Optimal. Leaf size=81 \[ \frac{a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.048033, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 88} \[ \frac{a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{a^5}{x^5}+\frac{a^4}{x^4}-\frac{2 a^3}{x^3}-\frac{2 a^2}{x^2}+\frac{a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{2 a \csc (c+d x)}{d}+\frac{a \csc ^2(c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \log (\sin (c+d x))}{d}+\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.197611, size = 87, normalized size = 1.07 \[ \frac{a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 136, normalized size = 1.7 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}+{\frac{8\,a\sin \left ( dx+c \right ) }{3\,d}}+{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{d}}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{3\,d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08056, size = 93, normalized size = 1.15 \begin{align*} \frac{12 \, a \log \left (\sin \left (d x + c\right )\right ) + 12 \, a \sin \left (d x + c\right ) + \frac{24 \, a \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 4 \, a \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57272, size = 292, normalized size = 3.6 \begin{align*} -\frac{12 \, a \cos \left (d x + c\right )^{2} - 12 \,{\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 ) - 4 \,{\left (3 \, a \cos \left (d x + c\right )^{4} - 12 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) - 9 \, a}{12 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \sin{\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int \cot ^{5}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38853, size = 111, normalized size = 1.37 \begin{align*} \frac{12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 12 \, a \sin \left (d x + c\right ) - \frac{25 \, a \sin \left (d x + c\right )^{4} - 24 \, a \sin \left (d x + c\right )^{3} - 12 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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