Optimal. Leaf size=25 \[ -\frac{i c (a+i a \tan (e+f x))^3}{3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0698879, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ -\frac{i c (a+i a \tan (e+f x))^3}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x)) \, dx &=(a c) \int \sec ^2(e+f x) (a+i a \tan (e+f x))^2 \, dx\\ &=-\frac{(i c) \operatorname{Subst}\left (\int (a+x)^2 \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=-\frac{i c (a+i a \tan (e+f x))^3}{3 f}\\ \end{align*}
Mathematica [B] time = 0.175807, size = 55, normalized size = 2.2 \[ \frac{a^3 c \left (-\tan ^3(e+f x)+3 i \tan ^2(e+f x)-3 \tan ^{-1}(\tan (e+f x))+3 \tan (e+f x)+3 f x\right )}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 37, normalized size = 1.5 \begin{align*}{\frac{{a}^{3}c}{f} \left ( i \left ( \tan \left ( fx+e \right ) \right ) ^{2}-{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{3}}+\tan \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.62986, size = 61, normalized size = 2.44 \begin{align*} -\frac{a^{3} c \tan \left (f x + e\right )^{3} - 3 i \, a^{3} c \tan \left (f x + e\right )^{2} - 3 \, a^{3} c \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.00079, size = 220, normalized size = 8.8 \begin{align*} \frac{24 i \, a^{3} c e^{\left (4 i \, f x + 4 i \, e\right )} + 24 i \, a^{3} c e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{3} c}{3 \,{\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, 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 = 2.16872, size = 119, normalized size = 4.76 \begin{align*} \frac{\frac{8 i a^{3} c e^{- 2 i e} e^{4 i f x}}{f} + \frac{8 i a^{3} c e^{- 4 i e} e^{2 i f x}}{f} + \frac{8 i a^{3} c e^{- 6 i e}}{3 f}}{e^{6 i f x} + 3 e^{- 2 i e} e^{4 i f x} + 3 e^{- 4 i e} e^{2 i f x} + e^{- 6 i e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.56279, size = 112, normalized size = 4.48 \begin{align*} \frac{24 i \, a^{3} c e^{\left (4 i \, f x + 4 i \, e\right )} + 24 i \, a^{3} c e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{3} c}{3 \,{\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, 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]