Optimal. Leaf size=73 \[ -\frac{\csc ^6(c+d x)}{6 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.0571898, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 75} \[ -\frac{\csc ^6(c+d x)}{6 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 75
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^4}{x^7}-\frac{2 a^3}{x^6}+\frac{2 a}{x^4}-\frac{1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\csc ^2(c+d x)}{2 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^6(c+d x)}{6 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.07344, size = 73, normalized size = 1. \[ -\frac{\csc ^6(c+d x)}{6 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.108, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{2}} \left ({\frac{2}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{2}{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07194, size = 62, normalized size = 0.85 \begin{align*} \frac{15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 12 \, \sin \left (d x + c\right ) - 5}{30 \, a^{2} 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 = 1.40071, size = 234, normalized size = 3.21 \begin{align*} -\frac{15 \, \cos \left (d x + c\right )^{4} - 30 \, \cos \left (d x + c\right )^{2} + 4 \,{\left (5 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 10}{30 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{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 = 2.36406, size = 62, normalized size = 0.85 \begin{align*} \frac{15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 12 \, \sin \left (d x + c\right ) - 5}{30 \, a^{2} d \sin \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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