Optimal. Leaf size=52 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{b^{3/2} d \sqrt{a+b}}-\frac{\cos (c+d x)}{b d} \]
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Rubi [A] time = 0.0696408, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3186, 388, 208} \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{b^{3/2} d \sqrt{a+b}}-\frac{\cos (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\cos (c+d x)}{b d}+\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{b d}\\ &=\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b} d}-\frac{\cos (c+d x)}{b d}\\ \end{align*}
Mathematica [C] time = 0.257717, size = 125, normalized size = 2.4 \[ -\frac{\sqrt{b} \sqrt{-a-b} \cos (c+d x)+a \tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a-b}}\right )+a \tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a-b}}\right )}{b^{3/2} d \sqrt{-a-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 45, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{\frac{\cos \left ( dx+c \right ) }{b}}+{\frac{a}{b}{\it Artanh} \left ({b\cos \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79428, size = 381, normalized size = 7.33 \begin{align*} \left [\frac{\sqrt{a b + b^{2}} a \log \left (\frac{b \cos \left (d x + c\right )^{2} + 2 \, \sqrt{a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) - 2 \,{\left (a b + b^{2}\right )} \cos \left (d x + c\right )}{2 \,{\left (a b^{2} + b^{3}\right )} d}, -\frac{\sqrt{-a b - b^{2}} a \arctan \left (\frac{\sqrt{-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right ) +{\left (a b + b^{2}\right )} \cos \left (d x + c\right )}{{\left (a b^{2} + b^{3}\right )} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16063, size = 77, normalized size = 1.48 \begin{align*} -\frac{a \arctan \left (\frac{b \cos \left (d x + c\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} b d} - \frac{\cos \left (d x + c\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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