Integrand size = 26, antiderivative size = 198 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^3 d} \]
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Time = 0.21 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5812, 5789, 4265, 2611, 2320, 6724, 5798, 8} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3 d}+\frac {2 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3 d}-\frac {2 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^3 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 b^2 x}{c^2 d} \]
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Rule 8
Rule 2320
Rule 2611
Rule 4265
Rule 5789
Rule 5798
Rule 5812
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx}{c^2}-\frac {(2 b) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{c d} \\ & = -\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {\text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 d}+\frac {\left (2 b^2\right ) \int 1 \, dx}{c^2 d} \\ & = \frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {(2 i b) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 d}-\frac {(2 i b) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 d} \\ & = \frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 d}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3 d} \\ & = \frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^3 d} \\ & = \frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^3 d} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.60 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {a^2 x}{c^2 d}-\frac {a^2 \arctan (c x)}{c^3 d}+\frac {2 a b \left (-\sqrt {1+c^2 x^2}+c x \text {arcsinh}(c x)+\frac {1}{2} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {1}{2} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )\right )}{c^3 d}+\frac {b^2 \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \left (2+\text {arcsinh}(c x)^2\right )-i \left (-\text {arcsinh}(c x)^2 \left (\log \left (1-i e^{-\text {arcsinh}(c x)}\right )-\log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right )-2 \text {arcsinh}(c x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )-2 \left (\operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )\right )\right )}{c^3 d} \]
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\[\int \frac {x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{c^{2} d \,x^{2}+d}d x\]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {\int \frac {a^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{d\,c^2\,x^2+d} \,d x \]
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