Integrand size = 25, antiderivative size = 75 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {x}{b}+\frac {2 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a b d}-\frac {\text {arctanh}(\cos (c+d x))}{a d} \]
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Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2968, 3137, 2739, 632, 210, 3855} \[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a b d}-\frac {\text {arctanh}(\cos (c+d x))}{a d}-\frac {x}{b} \]
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Rule 210
Rule 632
Rule 2739
Rule 2968
Rule 3137
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc (c+d x) \left (1-\sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx \\ & = -\frac {x}{b}+\frac {\int \csc (c+d x) \, dx}{a}-\left (-\frac {a}{b}+\frac {b}{a}\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx \\ & = -\frac {x}{b}-\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {\left (2 \left (\frac {a}{b}-\frac {b}{a}\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = -\frac {x}{b}-\frac {\text {arctanh}(\cos (c+d x))}{a d}-\frac {\left (4 \left (\frac {a}{b}-\frac {b}{a}\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = -\frac {x}{b}+\frac {2 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a b d}-\frac {\text {arctanh}(\cos (c+d x))}{a d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a c+a d x-2 \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a b d} \]
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Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {\frac {\left (2 a^{2}-2 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a b \sqrt {a^{2}-b^{2}}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(94\) |
default | \(\frac {\frac {\left (2 a^{2}-2 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a b \sqrt {a^{2}-b^{2}}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(94\) |
risch | \(-\frac {x}{b}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d b a}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d b a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}\) | \(155\) |
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Time = 0.41 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.49 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {2 \, a d x + b \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - b \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right )}{2 \, a b d}, -\frac {2 \, a d x + b \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - b \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right )}{2 \, a b d}\right ] \]
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\[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.25 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {d x + c}{b} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} \sqrt {a^{2} - b^{2}}}{a b}}{d} \]
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Time = 12.08 (sec) , antiderivative size = 896, normalized size of antiderivative = 11.95 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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