Optimal. Leaf size=73 \[ \frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^7(c+d x)}{7 a d}+\frac{2 \csc ^5(c+d x)}{5 a d}-\frac{\csc ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.138536, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2835, 2606, 270, 2607, 30} \[ \frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^7(c+d x)}{7 a d}+\frac{2 \csc ^5(c+d x)}{5 a d}-\frac{\csc ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2606
Rule 270
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^5(c+d x) \csc ^2(c+d x) \, dx}{a}+\frac{\int \cot ^5(c+d x) \csc ^3(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=\frac{\cot ^6(c+d x)}{6 a d}-\frac{\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a d}\\ &=\frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{2 \csc ^5(c+d x)}{5 a d}-\frac{\csc ^7(c+d x)}{7 a d}\\ \end{align*}
Mathematica [A] time = 0.151838, size = 68, normalized size = 0.93 \[ \frac{\csc ^2(c+d x) \left (-30 \csc ^5(c+d x)+35 \csc ^4(c+d x)+84 \csc ^3(c+d x)-105 \csc ^2(c+d x)-70 \csc (c+d x)+105\right )}{210 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 69, normalized size = 1. \begin{align*}{\frac{1}{da} \left ( -{\frac{1}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{2}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{2\, \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}}}+{\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 = 1.01652, size = 89, normalized size = 1.22 \begin{align*} \frac{105 \, \sin \left (d x + c\right )^{5} - 70 \, \sin \left (d x + c\right )^{4} - 105 \, \sin \left (d x + c\right )^{3} + 84 \, \sin \left (d x + c\right )^{2} + 35 \, \sin \left (d x + c\right ) - 30}{210 \, a d \sin \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13911, size = 270, normalized size = 3.7 \begin{align*} \frac{70 \, \cos \left (d x + c\right )^{4} - 56 \, \cos \left (d x + c\right )^{2} - 35 \,{\left (3 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 16}{210 \,{\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a 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 = 1.32596, size = 89, normalized size = 1.22 \begin{align*} \frac{105 \, \sin \left (d x + c\right )^{5} - 70 \, \sin \left (d x + c\right )^{4} - 105 \, \sin \left (d x + c\right )^{3} + 84 \, \sin \left (d x + c\right )^{2} + 35 \, \sin \left (d x + c\right ) - 30}{210 \, a d \sin \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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