Optimal. Leaf size=187 \[ -\frac {4 \tan ^9(c+d x)}{9 a^3 d}-\frac {15 \tan ^7(c+d x)}{7 a^3 d}-\frac {21 \tan ^5(c+d x)}{5 a^3 d}-\frac {13 \tan ^3(c+d x)}{3 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {\sec ^7(c+d x)}{7 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d}+\frac {\sec (c+d x)}{a^3 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
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Rubi [A] time = 0.36, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2875, 2873, 3767, 2622, 302, 207, 2606, 30, 2607, 270} \[ -\frac {4 \tan ^9(c+d x)}{9 a^3 d}-\frac {15 \tan ^7(c+d x)}{7 a^3 d}-\frac {21 \tan ^5(c+d x)}{5 a^3 d}-\frac {13 \tan ^3(c+d x)}{3 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {\sec ^7(c+d x)}{7 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d}+\frac {\sec (c+d x)}{a^3 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 207
Rule 270
Rule 302
Rule 2606
Rule 2607
Rule 2622
Rule 2873
Rule 2875
Rule 3767
Rubi steps
\begin {align*} \int \frac {\csc (c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \csc (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)+a^3 \csc (c+d x) \sec ^{10}(c+d x)+3 a^3 \sec ^9(c+d x) \tan (c+d x)-a^3 \sec ^8(c+d x) \tan ^2(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \csc (c+d x) \sec ^{10}(c+d x) \, dx}{a^3}-\frac {\int \sec ^8(c+d x) \tan ^2(c+d x) \, dx}{a^3}-\frac {3 \int \sec ^{10}(c+d x) \, dx}{a^3}+\frac {3 \int \sec ^9(c+d x) \tan (c+d x) \, dx}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^3 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int x^8 \, 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)}{3 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 (x^2+3 x^4+3 x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac {\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 {\sec (c+d x)}{a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}+\frac {\sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}-\frac {13 \tan ^3(c+d x)}{3 a^3 d}-\frac {21 \tan ^5(c+d x)}{5 a^3 d}-\frac {15 \tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {\sec (c+d x)}{a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}+\frac {\sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}-\frac {13 \tan ^3(c+d x)}{3 a^3 d}-\frac {21 \tan ^5(c+d x)}{5 a^3 d}-\frac {15 \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 = 1.38, size = 204, normalized size = 1.09 \[ \frac {322560 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-322560 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {196992 \sin (c+d x)-383157 \sin (2 (c+d x))+211648 \sin (3 (c+d x))-170292 \sin (4 (c+d x))+50496 \sin (5 (c+d x))+14191 \sin (6 (c+d x))-510876 \cos (c+d x)+317952 \cos (2 (c+d x))-28382 \cos (3 (c+d x))+20352 \cos (4 (c+d x))+85146 \cos (5 (c+d x))-11776 \cos (6 (c+d x))+357504}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^9}}{322560 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 250, normalized size = 1.34 \[ \frac {736 \, \cos \left (d x + c\right )^{6} - 1422 \, \cos \left (d x + c\right )^{4} - 510 \, \cos \left (d x + c\right )^{2} - 315 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (789 \, \cos \left (d x + c\right )^{4} + 235 \, \cos \left (d x + c\right )^{2} + 35\right )} \sin \left (d x + c\right ) - 140}{630 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 187, normalized size = 1.00 \[ \frac {\frac {10080 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {105 \, {\left (27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {63315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 412020 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1273440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2324700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2731302 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2097228 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1032552 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 297828 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 40127}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 271, normalized size = 1.45 \[ -\frac {1}{24 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {9}{32 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}+\frac {8}{9 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {72}{7 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {52}{3 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {219}{10 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {83}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {193}{12 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {75}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {201}{32 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 508, normalized size = 2.72 \[ \frac {\frac {2 \, {\left (\frac {3063 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4866 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1289 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {11736 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {10566 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5292 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {13482 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {6300 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2625 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3150 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {945 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 668\right )}}{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 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.56, size = 195, normalized size = 1.04 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {50\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {428\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\frac {168\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}-\frac {2348\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}-\frac {2608\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}-\frac {2578\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{315}+\frac {3244\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}+\frac {2042\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}+\frac {1336}{315}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^9} \]
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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