Optimal. Leaf size=235 \[ \frac{\left (3 a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac{4 a \left (a^2-b^2\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left (-6 a^2 b^2+5 a^4+b^4\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac{2 a \left (-4 a^2 b^2+3 a^4+b^4\right ) \sin (c+d x)}{b^7 d}+\frac{a^3 \left (a^2-b^2\right )^2}{b^8 d (a+b \sin (c+d x))}+\frac{a^2 \left (-10 a^2 b^2+7 a^4+3 b^4\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac{2 a \sin ^5(c+d x)}{5 b^3 d}+\frac{\sin ^6(c+d x)}{6 b^2 d} \]
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Rubi [A] time = 0.28051, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2837, 12, 948} \[ \frac{\left (3 a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac{4 a \left (a^2-b^2\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left (-6 a^2 b^2+5 a^4+b^4\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac{2 a \left (-4 a^2 b^2+3 a^4+b^4\right ) \sin (c+d x)}{b^7 d}+\frac{a^3 \left (a^2-b^2\right )^2}{b^8 d (a+b \sin (c+d x))}+\frac{a^2 \left (-10 a^2 b^2+7 a^4+3 b^4\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac{2 a \sin ^5(c+d x)}{5 b^3 d}+\frac{\sin ^6(c+d x)}{6 b^2 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 948
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (b^2-x^2\right )^2}{b^3 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (b^2-x^2\right )^2}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 a \left (3 a^4-4 a^2 b^2+b^4\right )+\left (5 a^4-6 a^2 b^2+b^4\right ) x-4 a \left (a^2-b^2\right ) x^2+\left (3 a^2-2 b^2\right ) x^3-2 a x^4+x^5-\frac{a^3 \left (a^2-b^2\right )^2}{(a+x)^2}+\frac{7 a^6-10 a^4 b^2+3 a^2 b^4}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=\frac{a^2 \left (7 a^4-10 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac{2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \sin (c+d x)}{b^7 d}+\frac{\left (5 a^4-6 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac{4 a \left (a^2-b^2\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left (3 a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac{2 a \sin ^5(c+d x)}{5 b^3 d}+\frac{\sin ^6(c+d x)}{6 b^2 d}+\frac{a^3 \left (a^2-b^2\right )^2}{b^8 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 2.12213, size = 264, normalized size = 1.12 \[ \frac{3 b^5 \left (7 a^2-10 b^2\right ) \sin ^5(c+d x)+\left (50 a b^6-35 a^3 b^4\right ) \sin ^4(c+d x)+10 b^3 \left (-10 a^2 b^2+7 a^4+3 b^4\right ) \sin ^3(c+d x)-30 a b^2 \left (-10 a^2 b^2+7 a^4+3 b^4\right ) \sin ^2(c+d x)+60 a^2 b \left (a^2-b^2\right ) \sin (c+d x) \left (\left (7 a^2-3 b^2\right ) \log (a+b \sin (c+d x))-6 a^2+2 b^2\right )+60 a^3 \left (a^2-b^2\right ) \left (\left (7 a^2-3 b^2\right ) \log (a+b \sin (c+d x))+a^2-b^2\right )-14 a b^6 \sin ^6(c+d x)+10 b^7 \sin ^7(c+d x)}{60 b^8 d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 342, normalized size = 1.5 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6\,{b}^{2}d}}-{\frac{2\,a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,{b}^{3}d}}+{\frac{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{2}}{4\,d{b}^{4}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2\,{b}^{2}d}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}{a}^{3}}{3\,d{b}^{5}}}+{\frac{4\,a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,{b}^{3}d}}+{\frac{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{2\,d{b}^{6}}}-3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{d{b}^{4}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{b}^{2}d}}-6\,{\frac{{a}^{5}\sin \left ( dx+c \right ) }{d{b}^{7}}}+8\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d{b}^{5}}}-2\,{\frac{a\sin \left ( dx+c \right ) }{{b}^{3}d}}+7\,{\frac{{a}^{6}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{8}}}-10\,{\frac{{a}^{4}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{6}}}+3\,{\frac{{a}^{2}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{4}}}+{\frac{{a}^{7}}{d{b}^{8} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-2\,{\frac{{a}^{5}}{d{b}^{6} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+{\frac{{a}^{3}}{d{b}^{4} \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996216, size = 294, normalized size = 1.25 \begin{align*} \frac{\frac{60 \,{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )}}{b^{9} \sin \left (d x + c\right ) + a b^{8}} + \frac{10 \, b^{5} \sin \left (d x + c\right )^{6} - 24 \, a b^{4} \sin \left (d x + c\right )^{5} + 15 \,{\left (3 \, a^{2} b^{3} - 2 \, b^{5}\right )} \sin \left (d x + c\right )^{4} - 80 \,{\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )^{3} + 30 \,{\left (5 \, a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{2} - 120 \,{\left (3 \, a^{5} - 4 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )}{b^{7}} + \frac{60 \,{\left (7 \, a^{6} - 10 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20311, size = 671, normalized size = 2.86 \begin{align*} \frac{112 \, a b^{6} \cos \left (d x + c\right )^{6} + 480 \, a^{7} - 3240 \, a^{5} b^{2} + 3185 \, a^{3} b^{4} - 487 \, a b^{6} - 8 \,{\left (35 \, a^{3} b^{4} - 8 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} + 16 \,{\left (105 \, a^{5} b^{2} - 115 \, a^{3} b^{4} + 16 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 480 \,{\left (7 \, a^{7} - 10 \, a^{5} b^{2} + 3 \, a^{3} b^{4} +{\left (7 \, a^{6} b - 10 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left (80 \, b^{7} \cos \left (d x + c\right )^{6} - 168 \, a^{2} b^{5} \cos \left (d x + c\right )^{4} + 2880 \, a^{6} b - 3800 \, a^{4} b^{3} + 1007 \, a^{2} b^{5} - 25 \, b^{7} + 16 \,{\left (35 \, a^{4} b^{3} - 29 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \,{\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} 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 [A] time = 1.18885, size = 405, normalized size = 1.72 \begin{align*} \frac{\frac{60 \,{\left (7 \, a^{6} - 10 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}} - \frac{60 \,{\left (7 \, a^{6} b \sin \left (d x + c\right ) - 10 \, a^{4} b^{3} \sin \left (d x + c\right ) + 3 \, a^{2} b^{5} \sin \left (d x + c\right ) + 6 \, a^{7} - 8 \, a^{5} b^{2} + 2 \, a^{3} b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{8}} + \frac{10 \, b^{10} \sin \left (d x + c\right )^{6} - 24 \, a b^{9} \sin \left (d x + c\right )^{5} + 45 \, a^{2} b^{8} \sin \left (d x + c\right )^{4} - 30 \, b^{10} \sin \left (d x + c\right )^{4} - 80 \, a^{3} b^{7} \sin \left (d x + c\right )^{3} + 80 \, a b^{9} \sin \left (d x + c\right )^{3} + 150 \, a^{4} b^{6} \sin \left (d x + c\right )^{2} - 180 \, a^{2} b^{8} \sin \left (d x + c\right )^{2} + 30 \, b^{10} \sin \left (d x + c\right )^{2} - 360 \, a^{5} b^{5} \sin \left (d x + c\right ) + 480 \, a^{3} b^{7} \sin \left (d x + c\right ) - 120 \, a b^{9} \sin \left (d x + c\right )}{b^{12}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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