Optimal. Leaf size=29 \[ \frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {\cot (c+d x)}{a d} \]
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Rubi [A] time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2706, 3767, 8, 3770} \[ \frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {\cot (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2706
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \csc (c+d x) \, dx}{a}+\frac {\int \csc ^2(c+d x) \, dx}{a}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a d}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {\cot (c+d x)}{a d}\\ \end {align*}
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Mathematica [B] time = 0.24, size = 69, normalized size = 2.38 \[ -\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\cos (c+d x)+\sin (c+d x) \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{2 a d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 62, normalized size = 2.14 \[ \frac {\log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )}{2 \, a d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 65, normalized size = 2.24 \[ -\frac {\frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 56, normalized size = 1.93 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {1}{2 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 70, normalized size = 2.41 \[ -\frac {\frac {2 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\cos \left (d x + c\right ) + 1}{a \sin \left (d x + c\right )} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.60, size = 25, normalized size = 0.86 \[ -\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+\mathrm {cot}\left (c+d\,x\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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