Optimal. Leaf size=115 \[ -\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}+\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}+\frac{3 a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0815798, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2836, 12, 88} \[ -\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}+\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}+\frac{3 a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \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)^4}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^4}{x^6} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a+\frac{a^7}{x^6}+\frac{a^6}{x^5}-\frac{3 a^5}{x^4}-\frac{3 a^4}{x^3}+\frac{3 a^3}{x^2}+\frac{3 a^2}{x}-x\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{3 a \csc (c+d x)}{d}+\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \log (\sin (c+d x))}{d}-\frac{a \sin (c+d x)}{d}-\frac{a \sin ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.185646, size = 102, normalized size = 0.89 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc (c+d x)}{d}+\frac{a \left (-2 \sin ^2(c+d x)-\csc ^4(c+d x)+6 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 239, normalized size = 2.1 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2\,d}}+{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{d\sin \left ( dx+c \right ) }}-{\frac{16\,a\sin \left ( dx+c \right ) }{5\,d}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}a}{d}}-{\frac{6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{5\,d}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{5\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04542, size = 123, normalized size = 1.07 \begin{align*} -\frac{10 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left (\sin \left (d x + c\right )\right ) + 20 \, a \sin \left (d x + c\right ) + \frac{60 \, a \sin \left (d x + c\right )^{4} - 30 \, a \sin \left (d x + c\right )^{3} - 20 \, a \sin \left (d x + c\right )^{2} + 5 \, a \sin \left (d x + c\right ) + 4 \, a}{\sin \left (d x + c\right )^{5}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38546, size = 419, normalized size = 3.64 \begin{align*} \frac{20 \, a \cos \left (d x + c\right )^{6} - 120 \, a \cos \left (d x + c\right )^{4} + 160 \, a \cos \left (d x + c\right )^{2} + 60 \,{\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 5 \,{\left (2 \, a \cos \left (d x + c\right )^{6} - 5 \, a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + 4 \, a\right )} \sin \left (d x + c\right ) - 64 \, a}{20 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + 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 = 1.18585, size = 139, normalized size = 1.21 \begin{align*} -\frac{10 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 20 \, a \sin \left (d x + c\right ) + \frac{137 \, a \sin \left (d x + c\right )^{5} + 60 \, a \sin \left (d x + c\right )^{4} - 30 \, a \sin \left (d x + c\right )^{3} - 20 \, a \sin \left (d x + c\right )^{2} + 5 \, a \sin \left (d x + c\right ) + 4 \, a}{\sin \left (d x + c\right )^{5}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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