Optimal. Leaf size=158 \[ -\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^3(c+d x)}{a^2 d}-\frac {6 \tan (c+d x)}{a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {9 \sec ^5(c+d x)}{10 a^2 d}+\frac {3 \sec ^3(c+d x)}{2 a^2 d}+\frac {9 \sec (c+d x)}{2 a^2 d}-\frac {9 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac {\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.35, antiderivative size = 158, 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} \[ -\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \tan ^3(c+d x)}{a^2 d}-\frac {6 \tan (c+d x)}{a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {9 \sec ^5(c+d x)}{10 a^2 d}+\frac {3 \sec ^3(c+d x)}{2 a^2 d}+\frac {9 \sec (c+d x)}{2 a^2 d}-\frac {9 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac {\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d} \]
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 ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \csc ^3(c+d x) \sec ^6(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \csc (c+d x) \sec ^6(c+d x)-2 a^2 \csc ^2(c+d x) \sec ^6(c+d x)+a^2 \csc ^3(c+d x) \sec ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \csc (c+d x) \sec ^6(c+d x) \, dx}{a^2}+\frac {\int \csc ^3(c+d x) \sec ^6(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^2(c+d x) \sec ^6(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \frac {x^6}{-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 )^3}{x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d}+\frac {\operatorname {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \left (3+\frac {1}{x^2}+3 x^2+x^4\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac {7 \operatorname {Subst}\left (\int \frac {x^6}{-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 {\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d}-\frac {6 \tan (c+d x)}{a^2 d}-\frac {2 \tan ^3(c+d x)}{a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {7 \operatorname {Subst}\left (\int \left (1+x^2+x^4+\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 {9 \sec (c+d x)}{2 a^2 d}+\frac {3 \sec ^3(c+d x)}{2 a^2 d}+\frac {9 \sec ^5(c+d x)}{10 a^2 d}-\frac {\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d}-\frac {6 \tan (c+d x)}{a^2 d}-\frac {2 \tan ^3(c+d x)}{a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a^2 d}\\ &=-\frac {9 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {9 \sec (c+d x)}{2 a^2 d}+\frac {3 \sec ^3(c+d x)}{2 a^2 d}+\frac {9 \sec ^5(c+d x)}{10 a^2 d}-\frac {\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d}-\frac {6 \tan (c+d x)}{a^2 d}-\frac {2 \tan ^3(c+d x)}{a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [B] time = 0.73, size = 328, normalized size = 2.08 \[ -\frac {\csc ^2(c+d x) \sec (c+d x) \left (-432 \sin (c+d x)+744 \sin (2 (c+d x))-176 \sin (3 (c+d x))-372 \sin (4 (c+d x))+128 \sin (5 (c+d x))+176 \cos (2 (c+d x))-651 \cos (3 (c+d x))+332 \cos (4 (c+d x))+93 \cos (5 (c+d x))-720 \sin (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+360 \sin (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-630 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+90 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+18 \cos (c+d x) \left (-30 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+31\right )+630 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-90 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+720 \sin (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-360 \sin (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-348\right )}{320 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 260, normalized size = 1.65 \[ -\frac {166 \, \cos \left (d x + c\right )^{4} - 144 \, \cos \left (d x + c\right )^{2} + 45 \, {\left (\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{3} - 2 \, {\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 45 \, {\left (\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{3} - 2 \, {\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4 \, {\left (32 \, \cos \left (d x + c\right )^{4} - 35 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 12}{20 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 3 \, a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right ) - 2 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 187, normalized size = 1.18 \[ \frac {\frac {180 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {5 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{4}} - \frac {10}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} - \frac {5 \, {\left (54 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left (245 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 870 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 810 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 211\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 219, normalized size = 1.39 \[ -\frac {1}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2} d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2}}-\frac {1}{8 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {9 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}+\frac {4}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {5}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {11}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {49}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 354, normalized size = 2.24 \[ \frac {\frac {\frac {20 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {567 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1448 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {985 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {820 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1355 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {520 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 5}{\frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {5 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {5 \, {\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{a^{2}} + \frac {180 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.09, size = 191, normalized size = 1.21 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}+\frac {9\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^2\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {271\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{8}-\frac {41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+\frac {197\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{8}+\frac {181\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {567\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{40}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {1}{8}\right )}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]
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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