Optimal. Leaf size=59 \[ \frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{2 a} \]
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Rubi [A] time = 0.0980408, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2839, 2592, 321, 206, 2635, 8} \[ \frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{2 a} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2592
Rule 321
Rule 206
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cos ^2(c+d x) \, dx}{a}+\frac{\int \cos (c+d x) \cot (c+d x) \, dx}{a}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\int 1 \, dx}{2 a}-\frac{\operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{x}{2 a}+\frac{\cos (c+d x)}{a d}-\frac{\cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{x}{2 a}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\cos (c+d x)}{a d}-\frac{\cos (c+d x) \sin (c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.178481, size = 60, normalized size = 1.02 \[ -\frac{\sin (2 (c+d x))-4 \cos (c+d x)+2 \left (-2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+c+d x\right )}{4 a d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.088, size = 159, normalized size = 2.7 \begin{align*}{\frac{1}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+2\,{\frac{1}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{1}{da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\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 = 1.66312, size = 211, normalized size = 3.58 \begin{align*} -\frac{\frac{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 2}{a + \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12896, size = 167, normalized size = 2.83 \begin{align*} -\frac{d x + \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) + \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, a d} \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*} \frac{\int \frac{\cos ^{4}{\left (c + d x \right )} \csc{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37715, size = 119, normalized size = 2.02 \begin{align*} -\frac{\frac{d x + c}{a} - \frac{2 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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