Optimal. Leaf size=98 \[ -\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d} \]
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Rubi [A] time = 0.192339, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2763, 2981, 2773, 206} \[ -\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2763
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac{2 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\frac{2}{3} \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{3 a^2}{2}+\frac{7}{2} a^2 \sin (c+d x)\right ) \, dx\\ &=-\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+a^2 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{\left (2 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 )}{d}\\ &=-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.37231, size = 143, normalized size = 1.46 \[ -\frac{(a (\sin (c+d x)+1))^{5/2} \left (-15 \sin \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{3}{2} (c+d x)\right )+15 \cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )+3 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.615, size = 103, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ) a}{3\,d\cos \left ( dx+c \right ) }\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 3\,{a}^{3/2}{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( dx+c \right ) }}{\sqrt{a}}} \right ) - \left ( a-a\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}+9\,a\sqrt{a-a\sin \left ( dx+c \right ) } \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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44608, size = 733, normalized size = 7.48 \begin{align*} \frac{3 \,{\left (a^{2} \cos \left (d x + c\right ) + a^{2} \sin \left (d x + c\right ) + a^{2}\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 (a^{2} \cos \left (d x + c\right )^{2} + 8 \, a^{2} \cos \left (d x + c\right ) + 7 \, a^{2} +{\left (a^{2} \cos \left (d x + c\right ) - 7 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{6 \,{\left (d \cos \left (d x + c\right ) + d \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 = 2.72137, size = 429, normalized size = 4.38 \begin{align*} \frac{\frac{6 \, a^{3} \arctan \left (-\frac{\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 ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{\sqrt{-a}} - 3 \, a^{\frac{5}{2}} \log \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 |}\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) - \frac{{\left (6 \, a^{3} \arctan \left (\frac{\sqrt{2} \sqrt{a} + \sqrt{a}}{\sqrt{-a}}\right ) - 3 \, \sqrt{-a} a^{\frac{5}{2}} \log \left (\sqrt{2} \sqrt{a} + \sqrt{a}\right ) - 14 \, \sqrt{2} \sqrt{-a} a^{\frac{5}{2}}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{\sqrt{-a}} - \frac{4 \,{\left (4 \, a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) -{\left (3 \, a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) +{\left (4 \, a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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