Optimal. Leaf size=194 \[ -\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {8 \tan ^5(c+d x)}{5 a^2 d}-\frac {4 \tan ^3(c+d x)}{a^2 d}-\frac {8 \tan (c+d x)}{a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {11 \sec ^7(c+d x)}{14 a^2 d}+\frac {11 \sec ^5(c+d x)}{10 a^2 d}+\frac {11 \sec ^3(c+d x)}{6 a^2 d}+\frac {11 \sec (c+d x)}{2 a^2 d}-\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac {\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.37, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2622, 302, 207, 2620, 270, 288} \[ -\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {8 \tan ^5(c+d x)}{5 a^2 d}-\frac {4 \tan ^3(c+d x)}{a^2 d}-\frac {8 \tan (c+d x)}{a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {11 \sec ^7(c+d x)}{14 a^2 d}+\frac {11 \sec ^5(c+d x)}{10 a^2 d}+\frac {11 \sec ^3(c+d x)}{6 a^2 d}+\frac {11 \sec (c+d x)}{2 a^2 d}-\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac {\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 270
Rule 288
Rule 302
Rule 2620
Rule 2622
Rule 2873
Rule 2875
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \csc ^3(c+d x) \sec ^8(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \csc (c+d x) \sec ^8(c+d x)-2 a^2 \csc ^2(c+d x) \sec ^8(c+d x)+a^2 \csc ^3(c+d x) \sec ^8(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \csc (c+d x) \sec ^8(c+d x) \, dx}{a^2}+\frac {\int \csc ^3(c+d x) \sec ^8(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^2(c+d x) \sec ^8(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \frac {x^8}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^4}{x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d}+\frac {\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 {2 \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 {9 \operatorname {Subst}\left (\int \frac {x^8}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a^2 d}\\ &=\frac {2 \cot (c+d x)}{a^2 d}+\frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}+\frac {\sec ^5(c+d x)}{5 a^2 d}+\frac {\sec ^7(c+d x)}{7 a^2 d}-\frac {\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d}-\frac {8 \tan (c+d x)}{a^2 d}-\frac {4 \tan ^3(c+d x)}{a^2 d}-\frac {8 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {9 \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 )}{2 a^2 d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {11 \sec (c+d x)}{2 a^2 d}+\frac {11 \sec ^3(c+d x)}{6 a^2 d}+\frac {11 \sec ^5(c+d x)}{10 a^2 d}+\frac {11 \sec ^7(c+d x)}{14 a^2 d}-\frac {\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d}-\frac {8 \tan (c+d x)}{a^2 d}-\frac {4 \tan ^3(c+d x)}{a^2 d}-\frac {8 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a^2 d}\\ &=-\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {11 \sec (c+d x)}{2 a^2 d}+\frac {11 \sec ^3(c+d x)}{6 a^2 d}+\frac {11 \sec ^5(c+d x)}{10 a^2 d}+\frac {11 \sec ^7(c+d x)}{14 a^2 d}-\frac {\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d}-\frac {8 \tan (c+d x)}{a^2 d}-\frac {4 \tan ^3(c+d x)}{a^2 d}-\frac {8 \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 = 0.57, size = 277, normalized size = 1.43 \[ \frac {-36960 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7+36960 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7+\frac {\csc ^2(c+d x) (4488 \sin (c+d x)-7536 \sin (2 (c+d x))+3836 \sin (3 (c+d x))-780 \sin (5 (c+d x))+2512 \sin (6 (c+d x))-768 \sin (7 (c+d x))-6908 \cos (c+d x)-563 \cos (2 (c+d x))+4396 \cos (3 (c+d x))-5390 \cos (4 (c+d x))+3140 \cos (5 (c+d x))-1917 \cos (6 (c+d x))-628 \cos (7 (c+d x))+4510)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}}{6720 d (a \sin (c+d x)+a)^2 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 292, normalized size = 1.51 \[ -\frac {3834 \, \cos \left (d x + c\right )^{6} - 3056 \, \cos \left (d x + c\right )^{4} - 468 \, \cos \left (d x + c\right )^{2} + 1155 \, {\left (\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 1155 \, {\left (\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4 \, {\left (768 \, \cos \left (d x + c\right )^{6} - 765 \, \cos \left (d x + c\right )^{4} - 98 \, \cos \left (d x + c\right )^{2} - 10\right )} \sin \left (d x + c\right ) - 100}{420 \, {\left (a^{2} d \cos \left (d x + c\right )^{7} - 3 \, a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 238, normalized size = 1.23 \[ \frac {\frac {4620 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {105 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{4}} - \frac {105 \, {\left (66 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {35 \, {\left (18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {14070 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 75705 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 177205 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 226450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 166488 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 66661 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11533}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 303, normalized size = 1.56 \[ -\frac {1}{12 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2} d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2}}-\frac {1}{8 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {11 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}+\frac {4}{7 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {26}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {8}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {139}{12 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {83}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {67}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 526, normalized size = 2.71 \[ \frac {\frac {\frac {420 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15173 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {38432 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {894 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {95344 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {77182 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {61992 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {101115 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {11340 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {33495 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {14280 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 105}{\frac {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 {3 \, 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 {14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3 \, 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 \, {\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{a^{2}} + \frac {4620 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.77, size = 243, normalized size = 1.25 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}+\frac {11\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^2\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-\frac {319\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{8}+\frac {27\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2}+\frac {963\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{8}+\frac {369\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {5513\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{60}-\frac {11918\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{105}+\frac {149\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{140}+\frac {4804\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {15173\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{840}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {1}{8}\right )}{a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7} \]
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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