Optimal. Leaf size=54 \[ -\frac {\cot (c+d x)}{a^2 d}+\frac {2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {2 \cot (c+d x)}{a^2 d (\csc (c+d x)+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2709, 3770, 3767, 8, 3777} \[ -\frac {\cot (c+d x)}{a^2 d}+\frac {2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {2 \cot (c+d x)}{a^2 d (\csc (c+d x)+1)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2709
Rule 3767
Rule 3770
Rule 3777
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \left (2-2 \csc (c+d x)+\csc ^2(c+d x)-\frac {2}{1+\csc (c+d x)}\right ) \, dx}{a^2}\\ &=\frac {2 x}{a^2}+\frac {\int \csc ^2(c+d x) \, dx}{a^2}-\frac {2 \int \csc (c+d x) \, dx}{a^2}-\frac {2 \int \frac {1}{1+\csc (c+d x)} \, dx}{a^2}\\ &=\frac {2 x}{a^2}+\frac {2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {2 \cot (c+d x)}{a^2 d (1+\csc (c+d x))}+\frac {2 \int -1 \, dx}{a^2}-\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}\\ &=\frac {2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a^2 d}-\frac {2 \cot (c+d x)}{a^2 d (1+\csc (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.80, size = 216, normalized size = 4.00 \[ -\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {3}{2} (c+d x)\right ) \left (-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+5\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \left (2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3\right )+2 \sin \left (\frac {1}{2} (c+d x)\right ) \left (2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\cos (c+d x) \left (2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1\right )\right )\right )}{4 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 160, normalized size = 2.96 \[ \frac {3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (3 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) - 2}{a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d - {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 90, normalized size = 1.67 \[ -\frac {\frac {4 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 77, normalized size = 1.43 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {1}{2 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}-\frac {4}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 116, normalized size = 2.15 \[ -\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}{\frac {a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}} + \frac {4 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {\sin \left (d x + c\right )}{a^{2} {\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.69, size = 87, normalized size = 1.61 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1}{d\,\left (2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d} \]
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 \[ \frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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