Integrand size = 24, antiderivative size = 131 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=2 b^2 c^2 d x-2 b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 c^2 d x (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}-4 b c d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-2 b^2 c d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+2 b^2 c d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5807, 5772, 5798, 8, 5806, 5816, 4267, 2317, 2438} \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=-4 b c d \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-2 b c d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}+2 c^2 d x (a+b \text {arcsinh}(c x))^2-2 b^2 c d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+2 b^2 c d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )+2 b^2 c^2 d x \]
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Rule 8
Rule 2317
Rule 2438
Rule 4267
Rule 5772
Rule 5798
Rule 5806
Rule 5807
Rule 5816
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}+(2 b c d) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\left (2 c^2 d\right ) \int (a+b \text {arcsinh}(c x))^2 \, dx \\ & = 2 b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 c^2 d x (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}+(2 b c d) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx-\left (2 b^2 c^2 d\right ) \int 1 \, dx-\left (4 b c^3 d\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -2 b^2 c^2 d x-2 b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 c^2 d x (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}+(2 b c d) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\left (4 b^2 c^2 d\right ) \int 1 \, dx \\ & = 2 b^2 c^2 d x-2 b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 c^2 d x (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}-4 b c d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\left (2 b^2 c d\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\left (2 b^2 c d\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right ) \\ & = 2 b^2 c^2 d x-2 b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 c^2 d x (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}-4 b c d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\left (2 b^2 c d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\left (2 b^2 c d\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right ) \\ & = 2 b^2 c^2 d x-2 b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 c^2 d x (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{x}-4 b c d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-2 b^2 c d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+2 b^2 c d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.47 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {d \left (-a^2+a^2 c^2 x^2+2 a b c x \left (-\sqrt {1+c^2 x^2}+c x \text {arcsinh}(c x)\right )+b^2 c x \left (2 c x-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2\right )-2 a b \left (\text {arcsinh}(c x)+c x \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )\right )-b^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)+2 c x \left (-\log \left (1-e^{-\text {arcsinh}(c x)}\right )+\log \left (1+e^{-\text {arcsinh}(c x)}\right )\right )\right )-2 c x \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+2 c x \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )\right )}{x} \]
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Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(c \left (d \,a^{2} \left (c x -\frac {1}{c x}\right )+d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 d \,b^{2} c x -\frac {d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 d \,b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d \,b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d a b \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(239\) |
default | \(c \left (d \,a^{2} \left (c x -\frac {1}{c x}\right )+d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 d \,b^{2} c x -\frac {d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}-2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 d \,b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d \,b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d \,b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d a b \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(239\) |
parts | \(d \,a^{2} \left (c^{2} x -\frac {1}{x}\right )+d \,b^{2} c^{2} \operatorname {arcsinh}\left (c x \right )^{2} x -2 d \,b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 b^{2} c^{2} d x -\frac {d \,b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{x}-2 d \,b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} c d \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 d \,b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} c d \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d a b c \left (\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\) | \(243\) |
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=d \left (\int a^{2} c^{2}\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right )}{x^2} \,d x \]
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