Optimal. Leaf size=96 \[ -\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d} \]
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Rubi [A] time = 0.140451, antiderivative size = 96, 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 = {2838, 2607, 30, 2611, 3770} \[ -\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2607
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^6(c+d x) \csc (c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{1}{6} (5 a) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{1}{8} (5 a) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a \cot ^7(c+d x)}{7 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{1}{16} (5 a) \int \csc (c+d x) \, dx\\ &=\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^7(c+d x)}{7 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.0461955, size = 175, normalized size = 1.82 \[ -\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}-\frac{11 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{11 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}+\frac{5 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{16 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 152, normalized size = 1.6 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}-{\frac{5\,a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}-{\frac{5\,\cos \left ( dx+c \right ) a}{16\,d}}-{\frac{5\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02275, size = 143, normalized size = 1.49 \begin{align*} \frac{7 \, a{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \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} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{96 \, a}{\tan \left (d x + c\right )^{7}}}{672 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26202, size = 562, normalized size = 5.85 \begin{align*} \frac{96 \, a \cos \left (d x + c\right )^{7} + 105 \,{\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 ) \sin \left (d x + c\right ) - 105 \,{\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 ) \sin \left (d x + c\right ) + 14 \,{\left (33 \, a \cos \left (d x + c\right )^{5} - 40 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \,{\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 [B] time = 1.18765, size = 308, normalized size = 3.21 \begin{align*} \frac{3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 7 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 21 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 840 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 105 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{2178 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 105 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 315 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 21 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{2688 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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