Optimal. Leaf size=18 \[ \frac{\sec ^3(c+d x)}{3 a^2 d} \]
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Rubi [A] time = 0.0421728, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3175, 2606, 30} \[ \frac{\sec ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac{\int \sec ^3(c+d x) \tan (c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{\sec ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0111101, size = 18, normalized size = 1. \[ \frac{\sec ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 17, normalized size = 0.9 \begin{align*}{\frac{1}{3\,{a}^{2}d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952217, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{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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Fricas [A] time = 1.59587, size = 38, normalized size = 2.11 \begin{align*} \frac{1}{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 = 19.495, size = 156, normalized size = 8.67 \begin{align*} \begin{cases} - \frac{6 \tan ^{4}{\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{2}{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{\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.
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Giac [A] time = 1.15784, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{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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