Optimal. Leaf size=76 \[ \frac {3 \cos (c+d x)}{a^3 d}+\frac {3 x}{a^3}+\frac {2 \cos ^3(c+d x)}{a d (a \sin (c+d x)+a)^2}-\frac {\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.18, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2871, 2680, 2682, 8} \[ \frac {3 \cos (c+d x)}{a^3 d}+\frac {3 x}{a^3}+\frac {2 \cos ^3(c+d x)}{a d (a \sin (c+d x)+a)^2}-\frac {\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2680
Rule 2682
Rule 2871
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=-\frac {\cos ^3(c+d x)}{3 d (a+a \sin (c+d x))^3}-\int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx\\ &=-\frac {\cos ^3(c+d x)}{3 d (a+a \sin (c+d x))^3}+\frac {2 \cos ^3(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {3 \int \frac {\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac {3 \cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{3 d (a+a \sin (c+d x))^3}+\frac {2 \cos ^3(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {3 \int 1 \, dx}{a^3}\\ &=\frac {3 x}{a^3}+\frac {3 \cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{3 d (a+a \sin (c+d x))^3}+\frac {2 \cos ^3(c+d x)}{a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 96, normalized size = 1.26 \[ \frac {3 \cos (c+d x)-\frac {2 \sin \left (\frac {1}{2} (c+d x)\right ) (13 \sin (c+d x)+11)}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {2}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+9 c+9 d x}{3 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 144, normalized size = 1.89 \[ \frac {{\left (9 \, d x - 16\right )} \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right )^{3} - 18 \, d x - {\left (9 \, d x + 17\right )} \cos \left (d x + c\right ) - {\left (18 \, d x + {\left (9 \, d x + 19\right )} \cos \left (d x + c\right ) + 3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 2}{3 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 80, normalized size = 1.05 \[ \frac {\frac {9 \, {\left (d x + c\right )}}{a^{3}} + \frac {6}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 106, normalized size = 1.39 \[ \frac {2}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {8}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {4}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {6}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 228, normalized size = 3.00 \[ \frac {2 \, {\left (\frac {\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {29 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {27 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {9 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 14}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.32, size = 94, normalized size = 1.24 \[ \frac {3\,x}{a^3}+\frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {58\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+22\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {28}{3}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.82, size = 1246, normalized size = 16.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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