Optimal. Leaf size=166 \[ \frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d} \]
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Rubi [A] time = 0.41, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2875, 2873, 2607, 14, 2611, 3768, 3770, 270} \[ \frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2607
Rule 2611
Rule 2873
Rule 2875
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \cot ^2(c+d x) \csc ^7(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \cot ^2(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^5(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^6(c+d x)+a^3 \cot ^2(c+d x) \csc ^7(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3}+\frac {\int \cot ^2(c+d x) \csc ^7(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a^3}\\ &=-\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {\int \csc ^7(c+d x) \, dx}{8 a^3}-\frac {\int \csc ^5(c+d x) \, dx}{2 a^3}-\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {5 \int \csc ^5(c+d x) \, dx}{48 a^3}-\frac {3 \int \csc ^3(c+d x) \, dx}{8 a^3}-\frac {\operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {5 \int \csc ^3(c+d x) \, dx}{64 a^3}-\frac {3 \int \csc (c+d x) \, dx}{16 a^3}\\ &=\frac {3 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac {5 \int \csc (c+d x) \, dx}{128 a^3}\\ &=\frac {29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \cot ^5(c+d x)}{5 a^3 d}+\frac {3 \cot ^7(c+d x)}{7 a^3 d}+\frac {29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}+\frac {29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac {23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}\\ \end {align*}
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Mathematica [A] time = 5.51, size = 317, normalized size = 1.91 \[ -\frac {\sin ^7(c+d x) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (15 (7 \csc (c+d x)-24) \csc ^8\left (\frac {1}{2} (c+d x)\right )+4 (455 \csc (c+d x)-276) \csc ^6\left (\frac {1}{2} (c+d x)\right )+(1328-210 \csc (c+d x)) \csc ^4\left (\frac {1}{2} (c+d x)\right )-4 (3045 \csc (c+d x)-4864) \csc ^2\left (\frac {1}{2} (c+d x)\right )-8 \left (\frac {1}{4} (4616 \cos (c+d x)+1907 \cos (2 (c+d x))+304 \cos (3 (c+d x))+2833) \sec ^8\left (\frac {1}{2} (c+d x)\right )+3360 \sin ^8\left (\frac {1}{2} (c+d x)\right ) \csc ^9(c+d x)+14560 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^7(c+d x)-420 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)-6090 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+6090 \csc (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )\right )}{13762560 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 249, normalized size = 1.50 \[ -\frac {6090 \, \cos \left (d x + c\right )^{7} - 22330 \, \cos \left (d x + c\right )^{5} + 13510 \, \cos \left (d x + c\right )^{3} - 3045 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3045 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 256 \, {\left (38 \, \cos \left (d x + c\right )^{7} - 133 \, \cos \left (d x + c\right )^{5} + 140 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) + 6090 \, \cos \left (d x + c\right )}{26880 \, {\left (a^{3} d \cos \left (d x + c\right )^{8} - 4 \, a^{3} d \cos \left (d x + c\right )^{6} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 274, normalized size = 1.65 \[ -\frac {\frac {48720 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {132414 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 38640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3920 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4368 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}} - \frac {105 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 720 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2240 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4368 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5880 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3920 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 38640 \, a^{21} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{24}}}{215040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.76, size = 322, normalized size = 1.94 \[ \frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2048 d \,a^{3}}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{896 d \,a^{3}}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{96 d \,a^{3}}-\frac {13 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 d \,a^{3}}+\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 d \,a^{3}}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d \,a^{3}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{3} d}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{3}}-\frac {1}{96 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {23}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {29 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{3}}+\frac {13}{640 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{896 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{32 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{2048 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {7}{256 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {7}{384 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 354, normalized size = 2.13 \[ \frac {\frac {\frac {38640 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6720 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3920 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5880 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4368 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2240 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {105 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{a^{3}} - \frac {48720 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {720 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4368 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5880 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3920 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {38640 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 105\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{8}}{a^{3} \sin \left (d x + c\right )^{8}}}{215040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.50, size = 435, normalized size = 2.62 \[ -\frac {105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+720\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4368\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-5880\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+3920\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-38640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+38640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3920\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+5880\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4368\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+48720\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{215040\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]
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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