Optimal. Leaf size=18 \[ \frac{\tan ^3(x)}{3 a}+\frac{\tan (x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0494037, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 3767} \[ \frac{\tan ^3(x)}{3 a}+\frac{\tan (x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3175
Rule 3767
Rubi steps
\begin{align*} \int \frac{\sec ^2(x)}{a-a \sin ^2(x)} \, dx &=\frac{\int \sec ^4(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (x)\right )}{a}\\ &=\frac{\tan (x)}{a}+\frac{\tan ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0037023, size = 21, normalized size = 1.17 \[ \frac{\frac{2 \tan (x)}{3}+\frac{1}{3} \tan (x) \sec ^2(x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 14, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ({\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3}}+\tan \left ( x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.967517, size = 19, normalized size = 1.06 \begin{align*} \frac{\tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80905, size = 57, normalized size = 3.17 \begin{align*} \frac{{\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{3 \, a \cos \left (x\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*} - \frac{\int \frac{\sec ^{2}{\left (x \right )}}{\sin ^{2}{\left (x \right )} - 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16791, size = 19, normalized size = 1.06 \begin{align*} \frac{\tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]