Optimal. Leaf size=100 \[ -\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^4(c+d x)}{4 a d}+\frac{2 \csc ^3(c+d x)}{3 a d}-\frac{\csc ^2(c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.104356, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^4(c+d x)}{4 a d}+\frac{2 \csc ^3(c+d x)}{3 a d}-\frac{\csc ^2(c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^6 (a-x)^3 (a+x)^2}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^2}{x^6} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}-\frac{a^4}{x^5}-\frac{2 a^3}{x^4}+\frac{2 a^2}{x^3}+\frac{a}{x^2}-\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{a d}+\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc ^4(c+d x)}{4 a d}-\frac{\csc ^5(c+d x)}{5 a d}-\frac{\log (\sin (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.102729, size = 68, normalized size = 0.68 \[ -\frac{12 \csc ^5(c+d x)-15 \csc ^4(c+d x)-40 \csc ^3(c+d x)+60 \csc ^2(c+d x)+60 \csc (c+d x)+60 \log (\sin (c+d x))}{60 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.144, size = 97, normalized size = 1. \begin{align*} -{\frac{1}{da\sin \left ( dx+c \right ) }}-{\frac{1}{5\,da \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{4\,da \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}+{\frac{2}{3\,da \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04601, size = 95, normalized size = 0.95 \begin{align*} -\frac{\frac{60 \, \log \left (\sin \left (d x + c\right )\right )}{a} + \frac{60 \, \sin \left (d x + c\right )^{4} + 60 \, \sin \left (d x + c\right )^{3} - 40 \, \sin \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) + 12}{a \sin \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15792, size = 321, normalized size = 3.21 \begin{align*} -\frac{60 \, \cos \left (d x + c\right )^{4} + 60 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 80 \, \cos \left (d x + c\right )^{2} - 15 \,{\left (4 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 32}{60 \,{\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 = 1.295, size = 111, normalized size = 1.11 \begin{align*} -\frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{137 \, \sin \left (d x + c\right )^{5} - 60 \, \sin \left (d x + c\right )^{4} - 60 \, \sin \left (d x + c\right )^{3} + 40 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) - 12}{a \sin \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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