Optimal. Leaf size=106 \[ -\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{19 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 d}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d} \]
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Rubi [A] time = 0.218477, antiderivative size = 106, 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 = {2762, 2980, 2773, 206} \[ -\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{19 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{4 d}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d} \]
Antiderivative was successfully verified.
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Rule 2762
Rule 2980
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \csc ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}-\frac{1}{2} a \int \csc ^2(c+d x) \left (-\frac{9 a}{2}-\frac{5}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}+\frac{1}{8} \left (19 a^2\right ) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}-\frac{\left (19 a^3\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 )}{4 d}\\ &=-\frac{19 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 d}-\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}\\ \end{align*}
Mathematica [B] time = 0.749989, size = 252, normalized size = 2.38 \[ \frac{a^2 \csc ^7\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (-14 \sin \left (\frac{1}{2} (c+d x)\right )-22 \sin \left (\frac{3}{2} (c+d x)\right )+14 \cos \left (\frac{1}{2} (c+d x)\right )-22 \cos \left (\frac{3}{2} (c+d x)\right )+19 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-19 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-19 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+19 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{4 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.623, size = 126, 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 ) }\sqrt{a} \left ( 19\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}+13\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{3/2}-11\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}\sqrt{a} \right ){\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{5}{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 = 1.53422, size = 911, normalized size = 8.59 \begin{align*} \frac{19 \,{\left (a^{2} \cos \left (d x + c\right )^{3} + a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right ) - a^{2} +{\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )\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 (11 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) - 9 \, a^{2} +{\left (11 \, a^{2} \cos \left (d x + c\right ) + 9 \, a^{2}\right )} \sin \left (d x + c\right )\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 [B] time = 2.60091, size = 740, normalized size = 6.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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