Integrand size = 24, antiderivative size = 187 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {1}{4} b c^3 d^2 x \sqrt {1+c^2 x^2}-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+\frac {1}{4} b c^2 d^2 \text {arcsinh}(c x)+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {c^2 d^2 (a+b \text {arcsinh}(c x))^2}{b}+2 c^2 d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b c^2 d^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5802, 283, 201, 221, 5801, 5775, 3797, 2221, 2317, 2438} \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=c^2 d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {c^2 d^2 (a+b \text {arcsinh}(c x))^2}{b}+2 c^2 d^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-b c^2 d^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{4} b c^2 d^2 \text {arcsinh}(c x)-\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2}}{2 x}+\frac {1}{4} b c^3 d^2 x \sqrt {c^2 x^2+1} \]
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Rule 201
Rule 221
Rule 283
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5801
Rule 5802
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\left (2 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{2} \left (b c d^2\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2} \, dx \\ & = -\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\left (2 c^2 d^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x} \, dx-\left (b c^3 d^2\right ) \int \sqrt {1+c^2 x^2} \, dx+\frac {1}{2} \left (3 b c^3 d^2\right ) \int \sqrt {1+c^2 x^2} \, dx \\ & = \frac {1}{4} b c^3 d^2 x \sqrt {1+c^2 x^2}-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {\left (2 c^2 d^2\right ) \text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{4} \left (3 b c^3 d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx \\ & = \frac {1}{4} b c^3 d^2 x \sqrt {1+c^2 x^2}-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+\frac {1}{4} b c^2 d^2 \text {arcsinh}(c x)+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {c^2 d^2 (a+b \text {arcsinh}(c x))^2}{b}+\frac {\left (4 c^2 d^2\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{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 c^3 d^2 x \sqrt {1+c^2 x^2}-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+\frac {1}{4} b c^2 d^2 \text {arcsinh}(c x)+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {c^2 d^2 (a+b \text {arcsinh}(c x))^2}{b}+2 c^2 d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\left (2 c^2 d^2\right ) \text {Subst}\left (\int \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 c^3 d^2 x \sqrt {1+c^2 x^2}-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+\frac {1}{4} b c^2 d^2 \text {arcsinh}(c x)+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {c^2 d^2 (a+b \text {arcsinh}(c x))^2}{b}+2 c^2 d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+\left (b c^2 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = \frac {1}{4} b c^3 d^2 x \sqrt {1+c^2 x^2}-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+\frac {1}{4} b c^2 d^2 \text {arcsinh}(c x)+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {c^2 d^2 (a+b \text {arcsinh}(c x))^2}{b}+2 c^2 d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b c^2 d^2 \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.81 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {1}{4} d^2 \left (2 a c^4 x^2-\frac {2 b c \sqrt {1+c^2 x^2}}{x}-b c^3 x \sqrt {1+c^2 x^2}+b c^2 \text {arcsinh}(c x)+2 b c^4 x^2 \text {arcsinh}(c x)-\frac {2 (a+b \text {arcsinh}(c x))}{x^2}-\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{b}+4 c^2 \left (2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )\right ) \]
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Time = 0.24 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.28
method | result | size |
parts | \(d^{2} a \left (\frac {c^{4} x^{2}}{2}-\frac {1}{2 x^{2}}+2 c^{2} \ln \left (x \right )\right )+d^{2} b \,c^{2} \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\) | \(240\) |
derivativedivides | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}+2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(243\) |
default | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}+2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(243\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=d^{2} \left (\int \frac {a}{x^{3}}\, dx + \int \frac {2 a c^{2}}{x}\, dx + \int a c^{4} x\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{4} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2}{x^3} \,d x \]
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