Optimal. Leaf size=74 \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d} \]
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Rubi [A] time = 0.107246, antiderivative size = 74, 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 = {2834, 2607, 30, 2606, 194} \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2834
Rule 2607
Rule 30
Rule 2606
Rule 194
Rubi steps
\begin{align*} \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^7(c+d x) \csc (c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^7 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot ^8(c+d x)}{8 d}+\frac{a \csc (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.0328331, size = 74, normalized size = 1. \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.068, size = 138, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{7\,\sin \left ( dx+c \right ) }}+{\frac{\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04174, size = 124, normalized size = 1.68 \begin{align*} \frac{280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, 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 = 1.13352, size = 351, normalized size = 4.74 \begin{align*} -\frac{140 \, a \cos \left (d x + c\right )^{6} - 210 \, a \cos \left (d x + c\right )^{4} + 140 \, a \cos \left (d x + c\right )^{2} + 8 \,{\left (35 \, a \cos \left (d x + c\right )^{6} - 70 \, a \cos \left (d x + c\right )^{4} + 56 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 35 \, a}{280 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, 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 = 1.35942, size = 124, normalized size = 1.68 \begin{align*} \frac{280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, d \sin \left (d x + c\right )^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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