Optimal. Leaf size=65 \[ -\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.115577, antiderivative size = 65, 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, 2606, 270, 2607, 30} \[ -\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2834
Rule 2606
Rule 270
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^2(c+d x) \, dx+a \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \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 \csc ^3(c+d x)}{3 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.0256111, size = 65, normalized size = 1. \[ -\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 128, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+a \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 = 1.06919, size = 95, normalized size = 1.46 \begin{align*} -\frac{105 \, a \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, a \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, a \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06936, size = 273, normalized size = 4.2 \begin{align*} \frac{70 \, a \cos \left (d x + c\right )^{4} - 56 \, a \cos \left (d x + c\right )^{2} + 35 \,{\left (3 \, a \cos \left (d x + c\right )^{4} - 3 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 16 \, a}{210 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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 = 1.2966, size = 95, normalized size = 1.46 \begin{align*} -\frac{105 \, a \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, a \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, a \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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