Optimal. Leaf size=115 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0594132, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 88} \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rubi steps
\begin{align*} \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^4}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+\frac{a^7}{x^7}+\frac{a^6}{x^6}-\frac{3 a^5}{x^5}-\frac{3 a^4}{x^4}+\frac{3 a^3}{x^3}+\frac{3 a^2}{x^2}-\frac{a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{3 a \csc (c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{a \csc ^3(c+d x)}{d}+\frac{3 a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \log (\sin (c+d x))}{d}-\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.398755, size = 111, normalized size = 0.97 \[ -\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \left (2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 195, normalized size = 1.7 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{d\sin \left ( dx+c \right ) }}-{\frac{16\,a\sin \left ( dx+c \right ) }{5\,d}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}a}{d}}-{\frac{6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{5\,d}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{5\,d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15296, size = 123, normalized size = 1.07 \begin{align*} -\frac{60 \, a \log \left (\sin \left (d x + c\right )\right ) + 60 \, a \sin \left (d x + c\right ) + \frac{180 \, a \sin \left (d x + c\right )^{5} + 90 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 45 \, a \sin \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) + 10 \, a}{\sin \left (d x + c\right )^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59933, size = 412, normalized size = 3.58 \begin{align*} \frac{90 \, a \cos \left (d x + c\right )^{4} - 135 \, a \cos \left (d x + c\right )^{2} - 60 \,{\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} \, \sin \left (d x + c\right )\right ) - 12 \,{\left (5 \, a \cos \left (d x + c\right )^{6} - 30 \, a \cos \left (d x + c\right )^{4} + 40 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) + 55 \, a}{60 \,{\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 [A] time = 1.38698, size = 140, normalized size = 1.22 \begin{align*} -\frac{60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a \sin \left (d x + c\right ) - \frac{147 \, a \sin \left (d x + c\right )^{6} - 180 \, a \sin \left (d x + c\right )^{5} - 90 \, a \sin \left (d x + c\right )^{4} + 60 \, a \sin \left (d x + c\right )^{3} + 45 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right ) - 10 \, a}{\sin \left (d x + c\right )^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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