Optimal. Leaf size=199 \[ -\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{34 a^2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{14 a^2 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d}+\frac{16 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}+\frac{388 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d} \]
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Rubi [A] time = 0.710798, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {2881, 2763, 21, 2770, 2759, 2751, 2646, 3046, 2976, 2981, 2773, 206} \[ -\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{34 a^2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{14 a^2 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d}+\frac{16 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}+\frac{388 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d} \]
Antiderivative was successfully verified.
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Rule 2881
Rule 2763
Rule 21
Rule 2770
Rule 2759
Rule 2751
Rule 2646
Rule 3046
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc (c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x)}{9 d \sqrt{a+a \sin (c+d x)}}+\frac{4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{2}{9} \int \frac{\sin ^3(c+d x) \left (\frac{17 a^2}{2}+\frac{17}{2} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx+\frac{2 \int \csc (c+d x) \left (\frac{5 a}{2}-3 a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{5 a}\\ &=-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x)}{9 d \sqrt{a+a \sin (c+d x)}}+\frac{4 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}+\frac{4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{4 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{15 a^2}{4}-\frac{9}{4} a^2 \sin (c+d x)\right ) \, dx}{15 a}+\frac{1}{9} (17 a) \int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{6 a^2 \cos (c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{34 a^2 \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x)}{9 d \sqrt{a+a \sin (c+d x)}}+\frac{4 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}+\frac{4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+a \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx+\frac{1}{21} (34 a) \int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{6 a^2 \cos (c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{34 a^2 \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x)}{9 d \sqrt{a+a \sin (c+d x)}}+\frac{4 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{5 d}+\frac{16 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{105 d}+\frac{68}{105} \int \left (\frac{3 a}{2}-a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx-\frac{\left (2 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{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{6 a^2 \cos (c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{34 a^2 \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x)}{9 d \sqrt{a+a \sin (c+d x)}}+\frac{388 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{315 d}+\frac{16 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{105 d}+\frac{1}{45} (34 a) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}-\frac{14 a^2 \cos (c+d x)}{45 d \sqrt{a+a \sin (c+d x)}}-\frac{34 a^2 \cos (c+d x) \sin ^3(c+d x)}{63 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x) \sin ^4(c+d x)}{9 d \sqrt{a+a \sin (c+d x)}}+\frac{388 a \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{315 d}+\frac{16 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{105 d}\\ \end{align*}
Mathematica [A] time = 0.48403, size = 219, normalized size = 1.1 \[ \frac{(a (\sin (c+d x)+1))^{3/2} \left (-1260 \sin \left (\frac{1}{2} (c+d x)\right )+1470 \sin \left (\frac{3}{2} (c+d x)\right )+126 \sin \left (\frac{5}{2} (c+d x)\right )+135 \sin \left (\frac{7}{2} (c+d x)\right )+35 \sin \left (\frac{9}{2} (c+d x)\right )+1260 \cos \left (\frac{1}{2} (c+d x)\right )+1470 \cos \left (\frac{3}{2} (c+d x)\right )-126 \cos \left (\frac{5}{2} (c+d x)\right )+135 \cos \left (\frac{7}{2} (c+d x)\right )-35 \cos \left (\frac{9}{2} (c+d x)\right )-2520 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+2520 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{2520 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.864, size = 159, normalized size = 0.8 \begin{align*} -{\frac{2+2\,\sin \left ( dx+c \right ) }{315\,{a}^{3}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 315\,{a}^{9/2}{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( dx+c \right ) }}{\sqrt{a}}} \right ) +35\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{9/2}-225\,a \left ( a-a\sin \left ( dx+c \right ) \right ) ^{7/2}+441\,{a}^{2} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}-105\,{a}^{3} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}-315\,{a}^{4}\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{3}{2}} \cos \left (d x + c\right )^{4} \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 [A] time = 1.25361, size = 903, normalized size = 4.54 \begin{align*} \frac{315 \,{\left (a \cos \left (d x + c\right ) + a \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 (35 \, a \cos \left (d x + c\right )^{5} - 50 \, a \cos \left (d x + c\right )^{4} - 46 \, a \cos \left (d x + c\right )^{3} - 118 \, a \cos \left (d x + c\right )^{2} - 158 \, a \cos \left (d x + c\right ) -{\left (35 \, a \cos \left (d x + c\right )^{4} + 85 \, a \cos \left (d x + c\right )^{3} + 39 \, a \cos \left (d x + c\right )^{2} + 157 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{630 \,{\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 [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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