Optimal. Leaf size=105 \[ \frac{7 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.128016, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2758, 2751, 2649, 206} \[ \frac{7 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2758
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac{\cos (c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac{\int \frac{-\frac{3 a}{2}+2 a \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{\cos (c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{2 \cos (c+d x)}{a d \sqrt{a+a \sin (c+d x)}}-\frac{7 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{4 a}\\ &=-\frac{\cos (c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{2 \cos (c+d x)}{a d \sqrt{a+a \sin (c+d x)}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{2 a d}\\ &=\frac{7 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\cos (c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{2 \cos (c+d x)}{a d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.271054, size = 134, normalized size = 1.28 \[ -\frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (-3 \sin \left (\frac{1}{2} (c+d x)\right )+2 \sin \left (\frac{3}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{3}{2} (c+d x)\right )+(7+7 i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{2 d (a (\sin (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.549, size = 143, normalized size = 1.4 \begin{align*}{\frac{1}{4\,d\cos \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \left ( 7\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) a-8\,\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{a} \right ) +7\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) a-10\,\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{a} \right ) \sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76778, size = 725, normalized size = 6.9 \begin{align*} \frac{7 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \,{\left (4 \, \cos \left (d x + c\right )^{2} +{\left (4 \, \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 5 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{8 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d -{\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\left (c + d x \right )}}{\left (a \left (\sin{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.30412, size = 486, normalized size = 4.63 \begin{align*} \frac{\frac{4 \,{\left (\frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )} - \frac{1}{a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}\right )}}{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}} - \frac{7 \, \sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} + \sqrt{a}\right )}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )} + \frac{2 \,{\left (3 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{3} +{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} \sqrt{a} -{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )} a + a^{\frac{3}{2}}\right )}}{{\left ({\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} + 2 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )} \sqrt{a} - a\right )}^{2} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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