Optimal. Leaf size=151 \[ \frac {\cos (c+d x)}{a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}+\frac {3 x}{a^3} \]
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Rubi [A] time = 0.34, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2875, 2873, 2607, 30, 2606, 194, 3473, 8, 2590, 270} \[ \frac {\cos (c+d x)}{a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}+\frac {3 x}{a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 194
Rule 270
Rule 2590
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \sec ^2(c+d x) (a-a \sin (c+d x))^3 \tan ^6(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (a^3 \sec ^2(c+d x) \tan ^6(c+d x)-3 a^3 \sec (c+d x) \tan ^7(c+d x)+3 a^3 \tan ^8(c+d x)-a^3 \sin (c+d x) \tan ^8(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^3}-\frac {\int \sin (c+d x) \tan ^8(c+d x) \, dx}{a^3}-\frac {3 \int \sec (c+d x) \tan ^7(c+d x) \, dx}{a^3}+\frac {3 \int \tan ^8(c+d x) \, dx}{a^3}\\ &=\frac {3 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \int \tan ^6(c+d x) \, dx}{a^3}+\frac {\operatorname {Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^8} \, dx,x,\cos (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 {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {3 \int \tan ^4(c+d x) \, dx}{a^3}+\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^8}-\frac {4}{x^6}+\frac {6}{x^4}-\frac {4}{x^2}\right ) \, dx,x,\cos (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 {\cos (c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {13 \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)}{a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \int \tan ^2(c+d x) \, dx}{a^3}\\ &=\frac {\cos (c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {3 \int 1 \, dx}{a^3}\\ &=\frac {3 x}{a^3}+\frac {\cos (c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \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 = 0.68, size = 224, normalized size = 1.48 \[ \frac {8008 \sin (c+d x)+11760 c \sin (2 (c+d x))+11760 d x \sin (2 (c+d x))-20762 \sin (2 (c+d x))+6588 \sin (3 (c+d x))-840 c \sin (4 (c+d x))-840 d x \sin (4 (c+d x))+1483 \sin (4 (c+d x))-140 \sin (5 (c+d x))+14 (840 c+840 d x-1483) \cos (c+d x)+5152 \cos (2 (c+d x))-5040 c \cos (3 (c+d x))-5040 d x \cos (3 (c+d x))+8898 \cos (3 (c+d x))-2288 \cos (4 (c+d x))+8400}{2240 a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 159, normalized size = 1.05 \[ \frac {315 \, d x \cos \left (d x + c\right )^{3} + 286 \, \cos \left (d x + c\right )^{4} - 420 \, d x \cos \left (d x + c\right ) - 447 \, \cos \left (d x + c\right )^{2} + {\left (105 \, d x \cos \left (d x + c\right )^{3} + 35 \, \cos \left (d x + c\right )^{4} - 420 \, d x \cos \left (d x + c\right ) - 438 \, \cos \left (d x + c\right )^{2} - 20\right )} \sin \left (d x + c\right ) - 15}{35 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 177, normalized size = 1.17 \[ \frac {\frac {840 \, {\left (d x + c\right )}}{a^{3}} - \frac {35 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )} a^{3}} + \frac {1715 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31815 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 45920 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35161 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 13832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2221}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 211, normalized size = 1.40 \[ -\frac {1}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}-\frac {8}{7 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {14}{5 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {17}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {49}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 421, normalized size = 2.79 \[ \frac {2 \, {\left (\frac {\frac {951 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2010 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1980 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {574 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {966 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1890 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1540 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {630 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {105 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 176}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {20 \, 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 )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {20 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \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 )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.50, size = 182, normalized size = 1.21 \[ \frac {3\,x}{a^3}-\frac {-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-88\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-108\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {276\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {164\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {792\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{7}+\frac {804\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}+\frac {1902\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+\frac {352}{35}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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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