Optimal. Leaf size=83 \[ -\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {3 \cos (c+d x)}{a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac {3 x}{a^2} \]
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Rubi [A] time = 0.22, antiderivative size = 83, 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, 2633, 2648} \[ -\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {3 \cos (c+d x)}{a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac {3 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2648
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))^2} \, dx &=\frac {\int \frac {\sin ^3(c+d x) (a-a \sin (c+d x))}{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac {\int \left (2-2 \sin (c+d x)+2 \sin ^2(c+d x)-\sin ^3(c+d x)-\frac {2}{1+\sin (c+d x)}\right ) \, dx}{a^2}\\ &=\frac {2 x}{a^2}-\frac {\int \sin ^3(c+d x) \, dx}{a^2}-\frac {2 \int \sin (c+d x) \, dx}{a^2}+\frac {2 \int \sin ^2(c+d x) \, dx}{a^2}-\frac {2 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^2}\\ &=\frac {2 x}{a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))}+\frac {\int 1 \, dx}{a^2}+\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {3 x}{a^2}+\frac {3 \cos (c+d x)}{a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.97, size = 165, normalized size = 1.99 \[ -\frac {-72 d x \sin \left (c+\frac {d x}{2}\right )-27 \sin \left (2 c+\frac {3 d x}{2}\right )+5 \sin \left (2 c+\frac {5 d x}{2}\right )+\sin \left (4 c+\frac {7 d x}{2}\right )-31 \cos \left (c+\frac {d x}{2}\right )-27 \cos \left (c+\frac {3 d x}{2}\right )-5 \cos \left (3 c+\frac {5 d x}{2}\right )+\cos \left (3 c+\frac {7 d x}{2}\right )+131 \sin \left (\frac {d x}{2}\right )-72 d x \cos \left (\frac {d x}{2}\right )}{24 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 119, normalized size = 1.43 \[ -\frac {\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 9 \, d x - 3 \, {\left (3 \, d x + 4\right )} \cos \left (d x + c\right ) - 9 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{3} - 9 \, d x + 3 \, \cos \left (d x + c\right )^{2} - 6 \, \cos \left (d x + c\right ) + 6\right )} \sin \left (d x + c\right ) - 6}{3 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 106, normalized size = 1.28 \[ \frac {\frac {9 \, {\left (d x + c\right )}}{a^{2}} + \frac {12}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 198, normalized size = 2.39 \[ \frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {16}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}+\frac {4}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 312, normalized size = 3.76 \[ \frac {2 \, {\left (\frac {\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {33 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {24 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {9 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {9 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 14}{a^{2} + \frac {a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.38, size = 120, normalized size = 1.45 \[ \frac {3\,x}{a^2}+\frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {28}{3}}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.33, size = 2263, normalized size = 27.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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