Optimal. Leaf size=124 \[ -\frac {\cot ^7(c+d x)}{7 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {\tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d} \]
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Rubi [A] time = 0.26, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2709, 3767, 8, 3768, 3770} \[ -\frac {\cot ^7(c+d x)}{7 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {\tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2709
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \left (a^6 \csc ^2(c+d x)-2 a^6 \csc ^3(c+d x)-a^6 \csc ^4(c+d x)+4 a^6 \csc ^5(c+d x)-a^6 \csc ^6(c+d x)-2 a^6 \csc ^7(c+d x)+a^6 \csc ^8(c+d x)\right ) \, dx}{a^8}\\ &=\frac {\int \csc ^2(c+d x) \, dx}{a^2}-\frac {\int \csc ^4(c+d x) \, dx}{a^2}-\frac {\int \csc ^6(c+d x) \, dx}{a^2}+\frac {\int \csc ^8(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^3(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^7(c+d x) \, dx}{a^2}+\frac {4 \int \csc ^5(c+d x) \, dx}{a^2}\\ &=\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {\int \csc (c+d x) \, dx}{a^2}-\frac {5 \int \csc ^5(c+d x) \, dx}{3 a^2}+\frac {3 \int \csc ^3(c+d x) \, dx}{a^2}-\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {5 \int \csc ^3(c+d x) \, dx}{4 a^2}+\frac {3 \int \csc (c+d x) \, dx}{2 a^2}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac {5 \int \csc (c+d x) \, dx}{8 a^2}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] time = 1.11, size = 251, normalized size = 2.02 \[ -\frac {\csc ^7(c+d x) \left (-2170 \sin (2 (c+d x))-3080 \sin (4 (c+d x))-210 \sin (6 (c+d x))+5880 \cos (c+d x)+2184 \cos (3 (c+d x))-168 \cos (5 (c+d x))-216 \cos (7 (c+d x))+3675 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2205 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+735 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3675 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2205 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-735 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 \sin (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{53760 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 216, normalized size = 1.74 \[ -\frac {432 \, \cos \left (d x + c\right )^{7} - 672 \, \cos \left (d x + c\right )^{5} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 245, normalized size = 1.98 \[ -\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {4356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} - \frac {15 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 70 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 525 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 210 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1155 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{14}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.74, size = 284, normalized size = 2.29 \[ \frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{896 d \,a^{2}}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d \,a^{2}}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 a^{2} d}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{2} d}-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2}}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{2} d}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{2}}+\frac {1}{192 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {11}{128 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}-\frac {3}{640 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{896 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{64 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{64 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {5}{128 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 314, normalized size = 2.53 \[ \frac {\frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {210 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {525 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {210 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {70 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2}} - \frac {1680 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {70 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {210 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {525 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {210 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1155 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{2} \sin \left (d x + c\right )^{7}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.84, size = 387, normalized size = 3.12 \[ -\frac {15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+70\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-70\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+210\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1680\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{13440\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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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