Integrand size = 17, antiderivative size = 35 \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {2 \text {arctanh}\left (\frac {b+2 c \cos (x)}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3340, 632, 212} \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {2 \text {arctanh}\left (\frac {b+2 c \cos (x)}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Rule 212
Rule 632
Rule 3340
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,\cos (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c \cos (x)\right ) \\ & = \frac {2 \text {arctanh}\left (\frac {b+2 c \cos (x)}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11 \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=-\frac {2 \arctan \left (\frac {b+2 c \cos (x)}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(-\frac {2 \arctan \left (\frac {b +2 c \cos \left (x \right )}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(36\) |
default | \(-\frac {2 \arctan \left (\frac {b +2 c \cos \left (x \right )}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(36\) |
risch | \(-\frac {i \ln \left ({\mathrm e}^{2 i x}+\frac {\left (4 i a c -i b^{2}+b \sqrt {4 a c -b^{2}}\right ) {\mathrm e}^{i x}}{c \sqrt {4 a c -b^{2}}}+1\right )}{\sqrt {4 a c -b^{2}}}+\frac {i \ln \left ({\mathrm e}^{2 i x}+\frac {\left (-4 i a c +i b^{2}+b \sqrt {4 a c -b^{2}}\right ) {\mathrm e}^{i x}}{c \sqrt {4 a c -b^{2}}}+1\right )}{\sqrt {4 a c -b^{2}}}\) | \(142\) |
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none
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.60 \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\left [\frac {\log \left (-\frac {2 \, c^{2} \cos \left (x\right )^{2} + 2 \, b c \cos \left (x\right ) + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c \cos \left (x\right ) + b\right )}}{c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a}\right )}{\sqrt {b^{2} - 4 \, a c}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c \cos \left (x\right ) + b\right )}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (34) = 68\).
Time = 1.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.83 \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\begin {cases} - \frac {\log {\left (\frac {a}{b} + \cos {\left (x \right )} \right )}}{b} & \text {for}\: c = 0 \\\frac {2}{b + 2 c \cos {\left (x \right )}} & \text {for}\: a = \frac {b^{2}}{4 c} \\- \frac {\log {\left (\frac {b}{2 c} + \cos {\left (x \right )} - \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt {- 4 a c + b^{2}}} + \frac {\log {\left (\frac {b}{2 c} + \cos {\left (x \right )} + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt {- 4 a c + b^{2}}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=-\frac {2 \, \arctan \left (\frac {2 \, c \cos \left (x\right ) + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} \]
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Time = 3.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int \frac {\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,\cos \left (x\right )}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}} \]
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