Optimal. Leaf size=137 \[ \frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.106017, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a (a-x)^2 (a+x)^5}{x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^5}{x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a^6+\frac{a^7}{x}+a^5 x-5 a^4 x^2-5 a^3 x^3+a^2 x^4+3 a x^5+x^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{a^3 \sin ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.104836, size = 88, normalized size = 0.64 \[ \frac{a^3 \left (60 \sin ^7(c+d x)+210 \sin ^6(c+d x)+84 \sin ^5(c+d x)-525 \sin ^4(c+d x)-700 \sin ^3(c+d x)+210 \sin ^2(c+d x)+1260 \sin (c+d x)+420 \log (\sin (c+d x))\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 144, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) }{7\,d}}+{\frac{176\,{a}^{3}\sin \left ( dx+c \right ) }{105\,d}}+{\frac{22\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) }{35\,d}}+{\frac{88\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{105\,d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2\,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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20914, size = 144, normalized size = 1.05 \begin{align*} \frac{60 \, a^{3} \sin \left (d x + c\right )^{7} + 210 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} - 525 \, a^{3} \sin \left (d x + c\right )^{4} - 700 \, a^{3} \sin \left (d x + c\right )^{3} + 210 \, a^{3} \sin \left (d x + c\right )^{2} + 420 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 1260 \, a^{3} \sin \left (d x + c\right )}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65629, size = 292, normalized size = 2.13 \begin{align*} -\frac{210 \, a^{3} \cos \left (d x + c\right )^{6} - 105 \, a^{3} \cos \left (d x + c\right )^{4} - 210 \, a^{3} \cos \left (d x + c\right )^{2} - 420 \, a^{3} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 4 \,{\left (15 \, a^{3} \cos \left (d x + c\right )^{6} - 66 \, a^{3} \cos \left (d x + c\right )^{4} - 88 \, a^{3} \cos \left (d x + c\right )^{2} - 176 \, a^{3}\right )} \sin \left (d x + c\right )}{420 \, d} \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 = 1.29708, size = 146, normalized size = 1.07 \begin{align*} \frac{60 \, a^{3} \sin \left (d x + c\right )^{7} + 210 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} - 525 \, a^{3} \sin \left (d x + c\right )^{4} - 700 \, a^{3} \sin \left (d x + c\right )^{3} + 210 \, a^{3} \sin \left (d x + c\right )^{2} + 420 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 1260 \, a^{3} \sin \left (d x + c\right )}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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