Integrand size = 25, antiderivative size = 115 \[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3 d} \]
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Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4796, 4750, 4268, 2317, 2438, 267} \[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3 d}+\frac {b \sqrt {1-c^2 x^2}}{c^3 d} \]
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Rule 267
Rule 2317
Rule 2438
Rule 4268
Rule 4750
Rule 4796
Rubi steps \begin{align*} \text {integral}& = -\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {\int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx}{c^2}-\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c d} \\ & = \frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}-\frac {\text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arccos (c x))}{c^3 d} \\ & = \frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^3 d}+\frac {b \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{c^3 d}-\frac {b \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{c^3 d} \\ & = \frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^3 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{c^3 d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{c^3 d} \\ & = \frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arccos (c x))}{c^2 d}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c^3 d} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.20 \[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=-\frac {2 a c x-2 b \sqrt {1-c^2 x^2}+2 b c x \arccos (c x)+2 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-2 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+a \log (1-c x)-a \log (1+c x)+2 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c^3 d} \]
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Time = 1.86 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.57
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) c x}{d}-\frac {i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}}{c^{3}}\) | \(180\) |
default | \(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) c x}{d}-\frac {i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}}{c^{3}}\) | \(180\) |
parts | \(-\frac {a \left (\frac {x}{c^{2}}+\frac {\ln \left (c x -1\right )}{2 c^{3}}-\frac {\ln \left (c x +1\right )}{2 c^{3}}\right )}{d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{c^{3} d}-\frac {b \arccos \left (c x \right ) x}{d \,c^{2}}-\frac {i b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{c^{3} d}+\frac {i b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{c^{3} d}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}\) | \(201\) |
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\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{2} \operatorname {acos}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
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\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]
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