Optimal. Leaf size=38 \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 a d} \]
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Rubi [A] time = 0.0431496, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3194, 36, 29, 31} \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sin ^2(c+d x)\right )}{2 a d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,\sin ^2(c+d x)\right )}{2 a d}\\ &=\frac{\log (\sin (c+d x))}{a d}-\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.0215795, size = 38, normalized size = 1. \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 37, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{\ln \left ( a+ \left ( \sin \left ( dx+c \right ) \right ) ^{2}b \right ) }{2\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01851, size = 50, normalized size = 1.32 \begin{align*} -\frac{\frac{\log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a} - \frac{\log \left (\sin \left (d x + c\right )^{2}\right )}{a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82779, size = 96, normalized size = 2.53 \begin{align*} -\frac{\log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 2 \, \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right )}{2 \, a 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{\cot{\left (c + d x \right )}}{a + b \sin ^{2}{\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 = 1.1071, size = 51, normalized size = 1.34 \begin{align*} \frac{\frac{\log \left (\sin \left (d x + c\right )^{2}\right )}{a} - \frac{\log \left ({\left | b \sin \left (d x + c\right )^{2} + a \right |}\right )}{a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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