Optimal. Leaf size=81 \[ -\frac{a \csc ^9(c+d x)}{9 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{b \cot ^8(c+d x)}{8 d}-\frac{b \cot ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.131018, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2606, 270, 2607, 14} \[ -\frac{a \csc ^9(c+d x)}{9 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{b \cot ^8(c+d x)}{8 d}-\frac{b \cot ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 2834
Rule 2606
Rule 270
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \cot ^5(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^5(c+d x) \, dx+b \int \cot ^5(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}-\frac{b \operatorname{Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{b \operatorname{Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{b \cot ^6(c+d x)}{6 d}-\frac{b \cot ^8(c+d x)}{8 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^9(c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.111764, size = 88, normalized size = 1.09 \[ -\frac{a \csc ^9(c+d x)}{9 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{b \left (3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 166, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{9\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{21\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{315\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{105\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{105} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) +b \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{24\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983895, size = 95, normalized size = 1.17 \begin{align*} -\frac{630 \, b \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, b \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, b \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68691, size = 308, normalized size = 3.8 \begin{align*} -\frac{504 \, a \cos \left (d x + c\right )^{4} - 288 \, a \cos \left (d x + c\right )^{2} + 105 \,{\left (6 \, b \cos \left (d x + c\right )^{4} - 4 \, b \cos \left (d x + c\right )^{2} + b\right )} \sin \left (d x + c\right ) + 64 \, a}{2520 \,{\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 )} \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 = 1.22367, size = 95, normalized size = 1.17 \begin{align*} -\frac{630 \, b \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, b \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, b \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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