Optimal. Leaf size=33 \[ \frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.0615918, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3175, 2606} \[ \frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2606
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac{\int \sec (c+d x) \tan ^3(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac{\sec (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0296034, size = 31, normalized size = 0.94 \[ \frac{\frac{\sec ^3(c+d x)}{3 d}-\frac{\sec (c+d x)}{d}}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 29, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{2}d} \left ( - \left ( \cos \left ( dx+c \right ) \right ) ^{-1}+{\frac{1}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972657, size = 38, normalized size = 1.15 \begin{align*} -\frac{3 \, \cos \left (d x + c\right )^{2} - 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 = 1.53299, size = 70, normalized size = 2.12 \begin{align*} -\frac{3 \, \cos \left (d x + c\right )^{2} - 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 = 25.345, size = 309, normalized size = 9.36 \begin{align*} \begin{cases} - \frac{4 \tan ^{6}{\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{12 \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{24 \tan ^{2}{\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{8}{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 ^{3}{\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 = 1.15713, size = 38, normalized size = 1.15 \begin{align*} -\frac{3 \, \cos \left (d x + c\right )^{2} - 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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