Integrand size = 25, antiderivative size = 124 \[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=-\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arcsin (c x))}{c^2 d}-\frac {2 i (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^3 d} \]
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Time = 0.09 (sec) , antiderivative size = 124, 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 = {4795, 4749, 4266, 2317, 2438, 267} \[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=-\frac {2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3 d}-\frac {x (a+b \arcsin (c x))}{c^2 d}+\frac {i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^3 d}-\frac {b \sqrt {1-c^2 x^2}}{c^3 d} \]
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Rule 267
Rule 2317
Rule 2438
Rule 4266
Rule 4749
Rule 4795
Rubi steps \begin{align*} \text {integral}& = -\frac {x (a+b \arcsin (c x))}{c^2 d}+\frac {\int \frac {a+b \arcsin (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 \arcsin (c x))}{c^2 d}+\frac {\text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{c^3 d} \\ & = -\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arcsin (c x))}{c^2 d}-\frac {2 i (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^3 d}-\frac {b \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{c^3 d}+\frac {b \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{c^3 d} \\ & = -\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arcsin (c x))}{c^2 d}-\frac {2 i (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^3 d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{c^3 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{c^3 d} \\ & = -\frac {b \sqrt {1-c^2 x^2}}{c^3 d}-\frac {x (a+b \arcsin (c x))}{c^2 d}-\frac {2 i (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^3 d} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.92 \[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=-\frac {2 a c x+2 b \sqrt {1-c^2 x^2}+i b \pi \arcsin (c x)+2 b c x \arcsin (c x)-b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-2 b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+2 b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+a \log (1-c x)-a \log (1+c x)+b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+2 i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c^3 d} \]
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Time = 0.13 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \sqrt {-c^{2} x^{2}+1}}{d}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {b \arcsin \left (c x \right ) c x}{d}-\frac {i b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {i b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{3}}\) | \(191\) |
default | \(\frac {-\frac {a \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \sqrt {-c^{2} x^{2}+1}}{d}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {b \arcsin \left (c x \right ) c x}{d}-\frac {i b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {i b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{3}}\) | \(191\) |
parts | \(-\frac {a \left (\frac {x}{c^{2}}+\frac {\ln \left (c x -1\right )}{2 c^{3}}-\frac {\ln \left (c x +1\right )}{2 c^{3}}\right )}{d}-\frac {b \sqrt {-c^{2} x^{2}+1}}{c^{3} d}-\frac {b \arcsin \left (c x \right ) x}{d \,c^{2}}+\frac {i b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{3}}-\frac {i b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{3}}-\frac {b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{3}}+\frac {b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{3}}\) | \(212\) |
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\[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{2} \operatorname {asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
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\[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
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\[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} - d} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]
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