Optimal. Leaf size=91 \[ -\frac{\cot ^8(c+d x)}{8 a d}-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{2 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.160092, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2835, 2607, 14, 2606, 270} \[ -\frac{\cot ^8(c+d x)}{8 a d}-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{2 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2607
Rule 14
Rule 2606
Rule 270
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^5(c+d x) \csc ^3(c+d x) \, dx}{a}+\frac{\int \cot ^5(c+d x) \csc ^4(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{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{\operatorname{Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{\cot ^6(c+d x)}{6 a d}-\frac{\cot ^8(c+d x)}{8 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 = 0.140122, size = 68, normalized size = 0.75 \[ \frac{\csc ^3(c+d x) \left (-105 \csc ^5(c+d x)+120 \csc ^4(c+d x)+280 \csc ^3(c+d x)-336 \csc ^2(c+d x)-210 \csc (c+d x)+280\right )}{840 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.167, size = 69, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ({\frac{1}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{2}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{3\, \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 = 1.03196, size = 89, normalized size = 0.98 \begin{align*} \frac{280 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} + 120 \, \sin \left (d x + c\right ) - 105}{840 \, a d \sin \left (d x + c\right )^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20344, size = 286, normalized size = 3.14 \begin{align*} -\frac{210 \, \cos \left (d x + c\right )^{4} - 140 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (35 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 35}{840 \,{\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a 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 = 1.32088, size = 89, normalized size = 0.98 \begin{align*} \frac{280 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 336 \, \sin \left (d x + c\right )^{3} + 280 \, \sin \left (d x + c\right )^{2} + 120 \, \sin \left (d x + c\right ) - 105}{840 \, a d \sin \left (d x + c\right )^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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