Optimal. Leaf size=87 \[ \frac {\cos ^3(c+d x)}{3 a^3 d}-\frac {5 \cos (c+d x)}{a^3 d}+\frac {3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac {11 x}{2 a^3} \]
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Rubi [A] time = 0.23, antiderivative size = 87, 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 = {2875, 2709, 2638, 2635, 8, 2633, 2648} \[ \frac {\cos ^3(c+d x)}{3 a^3 d}-\frac {5 \cos (c+d x)}{a^3 d}+\frac {3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac {11 x}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2648
Rule 2709
Rule 2875
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int (a-a \sin (c+d x))^3 \tan ^2(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (-4 a+4 a \sin (c+d x)-3 a \sin ^2(c+d x)+a \sin ^3(c+d x)+\frac {4 a}{1+\sin (c+d x)}\right ) \, dx}{a^4}\\ &=-\frac {4 x}{a^3}+\frac {\int \sin ^3(c+d x) \, dx}{a^3}-\frac {3 \int \sin ^2(c+d x) \, dx}{a^3}+\frac {4 \int \sin (c+d x) \, dx}{a^3}+\frac {4 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=-\frac {4 x}{a^3}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {3 \int 1 \, dx}{2 a^3}-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=-\frac {11 x}{2 a^3}-\frac {5 \cos (c+d x)}{a^3 d}+\frac {\cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 1.14, size = 181, normalized size = 2.08 \[ \frac {-660 d x \sin \left (c+\frac {d x}{2}\right )+\sin \left (c+\frac {d x}{2}\right )-240 \sin \left (2 c+\frac {3 d x}{2}\right )+40 \sin \left (2 c+\frac {5 d x}{2}\right )+5 \sin \left (4 c+\frac {7 d x}{2}\right )-286 \cos \left (c+\frac {d x}{2}\right )-240 \cos \left (c+\frac {3 d x}{2}\right )-40 \cos \left (3 c+\frac {5 d x}{2}\right )+5 \cos \left (3 c+\frac {7 d x}{2}\right )+1244 \sin \left (\frac {d x}{2}\right )+(1-660 d x) \cos \left (\frac {d x}{2}\right )}{120 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 123, normalized size = 1.41 \[ \frac {2 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{3} - 33 \, d x - 3 \, {\left (11 \, d x + 15\right )} \cos \left (d x + c\right ) - 30 \, \cos \left (d x + c\right )^{2} + {\left (2 \, \cos \left (d x + c\right )^{3} - 33 \, d x + 9 \, \cos \left (d x + c\right )^{2} - 21 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right ) - 24}{6 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 106, normalized size = 1.22 \[ -\frac {\frac {33 \, {\left (d x + c\right )}}{a^{3}} + \frac {48}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 28\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 198, normalized size = 2.28 \[ -\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {20 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {28}{3 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {8}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 312, normalized size = 3.59 \[ -\frac {\frac {\frac {19 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {123 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {60 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {96 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {33 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {33 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 52}{a^{3} + \frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {33 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.35, size = 121, normalized size = 1.39 \[ -\frac {11\,x}{2\,a^3}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {19\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {52}{3}}{a^3\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 95.06, size = 2264, normalized size = 26.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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