Optimal. Leaf size=131 \[ \frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}+\frac{5 a^3 \csc (c+d x)}{d}-\frac{5 a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.070055, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 88} \[ \frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}+\frac{5 a^3 \csc (c+d x)}{d}-\frac{5 a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^5}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2+\frac{a^7}{x^5}+\frac{3 a^6}{x^4}+\frac{a^5}{x^3}-\frac{5 a^4}{x^2}-\frac{5 a^3}{x}+3 a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{5 a^3 \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{a^3 \csc ^3(c+d x)}{d}-\frac{a^3 \csc ^4(c+d x)}{4 d}-\frac{5 a^3 \log (\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.523199, size = 86, normalized size = 0.66 \[ \frac{a^3 \left (4 \sin ^3(c+d x)+18 \sin ^2(c+d x)+12 \sin (c+d x)-3 \csc ^4(c+d x)-12 \csc ^3(c+d x)-6 \csc ^2(c+d x)+60 \csc (c+d x)-60 \log (\sin (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.091, size = 211, normalized size = 1.6 \begin{align*} 2\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}+{\frac{16\,{a}^{3}\sin \left ( dx+c \right ) }{3\,d}}+2\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) }{d}}+{\frac{8\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{3\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{2\,d}}-3\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}-5\,{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1401, size = 146, normalized size = 1.11 \begin{align*} \frac{4 \, a^{3} \sin \left (d x + c\right )^{3} + 18 \, a^{3} \sin \left (d x + c\right )^{2} - 60 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 12 \, a^{3} \sin \left (d x + c\right ) + \frac{3 \,{\left (20 \, a^{3} \sin \left (d x + c\right )^{3} - 2 \, a^{3} \sin \left (d x + c\right )^{2} - 4 \, a^{3} \sin \left (d x + c\right ) - a^{3}\right )}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55544, size = 397, normalized size = 3.03 \begin{align*} -\frac{18 \, a^{3} \cos \left (d x + c\right )^{6} - 45 \, a^{3} \cos \left (d x + c\right )^{4} + 30 \, a^{3} \cos \left (d x + c\right )^{2} + 60 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 4 \,{\left (a^{3} \cos \left (d x + c\right )^{6} - 6 \, a^{3} \cos \left (d x + c\right )^{4} + 24 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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 = 1.26071, size = 163, normalized size = 1.24 \begin{align*} \frac{4 \, a^{3} \sin \left (d x + c\right )^{3} + 18 \, a^{3} \sin \left (d x + c\right )^{2} - 60 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 12 \, a^{3} \sin \left (d x + c\right ) + \frac{125 \, a^{3} \sin \left (d x + c\right )^{4} + 60 \, a^{3} \sin \left (d x + c\right )^{3} - 6 \, a^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right ) - 3 \, a^{3}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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