Optimal. Leaf size=142 \[ -\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {\tan ^5(c+d x)}{5 a^3 d}-\frac {\tan ^3(c+d x)}{3 a^3 d}+\frac {\tan (c+d x)}{a^3 d}+\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {11 \sec ^5(c+d x)}{5 a^3 d}+\frac {10 \sec ^3(c+d x)}{3 a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {x}{a^3} \]
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Rubi [A] time = 0.34, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2875, 2873, 2606, 270, 2607, 30, 194, 3473, 8} \[ -\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {\tan ^5(c+d x)}{5 a^3 d}-\frac {\tan ^3(c+d x)}{3 a^3 d}+\frac {\tan (c+d x)}{a^3 d}+\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {11 \sec ^5(c+d x)}{5 a^3 d}+\frac {10 \sec ^3(c+d x)}{3 a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {x}{a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 194
Rule 270
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \sec ^3(c+d x) (a-a \sin (c+d x))^3 \tan ^5(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (a^3 \sec ^3(c+d x) \tan ^5(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^6(c+d x)+3 a^3 \sec (c+d x) \tan ^7(c+d x)-a^3 \tan ^8(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \sec ^3(c+d x) \tan ^5(c+d x) \, dx}{a^3}-\frac {\int \tan ^8(c+d x) \, dx}{a^3}-\frac {3 \int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^3}+\frac {3 \int \sec (c+d x) \tan ^7(c+d x) \, dx}{a^3}\\ &=-\frac {\tan ^7(c+d x)}{7 a^3 d}+\frac {\int \tan ^6(c+d x) \, dx}{a^3}+\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac {\tan ^5(c+d x)}{5 a^3 d}-\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {\int \tan ^4(c+d x) \, dx}{a^3}+\frac {\operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac {3 \sec (c+d x)}{a^3 d}+\frac {10 \sec ^3(c+d x)}{3 a^3 d}-\frac {11 \sec ^5(c+d x)}{5 a^3 d}+\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {\tan ^3(c+d x)}{3 a^3 d}+\frac {\tan ^5(c+d x)}{5 a^3 d}-\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {\int \tan ^2(c+d x) \, dx}{a^3}\\ &=-\frac {3 \sec (c+d x)}{a^3 d}+\frac {10 \sec ^3(c+d x)}{3 a^3 d}-\frac {11 \sec ^5(c+d x)}{5 a^3 d}+\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {\tan (c+d x)}{a^3 d}-\frac {\tan ^3(c+d x)}{3 a^3 d}+\frac {\tan ^5(c+d x)}{5 a^3 d}-\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {\int 1 \, dx}{a^3}\\ &=-\frac {x}{a^3}-\frac {3 \sec (c+d x)}{a^3 d}+\frac {10 \sec ^3(c+d x)}{3 a^3 d}-\frac {11 \sec ^5(c+d x)}{5 a^3 d}+\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {\tan (c+d x)}{a^3 d}-\frac {\tan ^3(c+d x)}{3 a^3 d}+\frac {\tan ^5(c+d x)}{5 a^3 d}-\frac {4 \tan ^7(c+d x)}{7 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.80, size = 214, normalized size = 1.51 \[ -\frac {2688 \sin (c+d x)+11760 c \sin (2 (c+d x))+11760 d x \sin (2 (c+d x))-23282 \sin (2 (c+d x))+5568 \sin (3 (c+d x))-840 c \sin (4 (c+d x))-840 d x \sin (4 (c+d x))+1663 \sin (4 (c+d x))+14 (840 c+840 d x-1663) \cos (c+d x)+6272 \cos (2 (c+d x))-5040 c \cos (3 (c+d x))-5040 d x \cos (3 (c+d x))+9978 \cos (3 (c+d x))-1768 \cos (4 (c+d x))+4200}{6720 a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 150, normalized size = 1.06 \[ -\frac {315 \, d x \cos \left (d x + c\right )^{3} + 221 \, \cos \left (d x + c\right )^{4} - 420 \, d x \cos \left (d x + c\right ) - 417 \, \cos \left (d x + c\right )^{2} + 3 \, {\left (35 \, d x \cos \left (d x + c\right )^{3} - 140 \, d x \cos \left (d x + c\right ) - 116 \, \cos \left (d x + c\right )^{2} + 15\right )} \sin \left (d x + c\right ) + 60}{105 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 129, normalized size = 0.91 \[ -\frac {\frac {840 \, {\left (d x + c\right )}}{a^{3}} + \frac {105}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {1575 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 10920 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31675 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36981 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 14392 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2281}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 187, normalized size = 1.32 \[ -\frac {1}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}+\frac {8}{7 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {18}{5 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {5}{6 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {15}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 335, normalized size = 2.36 \[ -\frac {2 \, {\left (\frac {\frac {711 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1274 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {469 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1260 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1435 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {630 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {105 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 136}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.65, size = 131, normalized size = 0.92 \[ \frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {82\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {134\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {364\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {474\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+\frac {272}{105}}{a^3\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7}-\frac {x}{a^3} \]
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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