Optimal. Leaf size=49 \[ -\frac{\cos (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a} \]
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Rubi [A] time = 0.116347, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 3473, 8, 2592, 321, 206} \[ -\frac{\cos (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 3473
Rule 8
Rule 2592
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cos (c+d x) \cot (c+d x) \, dx}{a}+\frac{\int \cot ^2(c+d x) \, dx}{a}\\ &=-\frac{\cot (c+d x)}{a d}-\frac{\int 1 \, dx}{a}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{x}{a}-\frac{\cos (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{x}{a}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cos (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.407286, size = 93, normalized size = 1.9 \[ -\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right )^2 \left (\cos (c+d x)+\sin (c+d x) \left (\cos (c+d x)+\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+c+d x\right )\right )}{2 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.104, size = 97, normalized size = 2. \begin{align*}{\frac{1}{2\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{1}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6583, size = 208, normalized size = 4.24 \begin{align*} -\frac{\frac{\frac{4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1}{\frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac{4 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{2 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13203, size = 225, normalized size = 4.59 \begin{align*} -\frac{2 \,{\left (d x + \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )}{2 \, a 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.43979, size = 153, normalized size = 3.12 \begin{align*} -\frac{\frac{6 \,{\left (d x + c\right )}}{a} + \frac{6 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} a}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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