Integrand size = 24, antiderivative size = 180 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 c^2 d \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 180, 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 = {5807, 5775, 3797, 2221, 2611, 2320, 6724, 5805, 29, 5783} \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-b c^2 d \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-\frac {d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {b c d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2+c^2 d \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b^2 c^2 d \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x) \]
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Rule 29
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 5783
Rule 5805
Rule 5807
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+(b c d) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx+\left (c^2 d\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x} \, dx \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {\left (c^2 d\right ) \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}+\left (b^2 c^2 d\right ) \int \frac {1}{x} \, dx+\left (b c^3 d\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+b^2 c^2 d \log (x)+\frac {\left (2 c^2 d\right ) \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 {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-\left (2 c^2 d\right ) \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 {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+\left (b c^2 d\right ) \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 {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \left (b^2 c^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 {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)-b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 c^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 = 0.23 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {a^2 d}{2 x^2}+2 a b c^2 d \left (-\frac {\sqrt {1+c^2 x^2}}{2 c x}-\frac {\text {arcsinh}(c x)}{2 c^2 x^2}\right )+a^2 c^2 d \log (x)+b^2 c^2 d \left (-\frac {\sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}-\frac {\text {arcsinh}(c x)^2}{2 c^2 x^2}+\log (c x)\right )+2 a b c^2 d \left (-\frac {1}{2} \text {arcsinh}(c x)^2+\text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+b^2 c^2 d \left (-\frac {1}{3} \text {arcsinh}(c x)^3+\text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(197)=394\).
Time = 0.20 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.24
method | result | size |
derivativedivides | \(c^{2} \left (d \,a^{2} \left (\ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(404\) |
default | \(c^{2} \left (d \,a^{2} \left (\ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(404\) |
parts | \(d \,a^{2} \left (-\frac {1}{2 x^{2}}+c^{2} \ln \left (x \right )\right )+d \,b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d a b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\) | \(405\) |
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=d \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \frac {a^{2} c^{2}}{x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right )}{x^3} \,d x \]
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