Integrand size = 24, antiderivative size = 108 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=-\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d} \]
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Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5812, 5789, 4265, 2317, 2438, 267} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))}{c^2 d}+\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {b \sqrt {c^2 x^2+1}}{c^3 d} \]
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Rule 267
Rule 2317
Rule 2438
Rule 4265
Rule 5789
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {\int \frac {a+b \text {arcsinh}(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 \text {arcsinh}(c x))}{c^2 d}-\frac {\text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{c^3 d} \\ & = -\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {(i b) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 d}-\frac {(i b) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 d} \\ & = -\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^3 d} \\ & = -\frac {b \sqrt {1+c^2 x^2}}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\frac {a c x-b \sqrt {1+c^2 x^2}+b c x \text {arcsinh}(c x)-a \arctan (c x)-i b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+i b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d} \]
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Time = 0.17 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.57
method | result | size |
derivativedivides | \(\frac {\frac {a \left (c x -\arctan \left (c x \right )\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )+\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}-\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}}{c^{3}}\) | \(170\) |
default | \(\frac {\frac {a \left (c x -\arctan \left (c x \right )\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )+\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}-\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}}{c^{3}}\) | \(170\) |
parts | \(\frac {a \left (\frac {x}{c^{2}}-\frac {\arctan \left (c x \right )}{c^{3}}\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )+\operatorname {arcsinh}\left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}-\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d \,c^{3}}\) | \(174\) |
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\frac {\int \frac {a x^{2}}{c^{2} x^{2} + 1}\, dx + \int \frac {b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{d\,c^2\,x^2+d} \,d x \]
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