^6(e+f x)}{6 f}-\frac{\left (a^4 c^3\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\frac{i a^4 c^3 \sec ^6(e+f x)}{6 f}+\frac{a^4 c^3 \tan (e+f x)}{f}+\frac{2 a^4 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^4 c^3 \tan ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 3.70786, size = 63, normalized size = 0.77 \[ \frac{a^4 c^3 \sec (e) \sec ^6(e+f x) (15 \sin (e+2 f x)+6 \sin (3 e+4 f x)+\sin (5 e+6 f x)-10 \sin (e)+10 i \cos (e))}{60 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 71, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}{c}^{3}}{f} \left ( \tan \left ( fx+e \right ) +{\frac{i}{6}} \left ( \tan \left ( fx+e \right ) \right ) ^{6}+{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{5}}{5}}+{\frac{i}{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{4}+{\frac{2\, \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{3}}+{\frac{i}{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.64739, size = 135, normalized size = 1.65 \begin{align*} -\frac{-10 i \, a^{4} c^{3} \tan \left (f x + e\right )^{6} - 12 \, a^{4} c^{3} \tan \left (f x + e\right )^{5} - 30 i \, a^{4} c^{3} \tan \left (f x + e\right )^{4} - 40 \, a^{4} c^{3} \tan \left (f x + e\right )^{3} - 30 i \, a^{4} c^{3} \tan \left (f x + e\right )^{2} - 60 \, a^{4} c^{3} \tan \left (f x + e\right )}{60 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.2331, size = 396, normalized size = 4.83 \begin{align*} \frac{320 i \, a^{4} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 240 i \, a^{4} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, a^{4} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{4} c^{3}}{15 \,{\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 11.4866, size = 206, normalized size = 2.51 \begin{align*} \frac{\frac{64 i a^{4} c^{3} e^{- 6 i e} e^{6 i f x}}{3 f} + \frac{16 i a^{4} c^{3} e^{- 8 i e} e^{4 i f x}}{f} + \frac{32 i a^{4} c^{3} e^{- 10 i e} e^{2 i f x}}{5 f} + \frac{16 i a^{4} c^{3} e^{- 12 i e}}{15 f}}{e^{12 i f x} + 6 e^{- 2 i e} e^{10 i f x} + 15 e^{- 4 i e} e^{8 i f x} + 20 e^{- 6 i e} e^{6 i f x} + 15 e^{- 8 i e} e^{4 i f x} + 6 e^{- 10 i e} e^{2 i f x} + e^{- 12 i e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.95439, size = 197, normalized size = 2.4 \begin{align*} \frac{320 i \, a^{4} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 240 i \, a^{4} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, a^{4} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{4} c^{3}}{15 \,{\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]