Optimal. Leaf size=72 \[ \frac{b^2 \log (\sin (c+d x))}{a^3 d}-\frac{b^2 \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.0913019, antiderivative size = 72, 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 = {2833, 12, 44} \[ \frac{b^2 \log (\sin (c+d x))}{a^3 d}-\frac{b^2 \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d} \]
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 ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^3}{x^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^2 \operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{1}{a^2 x^2}+\frac{1}{a^3 x}-\frac{1}{a^3 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{b^2 \log (\sin (c+d x))}{a^3 d}-\frac{b^2 \log (a+b \sin (c+d x))}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.0466824, size = 72, normalized size = 1. \[ \frac{b^2 \log (\sin (c+d x))}{a^3 d}-\frac{b^2 \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 73, normalized size = 1. \begin{align*} -{\frac{{b}^{2}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}+{\frac{b}{d{a}^{2}\sin \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977051, size = 89, normalized size = 1.24 \begin{align*} -\frac{\frac{2 \, b^{2} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{3}} - \frac{2 \, b^{2} \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac{2 \, b \sin \left (d x + c\right ) - a}{a^{2} \sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59283, size = 234, normalized size = 3.25 \begin{align*} -\frac{2 \, a b \sin \left (d x + c\right ) - a^{2} + 2 \,{\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \,{\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right )}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\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 ^{3}{\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 = 1.19649, size = 96, normalized size = 1.33 \begin{align*} -\frac{\frac{2 \, b^{2} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{3}} - \frac{2 \, b^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{2 \, a b \sin \left (d x + c\right ) - a^{2}}{a^{3} \sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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