Optimal. Leaf size=105 \[ \frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{4 \cos (c+d x)}{a^3 d}-\frac{3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac{13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{13 x}{8 a^3} \]
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Rubi [A] time = 0.218723, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2869, 2757, 2635, 8, 2633} \[ \frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{4 \cos (c+d x)}{a^3 d}-\frac{3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac{13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{13 x}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 2869
Rule 2757
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \sin ^2(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (a^3 \sin ^2(c+d x)-3 a^3 \sin ^3(c+d x)+3 a^3 \sin ^4(c+d x)-a^3 \sin ^5(c+d x)\right ) \, dx}{a^6}\\ &=\frac{\int \sin ^2(c+d x) \, dx}{a^3}-\frac{\int \sin ^5(c+d x) \, dx}{a^3}-\frac{3 \int \sin ^3(c+d x) \, dx}{a^3}+\frac{3 \int \sin ^4(c+d x) \, dx}{a^3}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac{\int 1 \, dx}{2 a^3}+\frac{9 \int \sin ^2(c+d x) \, dx}{4 a^3}+\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}+\frac{3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=\frac{x}{2 a^3}+\frac{4 \cos (c+d x)}{a^3 d}-\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac{9 \int 1 \, dx}{8 a^3}\\ &=\frac{13 x}{8 a^3}+\frac{4 \cos (c+d x)}{a^3 d}-\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}\\ \end{align*}
Mathematica [B] time = 1.75688, size = 310, normalized size = 2.95 \[ \frac{1560 d x \sin \left (\frac{c}{2}\right )-1380 \sin \left (\frac{c}{2}+d x\right )+1380 \sin \left (\frac{3 c}{2}+d x\right )-480 \sin \left (\frac{3 c}{2}+2 d x\right )-480 \sin \left (\frac{5 c}{2}+2 d x\right )+170 \sin \left (\frac{5 c}{2}+3 d x\right )-170 \sin \left (\frac{7 c}{2}+3 d x\right )+45 \sin \left (\frac{7 c}{2}+4 d x\right )+45 \sin \left (\frac{9 c}{2}+4 d x\right )-6 \sin \left (\frac{9 c}{2}+5 d x\right )+6 \sin \left (\frac{11 c}{2}+5 d x\right )+1560 d x \cos \left (\frac{c}{2}\right )+1380 \cos \left (\frac{c}{2}+d x\right )+1380 \cos \left (\frac{3 c}{2}+d x\right )-480 \cos \left (\frac{3 c}{2}+2 d x\right )+480 \cos \left (\frac{5 c}{2}+2 d x\right )-170 \cos \left (\frac{5 c}{2}+3 d x\right )-170 \cos \left (\frac{7 c}{2}+3 d x\right )+45 \cos \left (\frac{7 c}{2}+4 d x\right )-45 \cos \left (\frac{9 c}{2}+4 d x\right )+6 \cos \left (\frac{9 c}{2}+5 d x\right )+6 \cos \left (\frac{11 c}{2}+5 d x\right )+10 \sin \left (\frac{c}{2}\right )}{960 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.107, size = 279, normalized size = 2.7 \begin{align*}{\frac{13}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{25}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+12\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{5}}}+{\frac{116}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{25}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{76}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{13}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{76}{15\,d{a}^{3}} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{13}{4\,d{a}^{3}}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54764, size = 392, normalized size = 3.73 \begin{align*} -\frac{\frac{\frac{195 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1520 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{2320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{750 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{195 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 304}{a^{3} + \frac{5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac{195 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12579, size = 189, normalized size = 1.8 \begin{align*} \frac{24 \, \cos \left (d x + c\right )^{5} - 200 \, \cos \left (d x + c\right )^{3} + 195 \, d x + 15 \,{\left (6 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{120 \, a^{3} d} \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 = 1.26507, size = 171, normalized size = 1.63 \begin{align*} \frac{\frac{195 \,{\left (d x + c\right )}}{a^{3}} + \frac{2 \,{\left (195 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 750 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2320 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 750 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1520 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 195 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 304\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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