Optimal. Leaf size=142 \[ \frac {\cot ^5(c+d x)}{5 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x)}{a d}+\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac {x}{a} \]
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Rubi [A] time = 0.18, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2839, 2611, 3770, 3473, 8} \[ \frac {\cot ^5(c+d x)}{5 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x)}{a d}+\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2611
Rule 2839
Rule 3473
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^6(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc (c+d x) \, dx}{a}\\ &=\frac {\cot ^5(c+d x)}{5 a d}-\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 {\int \cot ^4(c+d x) \, dx}{a}\\ &=-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 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 {\int \cot ^2(c+d x) \, dx}{a}\\ &=\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 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 {\int 1 \, dx}{a}\\ &=\frac {x}{a}+\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 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] time = 1.02, size = 317, normalized size = 2.23 \[ -\frac {\csc ^6(c+d x) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-1200 \sin (2 (c+d x))+768 \sin (4 (c+d x))-368 \sin (6 (c+d x))-1440 c \cos (4 (c+d x))+240 c \cos (6 (c+d x))+900 \cos (c+d x)+50 \cos (3 (c+d x))-1440 d x \cos (4 (c+d x))+330 \cos (5 (c+d x))+240 d x \cos (6 (c+d x))+750 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-450 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+75 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-750 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+225 \cos (2 (c+d x)) \left (16 (c+d x)-5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+450 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-75 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2400 c-2400 d x\right )}{7680 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 236, normalized size = 1.66 \[ \frac {480 \, d x \cos \left (d x + c\right )^{6} - 1440 \, d x \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{5} + 1440 \, d x \cos \left (d x + c\right )^{2} - 400 \, \cos \left (d x + c\right )^{3} - 480 \, d x + 75 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 75 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (23 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 150 \, \cos \left (d x + c\right )}{480 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 224, normalized size = 1.58 \[ \frac {\frac {1920 \, {\left (d x + c\right )}}{a} - \frac {600 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1320 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} + \frac {1470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.55, size = 264, normalized size = 1.86 \[ \frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 a d}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 a d}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a d}+\frac {15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {1}{384 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {1}{160 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {7}{96 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {15}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {11}{16 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 298, normalized size = 2.10 \[ -\frac {\frac {\frac {1320 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {12 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a} - \frac {3840 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {600 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {225 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1320 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a \sin \left (d x + c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.39, size = 413, normalized size = 2.91 \[ -\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+12\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-225\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-1320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+225\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3840\,\mathrm {atan}\left (\frac {16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+600\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]
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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