Optimal. Leaf size=135 \[ -\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2} \]
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Rubi [A] time = 0.347352, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2565
Rule 14
Rule 2568
Rule 2635
Rule 8
Rule 270
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \cos ^2(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \cos ^2(c+d x) \sin ^3(c+d x)-2 a^2 \cos ^2(c+d x) \sin ^4(c+d x)+a^2 \cos ^2(c+d x) \sin ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a^2}+\frac{\int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a^2}-\frac{2 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a^2}\\ &=\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}-\frac{\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a^2}-\frac{\operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac{\operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}-\frac{\int \cos ^2(c+d x) \, dx}{4 a^2}-\frac{\operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac{\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}-\frac{\int 1 \, dx}{8 a^2}\\ &=-\frac{x}{8 a^2}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [B] time = 3.0872, size = 418, normalized size = 3.1 \[ -\frac{1680 d x \sin \left (\frac{c}{2}\right )-1365 \sin \left (\frac{c}{2}+d x\right )+1365 \sin \left (\frac{3 c}{2}+d x\right )-210 \sin \left (\frac{3 c}{2}+2 d x\right )-210 \sin \left (\frac{5 c}{2}+2 d x\right )-175 \sin \left (\frac{5 c}{2}+3 d x\right )+175 \sin \left (\frac{7 c}{2}+3 d x\right )-210 \sin \left (\frac{7 c}{2}+4 d x\right )-210 \sin \left (\frac{9 c}{2}+4 d x\right )+147 \sin \left (\frac{9 c}{2}+5 d x\right )-147 \sin \left (\frac{11 c}{2}+5 d x\right )+70 \sin \left (\frac{11 c}{2}+6 d x\right )+70 \sin \left (\frac{13 c}{2}+6 d x\right )-15 \sin \left (\frac{13 c}{2}+7 d x\right )+15 \sin \left (\frac{15 c}{2}+7 d x\right )+210 \cos \left (\frac{c}{2}\right ) (8 d x+1)+1365 \cos \left (\frac{c}{2}+d x\right )+1365 \cos \left (\frac{3 c}{2}+d x\right )-210 \cos \left (\frac{3 c}{2}+2 d x\right )+210 \cos \left (\frac{5 c}{2}+2 d x\right )+175 \cos \left (\frac{5 c}{2}+3 d x\right )+175 \cos \left (\frac{7 c}{2}+3 d x\right )-210 \cos \left (\frac{7 c}{2}+4 d x\right )+210 \cos \left (\frac{9 c}{2}+4 d x\right )-147 \cos \left (\frac{9 c}{2}+5 d x\right )-147 \cos \left (\frac{11 c}{2}+5 d x\right )+70 \cos \left (\frac{11 c}{2}+6 d x\right )-70 \cos \left (\frac{13 c}{2}+6 d x\right )+15 \cos \left (\frac{13 c}{2}+7 d x\right )+15 \cos \left (\frac{15 c}{2}+7 d x\right )-210 \sin \left (\frac{c}{2}\right )}{13440 a^2 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.098, size = 415, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.56853, size = 562, normalized size = 4.16 \begin{align*} \frac{\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1232 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{700 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{2016 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{3395 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{1120 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{7280 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{3395 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{1680 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac{700 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac{105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 176}{a^{2} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac{a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac{105 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10794, size = 220, normalized size = 1.63 \begin{align*} -\frac{120 \, \cos \left (d x + c\right )^{7} - 504 \, \cos \left (d x + c\right )^{5} + 560 \, \cos \left (d x + c\right )^{3} + 105 \, d x + 35 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, a^{2} 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 = 1.32163, size = 242, normalized size = 1.79 \begin{align*} -\frac{\frac{105 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 700 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1680 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 3395 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 7280 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3395 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 700 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1232 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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