Optimal. Leaf size=81 \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{b \csc ^7(c+d x)}{7 d}+\frac{2 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.129311, 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, 2607, 14, 2606, 270} \[ -\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{b \csc ^7(c+d x)}{7 d}+\frac{2 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2834
Rule 2607
Rule 14
Rule 2606
Rule 270
Rubi steps
\begin{align*} \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^4(c+d x) \, dx+b \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac{b \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac{b \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.0997403, size = 88, normalized size = 1.09 \[ -\frac{a \left (3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right )}{24 d}-\frac{b \csc ^7(c+d x)}{7 d}+\frac{2 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.067, size = 148, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ( a \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 ) +b \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{105\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{35\,\sin \left ( dx+c \right ) }}-{\frac{\sin \left ( dx+c \right ) }{35} \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972255, size = 95, normalized size = 1.17 \begin{align*} -\frac{280 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, b \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, b \sin \left (d x + c\right ) + 105 \, a}{840 \, 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 = 1.60943, size = 289, normalized size = 3.57 \begin{align*} -\frac{210 \, a \cos \left (d x + c\right )^{4} - 140 \, a \cos \left (d x + c\right )^{2} + 8 \,{\left (35 \, b \cos \left (d x + c\right )^{4} - 28 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) + 35 \, a}{840 \,{\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 = 1.28645, size = 95, normalized size = 1.17 \begin{align*} -\frac{280 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, b \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, b \sin \left (d x + c\right ) + 105 \, a}{840 \, d \sin \left (d x + c\right )^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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