Optimal. Leaf size=161 \[ -\frac {3 \cos ^7(c+d x)}{7 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac {29 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a^3 d}+\frac {29 \sin (c+d x) \cos ^3(c+d x)}{64 a^3 d}-\frac {29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac {29 x}{128 a^3} \]
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Rubi [A] time = 0.48, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac {3 \cos ^7(c+d x)}{7 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac {29 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a^3 d}+\frac {29 \sin (c+d x) \cos ^3(c+d x)}{64 a^3 d}-\frac {29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac {29 x}{128 a^3} \]
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 ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \cos ^2(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (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)+3 a^3 \cos ^2(c+d x) \sin ^5(c+d x)-a^3 \cos ^2(c+d x) \sin ^6(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a^3}-\frac {\int \cos ^2(c+d x) \sin ^6(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a^3}+\frac {3 \int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a^3}\\ &=\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{2 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{8 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 ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=\frac {3 \cos ^3(c+d x) \sin (c+d x)}{8 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{16 a^3}-\frac {3 \int \cos ^2(c+d x) \, dx}{8 a^3}-\frac {\operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {3 \cos ^7(c+d x)}{7 a^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int \cos ^2(c+d x) \, dx}{64 a^3}-\frac {3 \int 1 \, dx}{16 a^3}\\ &=-\frac {3 x}{16 a^3}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {3 \cos ^7(c+d x)}{7 a^3 d}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}+\frac {29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int 1 \, dx}{128 a^3}\\ &=-\frac {29 x}{128 a^3}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {3 \cos ^7(c+d x)}{7 a^3 d}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}+\frac {29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}\\ \end {align*}
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Mathematica [B] time = 3.99, size = 482, normalized size = 2.99 \[ \frac {-48720 d x \sin \left (\frac {c}{2}\right )+38640 \sin \left (\frac {c}{2}+d x\right )-38640 \sin \left (\frac {3 c}{2}+d x\right )+6720 \sin \left (\frac {3 c}{2}+2 d x\right )+6720 \sin \left (\frac {5 c}{2}+2 d x\right )+3920 \sin \left (\frac {5 c}{2}+3 d x\right )-3920 \sin \left (\frac {7 c}{2}+3 d x\right )+5880 \sin \left (\frac {7 c}{2}+4 d x\right )+5880 \sin \left (\frac {9 c}{2}+4 d x\right )-4368 \sin \left (\frac {9 c}{2}+5 d x\right )+4368 \sin \left (\frac {11 c}{2}+5 d x\right )-2240 \sin \left (\frac {11 c}{2}+6 d x\right )-2240 \sin \left (\frac {13 c}{2}+6 d x\right )+720 \sin \left (\frac {13 c}{2}+7 d x\right )-720 \sin \left (\frac {15 c}{2}+7 d x\right )+105 \sin \left (\frac {15 c}{2}+8 d x\right )+105 \sin \left (\frac {17 c}{2}+8 d x\right )+84 \cos \left (\frac {c}{2}\right ) (12870 c-580 d x-7)-38640 \cos \left (\frac {c}{2}+d x\right )-38640 \cos \left (\frac {3 c}{2}+d x\right )+6720 \cos \left (\frac {3 c}{2}+2 d x\right )-6720 \cos \left (\frac {5 c}{2}+2 d x\right )-3920 \cos \left (\frac {5 c}{2}+3 d x\right )-3920 \cos \left (\frac {7 c}{2}+3 d x\right )+5880 \cos \left (\frac {7 c}{2}+4 d x\right )-5880 \cos \left (\frac {9 c}{2}+4 d x\right )+4368 \cos \left (\frac {9 c}{2}+5 d x\right )+4368 \cos \left (\frac {11 c}{2}+5 d x\right )-2240 \cos \left (\frac {11 c}{2}+6 d x\right )+2240 \cos \left (\frac {13 c}{2}+6 d x\right )-720 \cos \left (\frac {13 c}{2}+7 d x\right )-720 \cos \left (\frac {15 c}{2}+7 d x\right )+105 \cos \left (\frac {15 c}{2}+8 d x\right )-105 \cos \left (\frac {17 c}{2}+8 d x\right )+1081080 c \sin \left (\frac {c}{2}\right )-998928 \sin \left (\frac {c}{2}\right )}{215040 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 90, normalized size = 0.56 \[ -\frac {5760 \, \cos \left (d x + c\right )^{7} - 18816 \, \cos \left (d x + c\right )^{5} + 17920 \, \cos \left (d x + c\right )^{3} + 3045 \, d x - 35 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 328 \, \cos \left (d x + c\right )^{5} + 454 \, \cos \left (d x + c\right )^{3} - 87 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 218, normalized size = 1.35 \[ -\frac {\frac {3045 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (3045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 23345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 26880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 51275 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 286720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 179095 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 170240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 179095 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 14336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 51275 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 109312 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 23345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 38912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4864\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a^{3}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 517, normalized size = 3.21 \[ -\frac {76}{105 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {608 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {667 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {244 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {1465 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {32 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5117 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {76 \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 )^{8}}+\frac {5117 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {128 \left (\tan ^{10}\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 )^{8}}+\frac {1465 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {4 \left (\tan ^{12}\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 )^{8}}-\frac {667 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {29 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {29 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 499, normalized size = 3.10 \[ \frac {\frac {\frac {3045 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {38912 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {23345 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {109312 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {51275 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {179095 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {170240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {179095 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {286720 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {51275 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {26880 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {23345 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {3045 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - 4864}{a^{3} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{3} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {3045 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.94, size = 212, normalized size = 1.32 \[ -\frac {29\,x}{128\,a^3}-\frac {\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {667\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {1465\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}-\frac {5117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {76\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {5117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {1465\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {244\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}-\frac {667\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {608\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}-\frac {29\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {76}{105}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]
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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