Optimal. Leaf size=32 \[ \frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.0245773, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3175, 3767} \[ \frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac{\int \sec ^4(c+d x) \, dx}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a^2 d}\\ &=\frac{\tan (c+d x)}{a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0382637, size = 26, normalized size = 0.81 \[ \frac{\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 25, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}d} \left ({\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}+\tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956355, size = 34, normalized size = 1.06 \begin{align*} \frac{\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57837, size = 86, normalized size = 2.69 \begin{align*} \frac{{\left (2 \, \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.
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Sympy [A] time = 13.4509, size = 238, normalized size = 7.44 \begin{align*} \begin{cases} - \frac{6 \tan ^{5}{\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} + \frac{4 \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} - \frac{6 \tan{\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}{\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.
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Giac [A] time = 1.129, size = 34, normalized size = 1.06 \begin{align*} \frac{\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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