Optimal. Leaf size=164 \[ \frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {7 \tan ^5(c+d x)}{5 a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}+\frac {2 \tanh ^{-1}(\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.33, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 3767, 2622, 302, 207, 2620, 270} \[ \frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {7 \tan ^5(c+d x)}{5 a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}+\frac {2 \tanh ^{-1}(\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 270
Rule 302
Rule 2620
Rule 2622
Rule 2873
Rule 2875
Rule 3767
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \csc ^2(c+d x) \sec ^8(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sec ^8(c+d x)-2 a^2 \csc (c+d x) \sec ^8(c+d x)+a^2 \csc ^2(c+d x) \sec ^8(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sec ^8(c+d x) \, dx}{a^2}+\frac {\int \csc ^2(c+d x) \sec ^8(c+d x) \, dx}{a^2}-\frac {2 \int \csc (c+d x) \sec ^8(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^4}{x^2} \, dx,x,\tan (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,-\tan (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \frac {x^8}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{a^2 d}+\frac {3 \tan ^5(c+d x)}{5 a^2 d}+\frac {\tan ^7(c+d x)}{7 a^2 d}+\frac {\operatorname {Subst}\left (\int \left (4+\frac {1}{x^2}+6 x^2+4 x^4+x^6\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \left (1+x^2+x^4+x^6+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {7 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {5 \tan (c+d x)}{a^2 d}+\frac {3 \tan ^3(c+d x)}{a^2 d}+\frac {7 \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 [B] time = 6.10, size = 442, normalized size = 2.70 \[ \frac {16 \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {13 \sin \left (\frac {1}{2} (c+d x)\right )}{384 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{384 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4777 \sin \left (\frac {1}{2} (c+d x)\right )}{13440 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {1}{768 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {997}{26880 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {997 \sin \left (\frac {1}{2} (c+d x)\right )}{13440 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {3}{280 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {3 \sin \left (\frac {1}{2} (c+d x)\right )}{140 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5}-\frac {1}{448 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{224 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7}\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 250, normalized size = 1.52 \[ -\frac {432 \, \cos \left (d x + c\right )^{6} - 660 \, \cos \left (d x + c\right )^{4} + 98 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (327 \, \cos \left (d x + c\right )^{4} - 41 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 25}{105 \, {\left (2 \, a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} + {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 204, normalized size = 1.24 \[ -\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} - \frac {420 \, {\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {35 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 14\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {7875 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 41055 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 94640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 119630 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 87507 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 34979 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6122}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 266, normalized size = 1.62 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {1}{12 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{2 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}-\frac {4}{7 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {2}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {24}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {7}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {107}{12 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {59}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {75}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 481, normalized size = 2.93 \[ -\frac {\frac {\frac {1828 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3847 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1656 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {12734 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {7952 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {9702 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {12600 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {315 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {5460 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2205 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 105}{\frac {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 {3 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {14 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {3 \, 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 {a^{2} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}} + \frac {420 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {105 \, \sin \left (d x + c\right )}{a^{2} {\left (\cos \left (d x + c\right ) + 1\right )}}}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.24, size = 331, normalized size = 2.02 \[ \frac {21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {1136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\frac {12734\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{105}+\frac {552\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{35}-\frac {3847\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}-\frac {1828\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-1}{d\,\left (-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+28\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d} \]
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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