Optimal. Leaf size=133 \[ \frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.123447, 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} \[ \frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a-x)^2 (a+x)^5}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^5}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^5+\frac{a^7}{x^2}+\frac{3 a^6}{x}-5 a^4 x-5 a^3 x^2+a^2 x^3+3 a x^4+x^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x)}{d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin ^4(c+d x)}{4 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.164678, size = 86, normalized size = 0.65 \[ -\frac{a^3 \left (-10 \sin ^6(c+d x)-36 \sin ^5(c+d x)-15 \sin ^4(c+d x)+100 \sin ^3(c+d x)+150 \sin ^2(c+d x)-60 \sin (c+d x)+60 \csc (c+d x)-180 \log (\sin (c+d x))\right )}{60 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 147, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{16\,{a}^{3}\sin \left ( dx+c \right ) }{15\,d}}-{\frac{2\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) }{5\,d}}-{\frac{8\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{15\,d}}+{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08016, size = 144, normalized size = 1.08 \begin{align*} \frac{10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 180 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac{60 \, a^{3}}{\sin \left (d x + c\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60819, size = 339, normalized size = 2.55 \begin{align*} -\frac{144 \, a^{3} \cos \left (d x + c\right )^{6} - 32 \, a^{3} \cos \left (d x + c\right )^{4} - 128 \, a^{3} \cos \left (d x + c\right )^{2} - 720 \, a^{3} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 256 \, a^{3} + 5 \,{\left (8 \, a^{3} \cos \left (d x + c\right )^{6} - 36 \, a^{3} \cos \left (d x + c\right )^{4} - 72 \, a^{3} \cos \left (d x + c\right )^{2} + 47 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \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 [A] time = 1.34305, size = 162, normalized size = 1.22 \begin{align*} \frac{10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 180 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac{60 \,{\left (3 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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