Optimal. Leaf size=127 \[ \frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 d}-\frac{\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}+\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{4 a b \sin (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.157214, antiderivative size = 127, 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 = {2837, 12, 948} \[ \frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 d}-\frac{\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}+\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{4 a b \sin (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 948
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^3 (a+x)^2 \left (b^2-x^2\right )^2}{x^3} \, 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^3} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-4 a b^2+\frac{a^2 b^4}{x^3}+\frac{2 a b^4}{x^2}+\frac{-2 a^2 b^2+b^4}{x}+\left (a^2-2 b^2\right ) x+2 a x^2+x^3\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=-\frac{2 a b \csc (c+d x)}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac{4 a b \sin (c+d x)}{d}+\frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 d}+\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{b^2 \sin ^4(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.282604, size = 103, normalized size = 0.81 \[ \frac{6 \left (a^2-2 b^2\right ) \sin ^2(c+d x)+12 \left (b^2-2 a^2\right ) \log (\sin (c+d x))-6 a^2 \csc ^2(c+d x)+8 a b \sin ^3(c+d x)-48 a b \sin (c+d x)-24 a b \csc (c+d x)+3 b^2 \sin ^4(c+d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 197, normalized size = 1.6 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{2\,d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-2\,{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}-{\frac{16\,ab\sin \left ( dx+c \right ) }{3\,d}}-2\,{\frac{ab\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{8\,ab\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{b}^{2}\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 = 0.995812, size = 140, normalized size = 1.1 \begin{align*} \frac{3 \, b^{2} \sin \left (d x + c\right )^{4} + 8 \, a b \sin \left (d x + c\right )^{3} - 48 \, a b \sin \left (d x + c\right ) + 6 \,{\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{2} - 12 \,{\left (2 \, a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) - \frac{6 \,{\left (4 \, a b \sin \left (d x + c\right ) + a^{2}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79847, size = 381, normalized size = 3. \begin{align*} \frac{24 \, b^{2} \cos \left (d x + c\right )^{6} - 24 \,{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 9 \,{\left (8 \, a^{2} - 9 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 24 \, a^{2} + 33 \, b^{2} - 96 \,{\left ({\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} + b^{2}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 64 \,{\left (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 )}{96 \,{\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 = 1.30277, size = 189, normalized size = 1.49 \begin{align*} \frac{3 \, b^{2} \sin \left (d x + c\right )^{4} + 8 \, a b \sin \left (d x + c\right )^{3} + 6 \, a^{2} \sin \left (d x + c\right )^{2} - 12 \, b^{2} \sin \left (d x + c\right )^{2} - 48 \, a b \sin \left (d x + c\right ) - 12 \,{\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac{6 \,{\left (6 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, b^{2} \sin \left (d x + c\right )^{2} - 4 \, a b \sin \left (d x + c\right ) - a^{2}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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