Optimal. Leaf size=59 \[ \frac{\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a b^2 d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{b d} \]
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Rubi [A] time = 0.107853, antiderivative size = 59, 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, 894} \[ \frac{\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a b^2 d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b \left (b^2-x^2\right )}{x (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{x (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+\frac{b^2}{a x}+\frac{a^2-b^2}{a (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac{\log (\sin (c+d x))}{a d}+\frac{\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a b^2 d}-\frac{\sin (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.0751889, size = 53, normalized size = 0.9 \[ \frac{\left (a^2-b^2\right ) \log (a+b \sin (c+d x))-a b \sin (c+d x)+b^2 \log (\sin (c+d x))}{a b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 68, normalized size = 1.2 \begin{align*} -{\frac{\sin \left ( dx+c \right ) }{bd}}+{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{2}d}}-{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{da}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96684, size = 73, normalized size = 1.24 \begin{align*} \frac{\frac{\log \left (\sin \left (d x + c\right )\right )}{a} - \frac{\sin \left (d x + c\right )}{b} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96669, size = 131, normalized size = 2.22 \begin{align*} \frac{b^{2} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right ) - a b \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (c + d x \right )} \cot{\left (c + d x \right )}}{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25766, size = 76, normalized size = 1.29 \begin{align*} \frac{\frac{\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{\sin \left (d x + c\right )}{b} + \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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