Optimal. Leaf size=109 \[ -\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{\csc ^9(c+d x)}{9 a d}+\frac{\csc ^8(c+d x)}{4 a d}-\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d} \]
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Rubi [A] time = 0.121184, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{\csc ^9(c+d x)}{9 a d}+\frac{\csc ^8(c+d x)}{4 a d}-\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^{11} (a-x)^3 (a+x)^2}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^2}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{11}}-\frac{a^4}{x^{10}}-\frac{2 a^3}{x^9}+\frac{2 a^2}{x^8}+\frac{a}{x^7}-\frac{1}{x^6}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\csc ^5(c+d x)}{5 a d}-\frac{\csc ^6(c+d x)}{6 a d}-\frac{2 \csc ^7(c+d x)}{7 a d}+\frac{\csc ^8(c+d x)}{4 a d}+\frac{\csc ^9(c+d x)}{9 a d}-\frac{\csc ^{10}(c+d x)}{10 a d}\\ \end{align*}
Mathematica [A] time = 0.106386, size = 68, normalized size = 0.62 \[ \frac{\csc ^5(c+d x) \left (-126 \csc ^5(c+d x)+140 \csc ^4(c+d x)+315 \csc ^3(c+d x)-360 \csc ^2(c+d x)-210 \csc (c+d x)+252\right )}{1260 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 69, normalized size = 0.6 \begin{align*}{\frac{1}{da} \left ( -{\frac{1}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{2}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}+{\frac{1}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995203, size = 89, normalized size = 0.82 \begin{align*} \frac{252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a d \sin \left (d x + c\right )^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13609, size = 321, normalized size = 2.94 \begin{align*} \frac{210 \, \cos \left (d x + c\right )^{4} - 105 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (63 \, \cos \left (d x + c\right )^{4} - 36 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 21}{1260 \,{\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, 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 = 1.41387, size = 89, normalized size = 0.82 \begin{align*} \frac{252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a d \sin \left (d x + c\right )^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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