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 = 0.178634, antiderivative size = 142, normalized size of antiderivative = 1., 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 2839
Rule 2611
Rule 3770
Rule 3473
Rule 8
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*}
Mathematica [B] time = 0.944289, 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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Maple [B] time = 0.156, size = 264, normalized size = 1.9 \begin{align*}{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{3}{128\,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{15}{128\,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{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{3}{128\,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{15}{128\,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{5}{16\,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 = 1.53579, size = 402, normalized size = 2.83 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2034, size = 660, normalized size = 4.65 \begin{align*} \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 )}} \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 = 1.36038, size = 302, normalized size = 2.13 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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