Optimal. Leaf size=106 \[ -\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}-\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d} \]
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Rubi [A]
time = 0.10, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2785, 2687, 30,
2691, 3855} \begin {gather*} -\frac {\cot ^7(c+d x)}{7 a d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2687
Rule 2691
Rule 2785
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^6(c+d x) \csc (c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac {5 \int \cot ^4(c+d x) \csc (c+d x) \, dx}{6 a}+\frac {\text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}-\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac {5 \int \cot ^2(c+d x) \csc (c+d x) \, dx}{8 a}\\ &=-\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}-\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac {5 \int \csc (c+d x) \, dx}{16 a}\\ &=-\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}-\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(284\) vs. \(2(106)=212\).
time = 0.63, size = 284, normalized size = 2.68 \begin {gather*} -\frac {\csc ^5(c+d x) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))+3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-1190 \sin (2 (c+d x))-2205 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+2205 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+392 \sin (4 (c+d x))+735 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-735 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-462 \sin (6 (c+d x))-105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{86016 a d (1+\sin (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(199\) vs.
\(2(96)=192\).
time = 0.25, size = 200, normalized size = 1.89
method | result | size |
risch | \(-\frac {-336 i {\mathrm e}^{12 i \left (d x +c \right )}+231 \,{\mathrm e}^{13 i \left (d x +c \right )}-196 \,{\mathrm e}^{11 i \left (d x +c \right )}-1680 i {\mathrm e}^{8 i \left (d x +c \right )}+595 \,{\mathrm e}^{9 i \left (d x +c \right )}-1008 i {\mathrm e}^{4 i \left (d x +c \right )}-595 \,{\mathrm e}^{5 i \left (d x +c \right )}+196 \,{\mathrm e}^{3 i \left (d x +c \right )}-48 i-231 \,{\mathrm e}^{i \left (d x +c \right )}}{168 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d a}\) | \(168\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {15}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}\) | \(200\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {15}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 315 vs.
\(2 (96) = 192\).
time = 0.31, size = 315, normalized size = 2.97 \begin {gather*} -\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {315 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {63 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {63 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {7 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {63 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {63 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a \sin \left (d x + c\right )^{7}}}{2688 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs.
\(2 (96) = 192\).
time = 0.37, size = 198, normalized size = 1.87 \begin {gather*} \frac {96 \, \cos \left (d x + c\right )^{7} - 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 ) - 14 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cot ^{8}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs.
\(2 (96) = 192\).
time = 12.65, size = 244, normalized size = 2.30 \begin {gather*} \frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{7}} - \frac {2178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{2688 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.22, size = 387, normalized size = 3.65 \begin {gather*} \frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\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}{2688\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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