Optimal. Leaf size=121 \[ \frac{11 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d}+\frac{5 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
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Rubi [A] time = 0.308013, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2716, 2976, 2981, 2773, 206} \[ \frac{11 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d}+\frac{5 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
Antiderivative was successfully verified.
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Rule 2716
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}+\frac{\int \csc (c+d x) \left (\frac{3 a}{2}-\frac{5}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{a}\\ &=\frac{5 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}+\frac{2 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{9 a^2}{4}-\frac{11}{4} a^2 \sin (c+d x)\right ) \, dx}{3 a}\\ &=\frac{11 a^2 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{5 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}+\frac{1}{2} (3 a) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{11 a^2 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{5 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{\left (3 a^2\right ) \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 )}{d}\\ &=-\frac{3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{11 a^2 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{5 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}\\ \end{align*}
Mathematica [A] time = 0.767613, size = 233, normalized size = 1.93 \[ -\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-14 \sin \left (\frac{1}{2} (c+d x)\right )-9 \sin \left (\frac{3}{2} (c+d x)\right )-\sin \left (\frac{5}{2} (c+d x)\right )+14 \cos \left (\frac{1}{2} (c+d x)\right )-9 \cos \left (\frac{3}{2} (c+d x)\right )+\cos \left (\frac{5}{2} (c+d x)\right )+9 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-9 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{3 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc \left (\frac{1}{4} (c+d x)\right )-\sec \left (\frac{1}{4} (c+d x)\right )\right ) \left (\csc \left (\frac{1}{4} (c+d x)\right )+\sec \left (\frac{1}{4} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.872, size = 144, normalized size = 1.2 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{3\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( \sin \left ( dx+c \right ) \left ( 2\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{a}-12\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{3/2}+9\,{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( dx+c \right ) }}{\sqrt{a}}} \right ){a}^{2} \right ) +3\,\sqrt{a-a\sin \left ( dx+c \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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73617, size = 833, normalized size = 6.88 \begin{align*} \frac{9 \,{\left (a \cos \left (d x + c\right )^{2} -{\left (a \cos \left (d x + c\right ) + 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 (2 \, a \cos \left (d x + c\right )^{3} - 8 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) -{\left (2 \, a \cos \left (d x + c\right )^{2} + 10 \, a \cos \left (d x + c\right ) + 11 \, a\right )} \sin \left (d x + c\right ) + 11 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{12 \,{\left (d \cos \left (d x + c\right )^{2} -{\left (d \cos \left (d x + c\right ) + 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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