Integrand size = 24, antiderivative size = 166 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}+d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5808, 5775, 3797, 2221, 2611, 2320, 6724, 5785, 5783, 30} \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {1}{2} d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c d x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-b d \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+d \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{4} b^2 c^2 d x^2 \]
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 5783
Rule 5785
Rule 5808
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+d \int \frac {(a+b \text {arcsinh}(c x))^2}{x} \, dx-(b c d) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {d \text {Subst}\left (\int x^2 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {1}{2} (b c d) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{2} \left (b^2 c^2 d\right ) \int x \, dx \\ & = \frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}+\frac {(2 d) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^2}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b} \\ & = \frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}+d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-(2 d) \text {Subst}\left (\int x \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = \frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}+d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+(b d) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = \frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}+d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \left (b^2 d\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = \frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}+d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.36 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {1}{2} d \left (a^2 c^2 x^2+2 a b c^2 x^2 \text {arcsinh}(c x)+\frac {1}{4} b^2 \left (1+2 \text {arcsinh}(c x)^2\right ) \cosh (2 \text {arcsinh}(c x))-2 a b \text {arcsinh}(c x) \left (\text {arcsinh}(c x)-2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )+2 a^2 \log (x)-a b \left (c x \sqrt {1+c^2 x^2}+\log \left (-c x+\sqrt {1+c^2 x^2}\right )\right )+2 a b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+2 b^2 \left (-\frac {1}{3} \text {arcsinh}(c x)^3+\text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {1}{2} b^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(414\) vs. \(2(179)=358\).
Time = 0.17 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.50
method | result | size |
derivativedivides | \(d \,a^{2} \left (\frac {c^{2} x^{2}}{2}+\ln \left (c x \right )\right )+d \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {\left (2 \operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+1\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )-d a b \operatorname {arcsinh}\left (c x \right )^{2}+d a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d a b c x \sqrt {c^{2} x^{2}+1}}{2}+\frac {d a b \,\operatorname {arcsinh}\left (c x \right )}{2}+2 d a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(415\) |
default | \(d \,a^{2} \left (\frac {c^{2} x^{2}}{2}+\ln \left (c x \right )\right )+d \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {\left (2 \operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+1\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )-d a b \operatorname {arcsinh}\left (c x \right )^{2}+d a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d a b c x \sqrt {c^{2} x^{2}+1}}{2}+\frac {d a b \,\operatorname {arcsinh}\left (c x \right )}{2}+2 d a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(415\) |
parts | \(d \,a^{2} \left (\frac {c^{2} x^{2}}{2}+\ln \left (x \right )\right )-\frac {d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}}{2}-\frac {d \,b^{2} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) x c}{2}+\frac {b^{2} c^{2} d \,x^{2}}{4}+\frac {d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{4}+\frac {d \,b^{2}}{8}+d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 d \,b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 d \,b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )-d a b \operatorname {arcsinh}\left (c x \right )^{2}+d a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d a b c x \sqrt {c^{2} x^{2}+1}}{2}+\frac {d a b \,\operatorname {arcsinh}\left (c x \right )}{2}+2 d a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(420\) |
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=d \left (\int \frac {a^{2}}{x}\, dx + \int a^{2} c^{2} x\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{2} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right )}{x} \,d x \]
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