Integrand size = 25, antiderivative size = 92 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 b \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cos (c+d x)}{a d (a+b \sin (c+d x))} \]
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Time = 0.17 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2968, 3135, 12, 2826, 3855, 2739, 632, 210} \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 b \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 d \sqrt {a^2-b^2}}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cos (c+d x)}{a d (a+b \sin (c+d x))} \]
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2826
Rule 2968
Rule 3135
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))^2} \, dx \\ & = \frac {\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac {\int \frac {\left (a^2-b^2\right ) \csc (c+d x)}{a+b \sin (c+d x)} \, dx}{a \left (a^2-b^2\right )} \\ & = \frac {\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = \frac {\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{a^2}-\frac {b \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cos (c+d x)}{a d (a+b \sin (c+d x))}-\frac {(2 b) \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 )}{a^2 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac {(4 b) \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 )}{a^2 d} \\ & = -\frac {2 b \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cos (c+d x)}{a d (a+b \sin (c+d x))} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-\frac {2 b \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a \cos (c+d x)}{a+b \sin (c+d x)}}{a^2 d} \]
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Time = 0.48 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {-\frac {4 \left (\frac {-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {b \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(116\) |
default | \(\frac {-\frac {4 \left (\frac {-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {b \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(116\) |
risch | \(\frac {2 i \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{a b d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{2}}-\frac {i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}\) | \(240\) |
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (87) = 174\).
Time = 0.34 (sec) , antiderivative size = 483, normalized size of antiderivative = 5.25 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {{\left (b^{2} \sin \left (d x + c\right ) + a b\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) + {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left ({\left (a^{4} b - a^{2} b^{3}\right )} d \sin \left (d x + c\right ) + {\left (a^{5} - a^{3} b^{2}\right )} d\right )}}, \frac {2 \, {\left (b^{2} \sin \left (d x + c\right ) + a b\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left ({\left (a^{4} b - a^{2} b^{3}\right )} d \sin \left (d x + c\right ) + {\left (a^{5} - a^{3} b^{2}\right )} d\right )}}\right ] \]
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\[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.41 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b}{\sqrt {a^{2} - b^{2}} a^{2}} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{2}}}{d} \]
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Time = 10.82 (sec) , antiderivative size = 523, normalized size of antiderivative = 5.68 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {a^3\,\cos \left (c+d\,x\right )-a\,b^2-b^3\,\sin \left (c+d\,x\right )+a^3+a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-a\,b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-b^3\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-a\,b^2\,\cos \left (c+d\,x\right )+a^2\,b\,\sin \left (c+d\,x\right )+a^2\,b\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+b^2\,\sin \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {-a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}+b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}\,4{}\mathrm {i}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}\,2{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )\,\sqrt {b^2-a^2}\,2{}\mathrm {i}+a\,b\,\mathrm {atan}\left (\frac {-a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}+b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}\,4{}\mathrm {i}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}\,2{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )\,\sqrt {b^2-a^2}\,2{}\mathrm {i}}{a^2\,d\,\left (a^2-b^2\right )\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \]
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