Optimal. Leaf size=138 \[ -\frac {15 \cos (c+d x)}{8 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {x}{a} \]
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Rubi [A] time = 0.15, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2839, 3473, 8, 2592, 288, 321, 206} \[ -\frac {15 \cos (c+d x)}{8 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 288
Rule 321
Rule 2592
Rule 2839
Rule 3473
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cos (c+d x) \cot ^5(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \, dx}{a}\\ &=-\frac {\cot ^5(c+d x)}{5 a d}-\frac {\int \cot ^4(c+d x) \, dx}{a}+\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {\int \cot ^2(c+d x) \, dx}{a}-\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 a d}\\ &=-\frac {\cot (c+d x)}{a d}-\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {\cot ^5(c+d x)}{5 a d}-\frac {\int 1 \, dx}{a}+\frac {15 \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 a d}\\ &=-\frac {x}{a}-\frac {15 \cos (c+d x)}{8 a d}-\frac {\cot (c+d x)}{a d}-\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {\cot ^5(c+d x)}{5 a d}+\frac {15 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 a d}\\ &=-\frac {x}{a}+\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {15 \cos (c+d x)}{8 a d}-\frac {\cot (c+d x)}{a d}-\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {\cot ^5(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 264, normalized size = 1.91 \[ -\frac {\csc ^5(c+d x) \left (1200 c \sin (c+d x)+1200 d x \sin (c+d x)+600 \sin (2 (c+d x))-600 c \sin (3 (c+d x))-600 d x \sin (3 (c+d x))-510 \sin (4 (c+d x))+120 c \sin (5 (c+d x))+120 d x \sin (5 (c+d x))+60 \sin (6 (c+d x))+400 \cos (c+d x)-200 \cos (3 (c+d x))+184 \cos (5 (c+d x))+2250 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1125 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+225 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2250 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1125 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-225 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{1920 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 211, normalized size = 1.53 \[ -\frac {368 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} - 225 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 225 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (8 \, d x \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{5} - 16 \, d x \cos \left (d x + c\right )^{2} - 25 \, \cos \left (d x + c\right )^{3} + 8 \, d x + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 240 \, \cos \left (d x + c\right )}{240 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 217, normalized size = 1.57 \[ -\frac {\frac {960 \, {\left (d x + c\right )}}{a} + \frac {1800 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {1920}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a} - \frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} - \frac {4110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 249, normalized size = 1.80 \[ \frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 a d}-\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a d}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}-\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {1}{160 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{64 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {7}{96 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{4 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {11}{16 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 319, normalized size = 2.31 \[ \frac {\frac {\frac {660 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {70 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a} + \frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {64 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {225 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {590 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2160 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {660 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 6}{\frac {a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac {1920 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {1800 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.03, size = 279, normalized size = 2.02 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a\,d}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a\,d}+\frac {2\,\mathrm {atan}\left (\frac {15\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {15}{2}\right )}+\frac {4}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {15}{2}}\right )}{a\,d}-\frac {15\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a\,d}-\frac {22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+72\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {59\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {1}{5}}{d\,\left (32\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a\,d} \]
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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