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 = 0.151258, antiderivative size = 138, normalized size of antiderivative = 1., 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 2839
Rule 3473
Rule 8
Rule 2592
Rule 288
Rule 321
Rule 206
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*}
Mathematica [A] time = 0.994794, 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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Maple [A] time = 0.154, size = 249, normalized size = 1.8 \begin{align*}{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{7}{96\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{11}{16\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{1}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{1}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{7}{96\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{11}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{15}{8\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52293, size = 431, normalized size = 3.12 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16632, size = 593, normalized size = 4.3 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38203, size = 293, normalized size = 2.12 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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