Optimal. Leaf size=99 \[ -\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^4(c+d x)}{4 a d}+\frac{2 \sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.10026, antiderivative size = 99, 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} \[ -\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^4(c+d x)}{4 a d}+\frac{2 \sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a (a-x)^3 (a+x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^4+\frac{a^5}{x}-2 a^3 x+2 a^2 x^2+a x^3-x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{\log (\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{a d}+\frac{2 \sin ^3(c+d x)}{3 a d}+\frac{\sin ^4(c+d x)}{4 a d}-\frac{\sin ^5(c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.0722493, size = 68, normalized size = 0.69 \[ \frac{-12 \sin ^5(c+d x)+15 \sin ^4(c+d x)+40 \sin ^3(c+d x)-60 \sin ^2(c+d x)-60 \sin (c+d x)+60 \log (\sin (c+d x))}{60 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.101, size = 94, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{\sin \left ( dx+c \right ) }{da}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{da}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,da}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,da}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03304, size = 96, normalized size = 0.97 \begin{align*} -\frac{\frac{12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 40 \, \sin \left (d x + c\right )^{3} + 60 \, \sin \left (d x + c\right )^{2} + 60 \, \sin \left (d x + c\right )}{a} - \frac{60 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16531, size = 186, normalized size = 1.88 \begin{align*} \frac{15 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 60 \, \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right )}{60 \, a d} \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 = 1.25604, size = 119, normalized size = 1.2 \begin{align*} \frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{12 \, a^{4} \sin \left (d x + c\right )^{5} - 15 \, a^{4} \sin \left (d x + c\right )^{4} - 40 \, a^{4} \sin \left (d x + c\right )^{3} + 60 \, a^{4} \sin \left (d x + c\right )^{2} + 60 \, a^{4} \sin \left (d x + c\right )}{a^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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