Integrand size = 23, antiderivative size = 57 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {b x}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4768, 197} \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {a+b \arccos (c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c d^2 \sqrt {1-c^2 x^2}} \]
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Rule 197
Rule 4768
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \arccos (c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2} \\ & = \frac {b x}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {a+b c x \sqrt {1-c^2 x^2}+b \arccos (c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \]
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Time = 0.70 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.72
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}\right )}{d^{2}}}{c^{2}}\) | \(98\) |
default | \(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}\right )}{d^{2}}}{c^{2}}\) | \(98\) |
parts | \(-\frac {a}{2 d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}\right )}{d^{2} c^{2}}\) | \(100\) |
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {a c^{2} x^{2} + \sqrt {-c^{2} x^{2} + 1} b c x + b \arccos \left (c x\right )}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]
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\[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (50) = 100\).
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.39 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {1}{4} \, {\left ({\left (\frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x + c^{6} d^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x - c^{6} d^{4}}\right )} c^{2} + \frac {2 \, \arccos \left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac {a}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.75 \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x^{2} \arccos \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {a x^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} b x}{2 \, {\left (c^{2} x^{2} - 1\right )} c d^{2}} + \frac {b \arccos \left (c x\right )}{2 \, c^{2} d^{2}} + \frac {a}{2 \, c^{2} d^{2}} \]
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Timed out. \[ \int \frac {x (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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