Integrand size = 22, antiderivative size = 14 \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\frac {c x}{d^2}-\frac {\sin (x)}{d} \]
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Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4482, 3095, 2717} \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\frac {c x}{d^2}-\frac {\sin (x)}{d} \]
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Rule 2717
Rule 3095
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \frac {\frac {c^2}{d^2}-\cos ^2(x)}{c+d \cos (x)} \, dx \\ & = -\frac {\int (-c+d \cos (x)) \, dx}{d^2} \\ & = \frac {c x}{d^2}-\frac {\int \cos (x) \, dx}{d} \\ & = \frac {c x}{d^2}-\frac {\sin (x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\frac {c x}{d^2}-\frac {\sin (x)}{d} \]
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Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {-d \sin \left (x \right )+c x}{d^{2}}\) | \(14\) |
risch | \(\frac {c x}{d^{2}}-\frac {\sin \left (x \right )}{d}\) | \(15\) |
default | \(\frac {-\frac {2 d \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}+2 c \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{d^{2}}\) | \(31\) |
norman | \(\frac {\frac {c x}{d}+\frac {c x \tan \left (\frac {x}{2}\right )^{4}}{d}-2 \tan \left (\frac {x}{2}\right )^{3}+\frac {2 c x \tan \left (\frac {x}{2}\right )^{2}}{d}-2 \tan \left (\frac {x}{2}\right )}{d \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{2}}\) | \(61\) |
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none
Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\frac {c x - d \sin \left (x\right )}{d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (10) = 20\).
Time = 28.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.36 \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\frac {c x \tan ^{2}{\left (\frac {x}{2} \right )}}{d^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + d^{2}} + \frac {c x}{d^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + d^{2}} - \frac {2 d \tan {\left (\frac {x}{2} \right )}}{d^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + d^{2}} \]
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Exception generated. \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\frac {c x}{d^{2}} - \frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )} d} \]
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Time = 27.95 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {-1+\frac {c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx=\frac {c\,x-d\,\sin \left (x\right )}{d^2} \]
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