Optimal. Leaf size=178 \[ -\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}+\frac {x}{a^3} \]
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Rubi [A] time = 0.37, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ -\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}+\frac {x}{a^3} \]
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 ^4(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 ^7(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (a^3 \sec ^3(c+d x) \tan ^7(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^8(c+d x)+3 a^3 \sec (c+d x) \tan ^9(c+d x)-a^3 \tan ^{10}(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \sec ^3(c+d x) \tan ^7(c+d x) \, dx}{a^3}-\frac {\int \tan ^{10}(c+d x) \, dx}{a^3}-\frac {3 \int \sec ^2(c+d x) \tan ^8(c+d x) \, dx}{a^3}+\frac {3 \int \sec (c+d x) \tan ^9(c+d x) \, dx}{a^3}\\ &=-\frac {\tan ^9(c+d x)}{9 a^3 d}+\frac {\int \tan ^8(c+d x) \, dx}{a^3}+\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^8 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}-\frac {\int \tan ^6(c+d x) \, dx}{a^3}+\frac {\operatorname {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\int \tan ^4(c+d x) \, dx}{a^3}\\ &=\frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}-\frac {\int \tan ^2(c+d x) \, dx}{a^3}\\ &=\frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 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 {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\int 1 \, dx}{a^3}\\ &=\frac {x}{a^3}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 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 {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 273, normalized size = 1.53 \[ \frac {93312 \sin (c+d x)+272160 (c+d x) \sin (2 (c+d x))-506277 \sin (2 (c+d x))+125248 \sin (3 (c+d x))+120960 (c+d x) \sin (4 (c+d x))-225012 \sin (4 (c+d x))+67776 \sin (5 (c+d x))-10080 (c+d x) \sin (6 (c+d x))+18751 \sin (6 (c+d x))+362880 (c+d x) \cos (c+d x)-675036 \cos (c+d x)+173952 \cos (2 (c+d x))+20160 (c+d x) \cos (3 (c+d x))-37502 \cos (3 (c+d x))+54912 \cos (4 (c+d x))-60480 (c+d x) \cos (5 (c+d x))+112506 \cos (5 (c+d x))-21376 \cos (6 (c+d x))+169344}{322560 d (a \sin (c+d x)+a)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 177, normalized size = 0.99 \[ \frac {945 \, d x \cos \left (d x + c\right )^{5} + 668 \, \cos \left (d x + c\right )^{6} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1431 \, \cos \left (d x + c\right )^{4} + 465 \, \cos \left (d x + c\right )^{2} + {\left (315 \, d x \cos \left (d x + c\right )^{5} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1059 \, \cos \left (d x + c\right )^{4} + 305 \, \cos \left (d x + c\right )^{2} - 35\right )} \sin \left (d x + c\right ) - 70}{315 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 181, normalized size = 1.02 \[ \frac {\frac {10080 \, {\left (d x + c\right )}}{a^{3}} + \frac {105 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 23\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {17955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 160020 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 624960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1387260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1884582 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1556268 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 774792 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 215748 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25967}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 272, normalized size = 1.53 \[ -\frac {1}{24 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{32 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}+\frac {8}{9 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {40}{7 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{3 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {21}{10 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {3}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {13}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {57}{32 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 487, normalized size = 2.74 \[ \frac {2 \, {\left (\frac {\frac {1893 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2526 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2939 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {9936 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3546 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {11172 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {9702 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3675 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1890 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 368}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {315 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 17.81, size = 169, normalized size = 0.95 \[ \frac {x}{a^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}-\frac {308\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {1064\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {788\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}+\frac {2208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}+\frac {5878\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{315}-\frac {1684\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}-\frac {1262\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {736}{315}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^9} \]
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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