Integrand size = 19, antiderivative size = 66 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {b x}{a^2+b^2}+\frac {\log (\sin (c+d x))}{a d}-\frac {b^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a \left (a^2+b^2\right ) d} \]
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Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3652, 3611, 3556} \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {b^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac {b x}{a^2+b^2}+\frac {\log (\sin (c+d x))}{a d} \]
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Rule 3556
Rule 3611
Rule 3652
Rubi steps \begin{align*} \text {integral}& = -\frac {b x}{a^2+b^2}+\frac {\int \cot (c+d x) \, dx}{a}-\frac {b^2 \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a \left (a^2+b^2\right )} \\ & = -\frac {b x}{a^2+b^2}+\frac {\log (\sin (c+d x))}{a d}-\frac {b^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a \left (a^2+b^2\right ) d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {\log (i-\tan (c+d x))}{a+i b}-\frac {2 \log (\tan (c+d x))}{a}+\frac {\log (i+\tan (c+d x))}{a-i b}+\frac {2 b^2 \log (a+b \tan (c+d x))}{a^3+a b^2}}{2 d} \]
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Time = 0.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.20
method | result | size |
parallelrisch | \(\frac {-2 a b d x +2 a^{2} \ln \left (\tan \left (d x +c \right )\right )+2 \ln \left (\tan \left (d x +c \right )\right ) b^{2}-\ln \left (\sec ^{2}\left (d x +c \right )\right ) a^{2}-2 b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{2 \left (a^{2}+b^{2}\right ) a d}\) | \(79\) |
derivativedivides | \(\frac {-\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{a \left (a^{2}+b^{2}\right )}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a}+\frac {-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\) | \(80\) |
default | \(\frac {-\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{a \left (a^{2}+b^{2}\right )}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a}+\frac {-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\) | \(80\) |
norman | \(-\frac {b x}{a^{2}+b^{2}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a d}\) | \(86\) |
risch | \(-\frac {i x}{i b -a}+\frac {2 i b^{2} x}{\left (a^{2}+b^{2}\right ) a}+\frac {2 i b^{2} c}{\left (a^{2}+b^{2}\right ) a d}-\frac {2 i x}{a}-\frac {2 i c}{a d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{2}+b^{2}\right ) a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}\) | \(142\) |
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Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.48 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {2 \, a b d x + b^{2} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \]
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Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 626, normalized size of antiderivative = 9.48 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=\begin {cases} \frac {\tilde {\infty } x \cot {\left (c \right )}}{\tan {\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {- \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\log {\left (\tan {\left (c + d x \right )} \right )}}{d}}{a} & \text {for}\: b = 0 \\\frac {- x - \frac {1}{d \tan {\left (c + d x \right )}}}{b} & \text {for}\: a = 0 \\\frac {d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i d x}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {2 i \log {\left (\tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {2 \log {\left (\tan {\left (c + d x \right )} \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {1}{2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = - i b \\\frac {d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i d x}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {2 i \log {\left (\tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {2 \log {\left (\tan {\left (c + d x \right )} \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {1}{2 b d \tan {\left (c + d x \right )} + 2 i b d} & \text {for}\: a = i b \\\frac {x \cot {\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d + 2 a b^{2} d} + \frac {2 a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{2 a^{3} d + 2 a b^{2} d} - \frac {2 a b d x}{2 a^{3} d + 2 a b^{2} d} - \frac {2 b^{2} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{3} d + 2 a b^{2} d} + \frac {2 b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{2 a^{3} d + 2 a b^{2} d} & \text {otherwise} \end {cases} \]
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Time = 0.46 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.27 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {2 \, b^{2} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3} + a b^{2}} + \frac {2 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} + \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{a}}{2 \, d} \]
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Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.33 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {2 \, b^{3} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3} b + a b^{3}} + \frac {2 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} + \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a}}{2 \, d} \]
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Time = 5.59 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.44 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,d\,\left (a^2+b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )} \]
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