Optimal. Leaf size=47 \[ \frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0646025, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3904, 3473, 8} \[ \frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3904
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \tan ^4(e+f x) \, dx\\ &=\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\left (a^2 c^2\right ) \int \tan ^2(e+f x) \, dx\\ &=-\frac{a^2 c^2 \tan (e+f x)}{f}+\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}+\left (a^2 c^2\right ) \int 1 \, dx\\ &=a^2 c^2 x-\frac{a^2 c^2 \tan (e+f x)}{f}+\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.0339771, size = 45, normalized size = 0.96 \[ a^2 c^2 \left (\frac{\tan ^3(e+f x)}{3 f}+\frac{\tan ^{-1}(\tan (e+f x))}{f}-\frac{\tan (e+f x)}{f}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.021, size = 58, normalized size = 1.2 \begin{align*}{\frac{1}{f} \left ( -2\,{c}^{2}{a}^{2}\tan \left ( fx+e \right ) +{c}^{2}{a}^{2} \left ( fx+e \right ) -{c}^{2}{a}^{2} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01444, size = 77, normalized size = 1.64 \begin{align*} \frac{{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{2} + 3 \,{\left (f x + e\right )} a^{2} c^{2} - 6 \, a^{2} c^{2} \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.07713, size = 144, normalized size = 3.06 \begin{align*} \frac{3 \, a^{2} c^{2} f x \cos \left (f x + e\right )^{3} -{\left (4 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} - a^{2} c^{2}\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} c^{2} \left (\int 1\, dx + \int - 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.43435, size = 69, normalized size = 1.47 \begin{align*} \frac{a^{2} c^{2} \tan \left (f x + e\right )^{3} + 3 \,{\left (f x + e\right )} a^{2} c^{2} - 3 \, a^{2} c^{2} \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]