Integrand size = 23, antiderivative size = 82 \[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {i (a+b \arccos (c x))^2}{2 b c^2 d}-\frac {(a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{c^2 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 c^2 d} \]
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Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4766, 3798, 2221, 2317, 2438} \[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {i (a+b \arccos (c x))^2}{2 b c^2 d}-\frac {\log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^2 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 c^2 d} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4766
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}(\int (a+b x) \cot (x) \, dx,x,\arccos (c x))}{c^2 d} \\ & = \frac {i (a+b \arccos (c x))^2}{2 b c^2 d}+\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\arccos (c x)\right )}{c^2 d} \\ & = \frac {i (a+b \arccos (c x))^2}{2 b c^2 d}-\frac {(a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{c^2 d}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arccos (c x)\right )}{c^2 d} \\ & = \frac {i (a+b \arccos (c x))^2}{2 b c^2 d}-\frac {(a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{c^2 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right )}{2 c^2 d} \\ & = \frac {i (a+b \arccos (c x))^2}{2 b c^2 d}-\frac {(a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{c^2 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 c^2 d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.40 \[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {i \left (b \arccos (c x)^2+2 i b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+2 i b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+i a \log \left (1-c^2 x^2\right )+2 b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+2 b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{2 c^2 d} \]
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Time = 1.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.73
method | result | size |
parts | \(-\frac {a \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{2}}-\frac {b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{2}}\) | \(142\) |
derivativedivides | \(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{2}}\) | \(147\) |
default | \(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{2}}\) | \(147\) |
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\[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a x}{c^{2} x^{2} - 1}\, dx + \int \frac {b x \operatorname {acos}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
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\[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]
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