Optimal. Leaf size=183 \[ -\frac{95 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{48 a^3 d}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{163 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{17 \sin ^2(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.385496, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2765, 2977, 2968, 3023, 2751, 2649, 206} \[ -\frac{95 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{48 a^3 d}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{163 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{17 \sin ^2(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac{\int \frac{\sin ^2(c+d x) \left (3 a-\frac{11}{2} a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{\int \frac{\sin (c+d x) \left (17 a^2-\frac{95}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{8 a^4}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{\int \frac{17 a^2 \sin (c+d x)-\frac{95}{4} a^2 \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{8 a^4}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{95 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{48 a^3 d}-\frac{\int \frac{-\frac{95 a^3}{8}+\frac{197}{4} a^3 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{12 a^5}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{95 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{48 a^3 d}+\frac{163 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{32 a^2}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{95 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{48 a^3 d}-\frac{163 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{16 a^2 d}\\ &=-\frac{163 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{95 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{48 a^3 d}\\ \end{align*}
Mathematica [C] time = 0.48224, size = 197, normalized size = 1.08 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (-279 \sin \left (\frac{1}{2} (c+d x)\right )+399 \sin \left (\frac{3}{2} (c+d x)\right )+88 \sin \left (\frac{5}{2} (c+d x)\right )+8 \sin \left (\frac{7}{2} (c+d x)\right )+279 \cos \left (\frac{1}{2} (c+d x)\right )+399 \cos \left (\frac{3}{2} (c+d x)\right )-88 \cos \left (\frac{5}{2} (c+d x)\right )+8 \cos \left (\frac{7}{2} (c+d x)\right )+(978+978 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{96 d (a (\sin (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.787, size = 269, normalized size = 1.5 \begin{align*} -{\frac{1}{ \left ( 96+96\,\sin \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) d} \left ( \sin \left ( dx+c \right ) \left ( 978\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}-768\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{3/2}-128\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{a} \right ) + \left ( -489\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}+384\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{3/2}+64\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{a} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+978\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}-1092\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{3/2}+46\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{a} \right ) \sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{-{\frac{9}{2}}}{\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 \frac{\sin \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88903, size = 976, normalized size = 5.33 \begin{align*} \frac{489 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) - 4 \,{\left (32 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} + 279 \, \cos \left (d x + c\right )^{2} +{\left (32 \, \cos \left (d x + c\right )^{3} + 192 \, \cos \left (d x + c\right )^{2} + 471 \, \cos \left (d x + c\right ) + 12\right )} \sin \left (d x + c\right ) + 459 \, \cos \left (d x + c\right ) - 12\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{192 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\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] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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