Optimal. Leaf size=50 \[ -\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.0724541, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 44} \[ -\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{\cot (c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2}{x^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{1}{a^2 x}+\frac{1}{a^2 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\csc (c+d x)}{a d}-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0415074, size = 50, normalized size = 1. \[ -\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 35, normalized size = 0.7 \begin{align*} -{\frac{\csc \left ( dx+c \right ) }{da}}+{\frac{b\ln \left ( a\csc \left ( dx+c \right ) +b \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971688, size = 63, normalized size = 1.26 \begin{align*} \frac{\frac{b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2}} - \frac{b \log \left (\sin \left (d x + c\right )\right )}{a^{2}} - \frac{1}{a \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37955, size = 143, normalized size = 2.86 \begin{align*} \frac{b \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) - b \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - a}{a^{2} d \sin \left (d x + c\right )} \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{\left (c + d x \right )} \csc ^{2}{\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 = 1.2312, size = 66, normalized size = 1.32 \begin{align*} \frac{\frac{b \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2}} - \frac{b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac{1}{a \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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