Optimal. Leaf size=200 \[ \frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {17 \tan ^7(c+d x)}{7 a^3 d}+\frac {28 \tan ^5(c+d x)}{5 a^3 d}+\frac {22 \tan ^3(c+d x)}{3 a^3 d}+\frac {8 \tan (c+d x)}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {3 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
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Rubi [A] time = 0.39, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2875, 2873, 3767, 2622, 302, 207, 2620, 270, 2606, 30} \[ \frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {17 \tan ^7(c+d x)}{7 a^3 d}+\frac {28 \tan ^5(c+d x)}{5 a^3 d}+\frac {22 \tan ^3(c+d x)}{3 a^3 d}+\frac {8 \tan (c+d x)}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {3 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 207
Rule 270
Rule 302
Rule 2606
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))^3} \, dx &=\frac {\int \csc ^2(c+d x) \sec ^{10}(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (3 a^3 \sec ^{10}(c+d x)-3 a^3 \csc (c+d x) \sec ^{10}(c+d x)+a^3 \csc ^2(c+d x) \sec ^{10}(c+d x)-a^3 \sec ^9(c+d x) \tan (c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \csc ^2(c+d x) \sec ^{10}(c+d x) \, dx}{a^3}-\frac {\int \sec ^9(c+d x) \tan (c+d x) \, dx}{a^3}+\frac {3 \int \sec ^{10}(c+d x) \, dx}{a^3}-\frac {3 \int \csc (c+d x) \sec ^{10}(c+d x) \, dx}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int x^8 \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^2} \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, 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,-\tan (c+d x)\right )}{a^3 d}\\ &=-\frac {\sec ^9(c+d x)}{9 a^3 d}+\frac {3 \tan (c+d x)}{a^3 d}+\frac {4 \tan ^3(c+d x)}{a^3 d}+\frac {18 \tan ^5(c+d x)}{5 a^3 d}+\frac {12 \tan ^7(c+d x)}{7 a^3 d}+\frac {\tan ^9(c+d x)}{3 a^3 d}+\frac {\operatorname {Subst}\left (\int \left (5+\frac {1}{x^2}+10 x^2+10 x^4+5 x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (1+x^2+x^4+x^6+x^8+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac {\cot (c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {3 \sec ^7(c+d x)}{7 a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {8 \tan (c+d x)}{a^3 d}+\frac {22 \tan ^3(c+d x)}{3 a^3 d}+\frac {28 \tan ^5(c+d x)}{5 a^3 d}+\frac {17 \tan ^7(c+d x)}{7 a^3 d}+\frac {4 \tan ^9(c+d x)}{9 a^3 d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^3(c+d x)}{a^3 d}-\frac {3 \sec ^5(c+d x)}{5 a^3 d}-\frac {3 \sec ^7(c+d x)}{7 a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {8 \tan (c+d x)}{a^3 d}+\frac {22 \tan ^3(c+d x)}{3 a^3 d}+\frac {28 \tan ^5(c+d x)}{5 a^3 d}+\frac {17 \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 = 0.64, size = 230, normalized size = 1.15 \[ \frac {-1935360 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1935360 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {\csc (c+d x) (-707328 \sin (c+d x)+1364182 \sin (2 (c+d x))-1161600 \sin (3 (c+d x))+320984 \sin (4 (c+d x))-329344 \sin (5 (c+d x))-240738 \sin (6 (c+d x))+53248 \sin (7 (c+d x))+1083321 \cos (c+d x)-653248 \cos (2 (c+d x))-601845 \cos (3 (c+d x))+340096 \cos (4 (c+d x))-521599 \cos (5 (c+d x))+259008 \cos (6 (c+d x))+40123 \cos (7 (c+d x))-590976)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^9}}{645120 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 297, normalized size = 1.48 \[ \frac {8094 \, \cos \left (d x + c\right )^{6} - 9484 \, \cos \left (d x + c\right )^{4} + 620 \, \cos \left (d x + c\right )^{2} + 945 \, {\left (\cos \left (d x + c\right )^{7} - 5 \, \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 945 \, {\left (\cos \left (d x + c\right )^{7} - 5 \, \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{3} - {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (1664 \, \cos \left (d x + c\right )^{6} - 4653 \, \cos \left (d x + c\right )^{4} + 285 \, \cos \left (d x + c\right )^{2} + 35\right )} \sin \left (d x + c\right ) + 140}{630 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} - 5 \, a^{3} d \cos \left (d x + c\right )^{5} + 4 \, a^{3} d \cos \left (d x + c\right )^{3} - {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 230, normalized size = 1.15 \[ -\frac {\frac {30240 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {5040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {5040 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {105 \, {\left (33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 31\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {157815 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1093680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3488940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6524280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7788186 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6052704 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2995596 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 864504 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 113591}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 308, normalized size = 1.54 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{3}}-\frac {1}{24 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {11}{32 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{2 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {8}{9 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {76}{7 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {58}{3 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {267}{10 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {111}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {25}{a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {67}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {501}{32 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 567, normalized size = 2.84 \[ -\frac {\frac {\frac {8786 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35076 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {43062 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {41753 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {152172 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {99072 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {93324 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {157689 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {44730 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {50820 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {42210 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {10395 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + 315}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {27 \, 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 )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {a^{3} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}} + \frac {1890 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {315 \, \sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{630 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.32, size = 390, normalized size = 1.95 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {-33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-134\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-\frac {484\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+142\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {2503\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}+\frac {4444\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15}-\frac {11008\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{35}-\frac {16908\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}-\frac {41753\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{315}+\frac {14354\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {11692\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}+\frac {8786\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{315}+1}{d\,\left (-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+72\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-72\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]
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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