Optimal. Leaf size=130 \[ \frac{\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 d}-\frac{\left (2 a^2-b^2\right ) \sin ^2(c+d x)}{2 d}+\frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.133519, antiderivative size = 130, 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 = {2837, 12, 948} \[ \frac{\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 d}-\frac{\left (2 a^2-b^2\right ) \sin ^2(c+d x)}{2 d}+\frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 948
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b (a+x)^2 \left (b^2-x^2\right )^2}{x} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^2 \left (b^2-x^2\right )^2}{x} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a b^4+\frac{a^2 b^4}{x}-b^2 \left (2 a^2-b^2\right ) x-4 a b^2 x^2+\left (a^2-2 b^2\right ) x^3+2 a x^4+x^5\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin (c+d x)}{d}-\frac{\left (2 a^2-b^2\right ) \sin ^2(c+d x)}{2 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 d}+\frac{2 a b \sin ^5(c+d x)}{5 d}+\frac{b^2 \sin ^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.156253, size = 105, normalized size = 0.81 \[ \frac{15 \left (a^2-2 b^2\right ) \sin ^4(c+d x)+30 \left (b^2-2 a^2\right ) \sin ^2(c+d x)+60 a^2 \log (\sin (c+d x))+24 a b \sin ^5(c+d x)-80 a b \sin ^3(c+d x)+120 a b \sin (c+d x)+10 b^2 \sin ^6(c+d x)}{60 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 119, normalized size = 0.9 \begin{align*}{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{16\,ab\sin \left ( dx+c \right ) }{15\,d}}+{\frac{2\,ab\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{8\,ab\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{15\,d}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}{b}^{2}}{6\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966875, size = 142, normalized size = 1.09 \begin{align*} \frac{10 \, b^{2} \sin \left (d x + c\right )^{6} + 24 \, a b \sin \left (d x + c\right )^{5} - 80 \, a b \sin \left (d x + c\right )^{3} + 15 \,{\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4} + 60 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + 120 \, a b \sin \left (d x + c\right ) - 30 \,{\left (2 \, a^{2} - b^{2}\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 = 1.82381, size = 247, normalized size = 1.9 \begin{align*} -\frac{10 \, b^{2} \cos \left (d x + c\right )^{6} - 15 \, a^{2} \cos \left (d x + c\right )^{4} - 30 \, a^{2} \cos \left (d x + c\right )^{2} - 60 \, a^{2} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 8 \,{\left (3 \, a b \cos \left (d x + c\right )^{4} + 4 \, a b \cos \left (d x + c\right )^{2} + 8 \, a b\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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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 = 1.27565, size = 159, normalized size = 1.22 \begin{align*} \frac{10 \, b^{2} \sin \left (d x + c\right )^{6} + 24 \, a b \sin \left (d x + c\right )^{5} + 15 \, a^{2} \sin \left (d x + c\right )^{4} - 30 \, b^{2} \sin \left (d x + c\right )^{4} - 80 \, a b \sin \left (d x + c\right )^{3} - 60 \, a^{2} \sin \left (d x + c\right )^{2} + 30 \, b^{2} \sin \left (d x + c\right )^{2} + 60 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 120 \, a b \sin \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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