Optimal. Leaf size=63 \[ \frac{(a+b) \log \left (a+b \sin ^2(c+d x)\right )}{2 a^2 d}-\frac{(a+b) \log (\sin (c+d x))}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.0749722, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3194, 77} \[ \frac{(a+b) \log \left (a+b \sin ^2(c+d x)\right )}{2 a^2 d}-\frac{(a+b) \log (\sin (c+d x))}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 77
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x}{x^2 (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^2}+\frac{-a-b}{a^2 x}+\frac{b (a+b)}{a^2 (a+b x)}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{\csc ^2(c+d x)}{2 a d}-\frac{(a+b) \log (\sin (c+d x))}{a^2 d}+\frac{(a+b) \log \left (a+b \sin ^2(c+d x)\right )}{2 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.141051, size = 50, normalized size = 0.79 \[ -\frac{(a+b) \left (2 \log (\sin (c+d x))-\log \left (a+b \sin ^2(c+d x)\right )\right )+a \csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.094, size = 161, normalized size = 2.6 \begin{align*}{\frac{1}{4\,da \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,da}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{2\,{a}^{2}d}}-{\frac{1}{4\,da \left ( 1+\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\cos \left ( dx+c \right ) \right ) }{2\,da}}-{\frac{\ln \left ( 1+\cos \left ( dx+c \right ) \right ) b}{2\,{a}^{2}d}}+{\frac{\ln \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b \right ) }{2\,da}}+{\frac{\ln \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b \right ) b}{2\,{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986388, size = 76, normalized size = 1.21 \begin{align*} \frac{\frac{{\left (a + b\right )} \log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a^{2}} - \frac{{\left (a + b\right )} \log \left (\sin \left (d x + c\right )^{2}\right )}{a^{2}} - \frac{1}{a \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 = 2.0216, size = 223, normalized size = 3.54 \begin{align*} \frac{{\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 2 \,{\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + a}{2 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} 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{\cot ^{3}{\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 = 1.18514, size = 146, normalized size = 2.32 \begin{align*} \frac{\frac{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}}{a} + \frac{4 \,{\left (a + b\right )} \log \left ({\left | -a{\left (\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 2 \, a + 4 \, b \right |}\right )}{a^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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