Integrand size = 24, antiderivative size = 158 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {10}{3} b c^3 d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5807, 5776, 5816, 4267, 2317, 2438, 5805, 30} \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {10}{3} b c^3 d \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-\frac {b c d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )-\frac {b^2 c^2 d}{3 x} \]
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Rule 30
Rule 2317
Rule 2438
Rule 4267
Rule 5776
Rule 5805
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}{3 x^3}+\frac {1}{3} (2 b c d) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx+\frac {1}{3} \left (2 c^2 d\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2} \, dx \\ & = -\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2 d\right ) \int \frac {1}{x^2} \, dx+\frac {1}{3} \left (b c^3 d\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx+\frac {1}{3} \left (4 b c^3 d\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {1}{3} \left (b c^3 d\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))+\frac {1}{3} \left (4 b c^3 d\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x)) \\ & = -\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {10}{3} b c^3 d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\frac {1}{3} \left (b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )-\frac {1}{3} \left (4 b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\frac {1}{3} \left (4 b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {10}{3} b c^3 d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\frac {1}{3} \left (b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )-\frac {1}{3} \left (4 b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\frac {1}{3} \left (4 b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right ) \\ & = -\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {10}{3} b c^3 d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.55 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {d \left (a^2+3 a^2 c^2 x^2+b^2 c^2 x^2+a b c x \sqrt {1+c^2 x^2}+2 a b \text {arcsinh}(c x)+6 a b c^2 x^2 \text {arcsinh}(c x)+b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b^2 \text {arcsinh}(c x)^2+3 b^2 c^2 x^2 \text {arcsinh}(c x)^2+5 a b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )-5 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+5 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-5 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+5 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{3 x^3} \]
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Time = 0.17 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.52
method | result | size |
parts | \(d \,a^{2} \left (-\frac {c^{2}}{x}-\frac {1}{3 x^{3}}\right )+d \,b^{2} c^{3} \left (-\frac {3 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}+\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {5 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}\right )+2 d a b \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\) | \(240\) |
derivativedivides | \(c^{3} \left (d \,a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )+d \,b^{2} \left (-\frac {3 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}+\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {5 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}\right )+2 d a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(241\) |
default | \(c^{3} \left (d \,a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )+d \,b^{2} \left (-\frac {3 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}+\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {5 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{3}\right )+2 d a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(241\) |
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=d \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {a^{2} c^{2}}{x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right )}{x^4} \,d x \]
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