Optimal. Leaf size=98 \[ -\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.165285, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2838, 2611, 3768, 3770, 2607, 30} \[ -\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+b \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{1}{2} a \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{b \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{b \cot ^5(c+d x)}{5 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{1}{8} a \int \csc ^3(c+d x) \, dx\\ &=-\frac{b \cot ^5(c+d x)}{5 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{1}{16} a \int \csc (c+d x) \, dx\\ &=-\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{b \cot ^5(c+d x)}{5 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.0426912, size = 175, normalized size = 1.79 \[ -\frac{a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{b \cot ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 138, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{\cos \left ( dx+c \right ) a}{16\,d}}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14976, size = 143, normalized size = 1.46 \begin{align*} \frac{5 \, a{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{96 \, b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81841, size = 497, normalized size = 5.07 \begin{align*} \frac{96 \, b \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + 30 \, a \cos \left (d x + c\right )^{5} + 80 \, a \cos \left (d x + c\right )^{3} - 30 \, a \cos \left (d x + c\right ) - 15 \,{\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 15 \,{\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{480 \,{\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 )}} \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 [B] time = 1.42804, size = 271, normalized size = 2.77 \begin{align*} \frac{5 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 12 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 60 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 120 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 120 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{294 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 120 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 60 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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