Optimal. Leaf size=97 \[ -\frac {3 \cos (c+d x)}{a^3 d}+\frac {\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {19 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac {2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac {11 x}{2 a^3} \]
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Rubi [A] time = 0.26, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2874, 2966, 2638, 2635, 8, 2650, 2648} \[ -\frac {3 \cos (c+d x)}{a^3 d}+\frac {\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {19 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac {2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac {11 x}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 2648
Rule 2650
Rule 2874
Rule 2966
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \frac {\sin ^3(c+d x) (a-a \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx}{a^2}\\ &=\frac {\int \left (-\frac {5}{a}+\frac {3 \sin (c+d x)}{a}-\frac {\sin ^2(c+d x)}{a}-\frac {2}{a (1+\sin (c+d x))^2}+\frac {7}{a (1+\sin (c+d x))}\right ) \, dx}{a^2}\\ &=-\frac {5 x}{a^3}-\frac {\int \sin ^2(c+d x) \, dx}{a^3}-\frac {2 \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}+\frac {3 \int \sin (c+d x) \, dx}{a^3}+\frac {7 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=-\frac {5 x}{a^3}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}-\frac {7 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {\int 1 \, dx}{2 a^3}-\frac {2 \int \frac {1}{1+\sin (c+d x)} \, dx}{3 a^3}\\ &=-\frac {11 x}{2 a^3}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}-\frac {19 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 1.09, size = 197, normalized size = 2.03 \[ -\frac {1980 d x \sin \left (c+\frac {d x}{2}\right )+660 d x \sin \left (c+\frac {3 d x}{2}\right )+498 \sin \left (2 c+\frac {3 d x}{2}\right )+135 \sin \left (2 c+\frac {5 d x}{2}\right )+15 \sin \left (4 c+\frac {7 d x}{2}\right )-1326 \cos \left (c+\frac {d x}{2}\right )+2012 \cos \left (c+\frac {3 d x}{2}\right )-660 d x \cos \left (2 c+\frac {3 d x}{2}\right )-135 \cos \left (3 c+\frac {5 d x}{2}\right )+15 \cos \left (3 c+\frac {7 d x}{2}\right )-3216 \sin \left (\frac {d x}{2}\right )+1980 d x \cos \left (\frac {d x}{2}\right )}{240 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 163, normalized size = 1.68 \[ \frac {3 \, \cos \left (d x + c\right )^{4} - {\left (33 \, d x - 53\right )} \cos \left (d x + c\right )^{2} - 12 \, \cos \left (d x + c\right )^{3} + 66 \, d x + {\left (33 \, d x + 64\right )} \cos \left (d x + c\right ) + {\left (3 \, \cos \left (d x + c\right )^{3} + 66 \, d x + {\left (33 \, d x + 68\right )} \cos \left (d x + c\right ) + 15 \, \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) - 4}{6 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 117, normalized size = 1.21 \[ -\frac {\frac {33 \, {\left (d x + c\right )}}{a^{3}} + \frac {6 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {4 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 205, normalized size = 2.11 \[ -\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {6}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}+\frac {8}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {10}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 314, normalized size = 3.24 \[ -\frac {\frac {\frac {123 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {161 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {154 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {99 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {33 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 52}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {7 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {7 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {33 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.07, size = 121, normalized size = 1.25 \[ -\frac {11\,x}{2\,a^3}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {154\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {161\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+41\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {52}{3}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 63.55, size = 2264, normalized size = 23.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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