Optimal. Leaf size=18 \[ \frac{\tan ^3(c+d x)}{3 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0674126, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2607, 30} \[ \frac{\tan ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3175
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac{\int \sec ^2(c+d x) \tan ^2(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac{\tan ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0176636, size = 18, normalized size = 1. \[ \frac{\tan ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 17, normalized size = 0.9 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.13808, size = 22, normalized size = 1.22 \begin{align*} \frac{\tan \left (d x + c\right )^{3}}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53549, size = 85, normalized size = 4.72 \begin{align*} -\frac{{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 31.7744, size = 94, normalized size = 5.22 \begin{align*} \begin{cases} - \frac{8 \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 3 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{2}{\left (c \right )}}{\left (- a \sin ^{2}{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1213, size = 22, normalized size = 1.22 \begin{align*} \frac{\tan \left (d x + c\right )^{3}}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]