Optimal. Leaf size=38 \[ \frac{\tan ^3(e+f x)}{3 a^2 c^2 f}+\frac{\tan (e+f x)}{a^2 c^2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0632528, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2736, 3767} \[ \frac{\tan ^3(e+f x)}{3 a^2 c^2 f}+\frac{\tan (e+f x)}{a^2 c^2 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2736
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx &=\frac{\int \sec ^4(e+f x) \, dx}{a^2 c^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{a^2 c^2 f}\\ &=\frac{\tan (e+f x)}{a^2 c^2 f}+\frac{\tan ^3(e+f x)}{3 a^2 c^2 f}\\ \end{align*}
Mathematica [A] time = 0.0537988, size = 29, normalized size = 0.76 \[ \frac{\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)}{a^2 c^2 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{2} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.43591, size = 38, normalized size = 1. \begin{align*} \frac{\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )}{3 \, a^{2} c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.20854, size = 92, normalized size = 2.42 \begin{align*} \frac{{\left (2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sin \left (f x + e\right )}{3 \, a^{2} c^{2} f \cos \left (f x + e\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 13.2746, size = 286, normalized size = 7.53 \begin{align*} \begin{cases} - \frac{6 \tan ^{5}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 3 a^{2} c^{2} f} + \frac{4 \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 3 a^{2} c^{2} f} - \frac{6 \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{3 a^{2} c^{2} f \tan ^{6}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 9 a^{2} c^{2} f \tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 9 a^{2} c^{2} f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 3 a^{2} c^{2} f} & \text{for}\: f \neq 0 \\\frac{x}{\left (a \sin{\left (e \right )} + a\right )^{2} \left (- c \sin{\left (e \right )} + c\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.23937, size = 41, normalized size = 1.08 \begin{align*} \frac{\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )}{3 \, a^{2} c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]