Optimal. Leaf size=155 \[ -\frac {\cos (c+d x)}{a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {7 \sec ^5(c+d x)}{5 a^2 d}+\frac {3 \sec ^3(c+d x)}{a^2 d}-\frac {5 \sec (c+d x)}{a^2 d}-\frac {2 x}{a^2} \]
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Rubi [A] time = 0.29, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2606, 194, 3473, 8, 2590, 270} \[ -\frac {\cos (c+d x)}{a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {7 \sec ^5(c+d x)}{5 a^2 d}+\frac {3 \sec ^3(c+d x)}{a^2 d}-\frac {5 \sec (c+d x)}{a^2 d}-\frac {2 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 270
Rule 2590
Rule 2606
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))^2} \, dx &=\frac {\int \sec (c+d x) (a-a \sin (c+d x))^2 \tan ^7(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sec (c+d x) \tan ^7(c+d x)-2 a^2 \tan ^8(c+d x)+a^2 \sin (c+d x) \tan ^8(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sec (c+d x) \tan ^7(c+d x) \, dx}{a^2}+\frac {\int \sin (c+d x) \tan ^8(c+d x) \, dx}{a^2}-\frac {2 \int \tan ^8(c+d x) \, dx}{a^2}\\ &=-\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {2 \int \tan ^6(c+d x) \, dx}{a^2}-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^8} \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {2 \int \tan ^4(c+d x) \, dx}{a^2}-\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^2 d}+\frac {\operatorname {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {\cos (c+d x)}{a^2 d}-\frac {5 \sec (c+d x)}{a^2 d}+\frac {3 \sec ^3(c+d x)}{a^2 d}-\frac {7 \sec ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {2 \tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {2 \int \tan ^2(c+d x) \, dx}{a^2}\\ &=-\frac {\cos (c+d x)}{a^2 d}-\frac {5 \sec (c+d x)}{a^2 d}+\frac {3 \sec ^3(c+d x)}{a^2 d}-\frac {7 \sec ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {2 \tan (c+d x)}{a^2 d}-\frac {2 \tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {2 \int 1 \, dx}{a^2}\\ &=-\frac {2 x}{a^2}-\frac {\cos (c+d x)}{a^2 d}-\frac {5 \sec (c+d x)}{a^2 d}+\frac {3 \sec ^3(c+d x)}{a^2 d}-\frac {7 \sec ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {2 \tan (c+d x)}{a^2 d}-\frac {2 \tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 267, normalized size = 1.72 \[ -\frac {5488 \sin (c+d x)+6720 c \sin (2 (c+d x))+6720 d x \sin (2 (c+d x))-13224 \sin (2 (c+d x))+8376 \sin (3 (c+d x))+3360 c \sin (4 (c+d x))+3360 d x \sin (4 (c+d x))-6612 \sin (4 (c+d x))+2248 \sin (5 (c+d x))+42 (280 c+280 d x-551) \cos (c+d x)+14834 \cos (2 (c+d x))+2520 c \cos (3 (c+d x))+2520 d x \cos (3 (c+d x))-4959 \cos (3 (c+d x))+1852 \cos (4 (c+d x))-840 c \cos (5 (c+d x))-840 d x \cos (5 (c+d x))+1653 \cos (5 (c+d x))-210 \cos (6 (c+d x))+11172}{6720 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 150, normalized size = 0.97 \[ -\frac {210 \, d x \cos \left (d x + c\right )^{5} + 105 \, \cos \left (d x + c\right )^{6} - 420 \, d x \cos \left (d x + c\right )^{3} - 389 \, \cos \left (d x + c\right )^{4} - 173 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (210 \, d x \cos \left (d x + c\right )^{3} + 281 \, \cos \left (d x + c\right )^{4} + 51 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 25}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 175, normalized size = 1.13 \[ -\frac {\frac {1680 \, {\left (d x + c\right )}}{a^{2}} + \frac {1680}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{2}} - \frac {35 \, {\left (12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {3780 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 25095 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 68845 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 98350 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75222 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 29659 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4777}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 253, normalized size = 1.63 \[ -\frac {1}{12 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}+\frac {4}{7 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {6}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{12 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {23}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {9}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 507, normalized size = 3.27 \[ -\frac {4 \, {\left (\frac {\frac {759 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {444 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1249 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1816 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {454 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {616 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1274 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {560 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {420 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 216}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {11 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {11 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 17.73, size = 198, normalized size = 1.28 \[ \frac {-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {728\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15}+\frac {352\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}-\frac {1816\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{105}-\frac {7264\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{105}-\frac {4996\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {592\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+\frac {1012\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+\frac {288}{35}}{a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {2\,x}{a^2} \]
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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