Optimal. Leaf size=73 \[ -\frac{a^3 \csc ^6(c+d x)}{6 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0719408, antiderivative size = 73, 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 = {2833, 12, 43} \[ -\frac{a^3 \csc ^6(c+d x)}{6 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^7 (a+x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \frac{(a+x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \left (\frac{a^3}{x^7}+\frac{3 a^2}{x^6}+\frac{3 a}{x^5}+\frac{1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{a^3 \csc ^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.0303022, size = 73, normalized size = 1. \[ -\frac{a^3 \csc ^6(c+d x)}{6 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 49, normalized size = 0.7 \begin{align*}{\frac{{a}^{3}}{d} \left ( -{\frac{3}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{1}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23501, size = 76, normalized size = 1.04 \begin{align*} -\frac{20 \, a^{3} \sin \left (d x + c\right )^{3} + 45 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{60 \, d \sin \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77253, size = 208, normalized size = 2.85 \begin{align*} -\frac{45 \, a^{3} \cos \left (d x + c\right )^{2} - 55 \, a^{3} + 4 \,{\left (5 \, a^{3} \cos \left (d x + c\right )^{2} - 14 \, a^{3}\right )} \sin \left (d x + c\right )}{60 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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(-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 = 1.26843, size = 76, normalized size = 1.04 \begin{align*} -\frac{20 \, a^{3} \sin \left (d x + c\right )^{3} + 45 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{60 \, d \sin \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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