Optimal. Leaf size=131 \[ \frac{13 a^2 \cos (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
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Rubi [A] time = 0.518467, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2874, 2975, 2981, 2773, 206} \[ \frac{13 a^2 \cos (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 2874
Rule 2975
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{\int \csc ^3(c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{5/2} \, dx}{a^2}\\ &=-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{\int \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \left (\frac{3 a^2}{2}-\frac{5}{2} a^2 \sin (c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \left (-\frac{a^3}{4}-\frac{13}{4} a^3 \sin (c+d x)\right ) \, dx}{2 a^2}\\ &=\frac{13 a^2 \cos (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}-\frac{1}{8} a \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{13 a^2 \cos (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}\\ &=\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}+\frac{13 a^2 \cos (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a \cot (c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}\\ \end{align*}
Mathematica [B] time = 0.659407, size = 271, normalized size = 2.07 \[ -\frac{a \csc ^7\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (22 \sin \left (\frac{1}{2} (c+d x)\right )+22 \sin \left (\frac{3}{2} (c+d x)\right )-8 \sin \left (\frac{5}{2} (c+d x)\right )-22 \cos \left (\frac{1}{2} (c+d x)\right )+22 \cos \left (\frac{3}{2} (c+d x)\right )+8 \cos \left (\frac{5}{2} (c+d x)\right )+\cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-\log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-\cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+\log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{4 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.987, size = 151, normalized size = 1.2 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 8\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{3/2}+{\it Artanh} \left ({\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{\frac{1}{\sqrt{a}}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}+7\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}\sqrt{a}-9\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{3/2} \right ){\frac{1}{\sqrt{a}}}{\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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{2} \csc \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74725, size = 936, normalized size = 7.15 \begin{align*} \frac{{\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} - 9 \, a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \,{\left (8 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )^{2} - 6 \, a \cos \left (d x + c\right ) -{\left (8 \, a \cos \left (d x + c\right )^{2} - 7 \, a \cos \left (d x + c\right ) - 13 \, a\right )} \sin \left (d x + c\right ) - 13 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{16 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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