Optimal. Leaf size=129 \[ -\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {\sin ^5(c+d x) \cos (c+d x)}{6 a^3 d}+\frac {23 \sin ^3(c+d x) \cos (c+d x)}{24 a^3 d}+\frac {23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {23 x}{16 a^3} \]
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Rubi [A] time = 0.24, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2869, 2757, 2633, 2635, 8} \[ -\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {\sin ^5(c+d x) \cos (c+d x)}{6 a^3 d}+\frac {23 \sin ^3(c+d x) \cos (c+d x)}{24 a^3 d}+\frac {23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {23 x}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2757
Rule 2869
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \sin ^3(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (a^3 \sin ^3(c+d x)-3 a^3 \sin ^4(c+d x)+3 a^3 \sin ^5(c+d x)-a^3 \sin ^6(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \sin ^3(c+d x) \, dx}{a^3}-\frac {\int \sin ^6(c+d x) \, dx}{a^3}-\frac {3 \int \sin ^4(c+d x) \, dx}{a^3}+\frac {3 \int \sin ^5(c+d x) \, dx}{a^3}\\ &=\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac {5 \int \sin ^4(c+d x) \, dx}{6 a^3}-\frac {9 \int \sin ^2(c+d x) \, dx}{4 a^3}-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=-\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac {5 \int \sin ^2(c+d x) \, dx}{8 a^3}-\frac {9 \int 1 \, dx}{8 a^3}\\ &=-\frac {9 x}{8 a^3}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}-\frac {5 \int 1 \, dx}{16 a^3}\\ &=-\frac {23 x}{16 a^3}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d}\\ \end {align*}
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Mathematica [B] time = 2.19, size = 366, normalized size = 2.84 \[ \frac {-2760 d x \sin \left (\frac {c}{2}\right )+2520 \sin \left (\frac {c}{2}+d x\right )-2520 \sin \left (\frac {3 c}{2}+d x\right )+945 \sin \left (\frac {3 c}{2}+2 d x\right )+945 \sin \left (\frac {5 c}{2}+2 d x\right )-380 \sin \left (\frac {5 c}{2}+3 d x\right )+380 \sin \left (\frac {7 c}{2}+3 d x\right )-135 \sin \left (\frac {7 c}{2}+4 d x\right )-135 \sin \left (\frac {9 c}{2}+4 d x\right )+36 \sin \left (\frac {9 c}{2}+5 d x\right )-36 \sin \left (\frac {11 c}{2}+5 d x\right )+5 \sin \left (\frac {11 c}{2}+6 d x\right )+5 \sin \left (\frac {13 c}{2}+6 d x\right )-3 \cos \left (\frac {c}{2}\right ) (920 d x+3)-2520 \cos \left (\frac {c}{2}+d x\right )-2520 \cos \left (\frac {3 c}{2}+d x\right )+945 \cos \left (\frac {3 c}{2}+2 d x\right )-945 \cos \left (\frac {5 c}{2}+2 d x\right )+380 \cos \left (\frac {5 c}{2}+3 d x\right )+380 \cos \left (\frac {7 c}{2}+3 d x\right )-135 \cos \left (\frac {7 c}{2}+4 d x\right )+135 \cos \left (\frac {9 c}{2}+4 d x\right )-36 \cos \left (\frac {9 c}{2}+5 d x\right )-36 \cos \left (\frac {11 c}{2}+5 d x\right )+5 \cos \left (\frac {11 c}{2}+6 d x\right )-5 \cos \left (\frac {13 c}{2}+6 d x\right )+9 \sin \left (\frac {c}{2}\right )}{1920 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 78, normalized size = 0.60 \[ -\frac {144 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} + 345 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 62 \, \cos \left (d x + c\right )^{3} + 123 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 960 \, \cos \left (d x + c\right )}{240 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 166, normalized size = 1.29 \[ -\frac {\frac {345 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3264 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 544\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 381, normalized size = 2.95 \[ -\frac {23 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {391 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {4 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {75 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {136 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {75 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {64 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {391 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {136 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {68}{15 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 373, normalized size = 2.89 \[ \frac {\frac {\frac {345 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3264 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1955 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {7680 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2250 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5440 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2250 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1955 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {345 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 544}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {345 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.56, size = 160, normalized size = 1.24 \[ -\frac {23\,x}{16\,a^3}-\frac {\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {75\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {75\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {23\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {68}{15}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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