Optimal. Leaf size=133 \[ \frac{a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.127574, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ \frac{a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^3 (a-x)^2 (a+x)^5}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^5}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-5 a^4+\frac{a^7}{x^3}+\frac{3 a^6}{x^2}+\frac{a^5}{x}-5 a^3 x+a^2 x^2+3 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac{3 a^3 \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}+\frac{a^3 \log (\sin (c+d x))}{d}-\frac{5 a^3 \sin (c+d x)}{d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.147773, size = 86, normalized size = 0.65 \[ -\frac{a^3 \left (-12 \sin ^5(c+d x)-45 \sin ^4(c+d x)-20 \sin ^3(c+d x)+150 \sin ^2(c+d x)+300 \sin (c+d x)+30 \csc ^2(c+d x)+180 \csc (c+d x)-60 \log (\sin (c+d x))\right )}{60 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 154, normalized size = 1.2 \begin{align*} -{\frac{112\,{a}^{3}\sin \left ( dx+c \right ) }{15\,d}}-{\frac{14\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) }{5\,d}}-{\frac{56\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{15\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17822, size = 143, normalized size = 1.08 \begin{align*} \frac{12 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 60 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - 300 \, a^{3} \sin \left (d x + c\right ) - \frac{30 \,{\left (6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57352, size = 363, normalized size = 2.73 \begin{align*} \frac{360 \, a^{3} \cos \left (d x + c\right )^{6} + 120 \, a^{3} \cos \left (d x + c\right )^{4} - 855 \, a^{3} \cos \left (d x + c\right )^{2} + 615 \, a^{3} + 480 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 32 \,{\left (3 \, a^{3} \cos \left (d x + c\right )^{6} - 14 \, a^{3} \cos \left (d x + c\right )^{4} - 56 \, a^{3} \cos \left (d x + c\right )^{2} + 112 \, a^{3}\right )} \sin \left (d x + c\right )}{480 \,{\left (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 = 1.33728, size = 162, normalized size = 1.22 \begin{align*} \frac{12 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 60 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 300 \, a^{3} \sin \left (d x + c\right ) - \frac{30 \,{\left (3 \, a^{3} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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