Integrand size = 24, antiderivative size = 61 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]
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Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5799, 5569, 4267, 2317, 2438} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]
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Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5799
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{d} \\ & = \frac {2 \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x))}{d} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d}+\frac {b \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(61)=122\).
Time = 0.07 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.39 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {a \text {arcsinh}(c x)}{d}-\frac {b \text {arcsinh}(c x) \log \left (1-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}-\frac {b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}+\frac {a \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {a \log \left (1+c^2 x^2\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.64
method | result | size |
parts | \(\frac {a \left (-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (x \right )\right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\) | \(161\) |
derivativedivides | \(\frac {a \left (\ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\) | \(163\) |
default | \(\frac {a \left (\ln \left (c x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\) | \(163\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a}{c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{3} + x}\, dx}{d} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,\left (d\,c^2\,x^2+d\right )} \,d x \]
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