Optimal. Leaf size=60 \[ -\frac{\left (1-\frac{b^2}{a^2}\right ) \log (a+b \sin (c+d x))}{b d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.123461, antiderivative size = 60, 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} \[ -\frac{\left (1-\frac{b^2}{a^2}\right ) \log (a+b \sin (c+d x))}{b d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2 \left (b^2-x^2\right )}{x^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{x^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{b^2}{a x^2}-\frac{b^2}{a^2 x}+\frac{-a^2+b^2}{a^2 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac{\csc (c+d x)}{a d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\left (1-\frac{b^2}{a^2}\right ) \log (a+b \sin (c+d x))}{b d}\\ \end{align*}
Mathematica [A] time = 0.0937522, size = 54, normalized size = 0.9 \[ \frac{\left (b^2-a^2\right ) \log (a+b \sin (c+d x))-a b \csc (c+d x)+b^2 (-\log (\sin (c+d x)))}{a^2 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 72, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{bd}}+{\frac{b\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{da\sin \left ( dx+c \right ) }}-{\frac{b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973981, size = 77, normalized size = 1.28 \begin{align*} -\frac{\frac{b \log \left (\sin \left (d x + c\right )\right )}{a^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} b} + \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 = 1.95147, size = 166, normalized size = 2.77 \begin{align*} -\frac{b^{2} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) + a b}{a^{2} b 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 )} \cot ^{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 = 1.15671, size = 97, normalized size = 1.62 \begin{align*} -\frac{\frac{b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2} b} - \frac{b \sin \left (d x + c\right ) - a}{a^{2} \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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