Optimal. Leaf size=133 \[ \frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {\cos ^5(c+d x)}{a^3 d}+\frac {4 \cos ^3(c+d x)}{3 a^3 d}-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{2 a^3 d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac {5 x}{16 a^3} \]
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Rubi [A] time = 0.40, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2568, 2635, 8, 2565, 14, 270} \[ \frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {\cos ^5(c+d x)}{a^3 d}+\frac {4 \cos ^3(c+d x)}{3 a^3 d}-\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{2 a^3 d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac {5 x}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2873
Rule 2875
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (a^3 \cos ^2(c+d x) \sin ^2(c+d x)-3 a^3 \cos ^2(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^2(c+d x) \sin ^4(c+d x)-a^3 \cos ^2(c+d x) \sin ^5(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a^3}-\frac {\int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a^3}+\frac {3 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a^3}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^3 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{2 a^3 d}+\frac {\int \cos ^2(c+d x) \, dx}{4 a^3}+\frac {3 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{2 a^3}+\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=\frac {\cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{2 a^3 d}+\frac {\int 1 \, dx}{8 a^3}+\frac {3 \int \cos ^2(c+d x) \, dx}{8 a^3}+\frac {\operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=\frac {x}{8 a^3}+\frac {4 \cos ^3(c+d x)}{3 a^3 d}-\frac {\cos ^5(c+d x)}{a^3 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{2 a^3 d}+\frac {3 \int 1 \, dx}{16 a^3}\\ &=\frac {5 x}{16 a^3}+\frac {4 \cos ^3(c+d x)}{3 a^3 d}-\frac {\cos ^5(c+d x)}{a^3 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{2 a^3 d}\\ \end {align*}
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Mathematica [B] time = 8.92, size = 429, normalized size = 3.23 \[ \frac {840 d x \sin \left (\frac {c}{2}\right )-609 \sin \left (\frac {c}{2}+d x\right )+609 \sin \left (\frac {3 c}{2}+d x\right )-63 \sin \left (\frac {3 c}{2}+2 d x\right )-63 \sin \left (\frac {5 c}{2}+2 d x\right )-91 \sin \left (\frac {5 c}{2}+3 d x\right )+91 \sin \left (\frac {7 c}{2}+3 d x\right )-105 \sin \left (\frac {7 c}{2}+4 d x\right )-105 \sin \left (\frac {9 c}{2}+4 d x\right )+63 \sin \left (\frac {9 c}{2}+5 d x\right )-63 \sin \left (\frac {11 c}{2}+5 d x\right )+21 \sin \left (\frac {11 c}{2}+6 d x\right )+21 \sin \left (\frac {13 c}{2}+6 d x\right )-3 \sin \left (\frac {13 c}{2}+7 d x\right )+3 \sin \left (\frac {15 c}{2}+7 d x\right )-168 \cos \left (\frac {c}{2}\right ) (99 c-5 d x)+609 \cos \left (\frac {c}{2}+d x\right )+609 \cos \left (\frac {3 c}{2}+d x\right )-63 \cos \left (\frac {3 c}{2}+2 d x\right )+63 \cos \left (\frac {5 c}{2}+2 d x\right )+91 \cos \left (\frac {5 c}{2}+3 d x\right )+91 \cos \left (\frac {7 c}{2}+3 d x\right )-105 \cos \left (\frac {7 c}{2}+4 d x\right )+105 \cos \left (\frac {9 c}{2}+4 d x\right )-63 \cos \left (\frac {9 c}{2}+5 d x\right )-63 \cos \left (\frac {11 c}{2}+5 d x\right )+21 \cos \left (\frac {11 c}{2}+6 d x\right )-21 \cos \left (\frac {13 c}{2}+6 d x\right )+3 \cos \left (\frac {13 c}{2}+7 d x\right )+3 \cos \left (\frac {15 c}{2}+7 d x\right )-16632 c \sin \left (\frac {c}{2}\right )+16996 \sin \left (\frac {c}{2}\right )}{2688 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 80, normalized size = 0.60 \[ \frac {48 \, \cos \left (d x + c\right )^{7} - 336 \, \cos \left (d x + c\right )^{5} + 448 \, \cos \left (d x + c\right )^{3} + 105 \, d x + 21 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 18 \, \cos \left (d x + c\right )^{3} + 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{336 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 179, normalized size = 1.35 \[ \frac {\frac {105 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 252 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 2499 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5152 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 448 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2499 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1344 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 252 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 160\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{3}}}{336 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 415, normalized size = 3.12 \[ \frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {3 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {12 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {119 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {92 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {119 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {20 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {20}{21 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 416, normalized size = 3.13 \[ -\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1120 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {252 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1344 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2499 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {448 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {5152 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {2499 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2016 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {252 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 160}{a^{3} + \frac {7 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{3} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.64, size = 172, normalized size = 1.29 \[ \frac {5\,x}{16\,a^3}+\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{2}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {119\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}+\frac {92\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {119\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {20}{21}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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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