Integrand size = 24, antiderivative size = 146 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2} \]
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Time = 0.18 (sec) , antiderivative size = 146, 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 = {5809, 5811, 5799, 5569, 4267, 2317, 2438, 197, 277} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\frac {4 c^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (c^2 x^2+1\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b c}{2 d^2 x \sqrt {c^2 x^2+1}} \]
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Rule 197
Rule 277
Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5799
Rule 5809
Rule 5811
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\left (2 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2} \\ & = -\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac {\left (2 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx}{d} \\ & = -\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac {\left (2 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{d^2} \\ & = -\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac {\left (4 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arcsinh}(c x))}{d^2} \\ & = -\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}-\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2} \\ & = -\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{d^2} \\ & = -\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {c^2 (a+b \text {arcsinh}(c x))}{d^2 \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(326\) vs. \(2(146)=292\).
Time = 0.31 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.23 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\frac {2 a^2 c^2}{b}-\frac {2 a}{x^2}+\frac {b c}{x \sqrt {1+c^2 x^2}}+\frac {2 b c^3 x}{\sqrt {1+c^2 x^2}}-\frac {2 b c \sqrt {1+c^2 x^2}}{x}+\frac {a}{x^2+c^2 x^4}+4 a c^2 \text {arcsinh}(c x)-\frac {2 b \text {arcsinh}(c x)}{x^2}+\frac {b \text {arcsinh}(c x)}{x^2+c^2 x^4}+4 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+4 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-4 a c^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )-4 b c^2 \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+2 a c^2 \log \left (1+c^2 x^2\right )+4 b c^2 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+4 b c^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-2 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \]
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Time = 0.23 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.66
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (c x \right )-\frac {1}{2 \left (c^{2} x^{2}+1\right )}+\ln \left (c^{2} x^{2}+1\right )\right )}{d^{2}}+\frac {b \left (-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}+1\right )}-2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\) | \(242\) |
default | \(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (c x \right )-\frac {1}{2 \left (c^{2} x^{2}+1\right )}+\ln \left (c^{2} x^{2}+1\right )\right )}{d^{2}}+\frac {b \left (-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}+1\right )}-2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\) | \(242\) |
parts | \(\frac {a \left (\frac {c^{4} \left (-\frac {1}{c^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 \ln \left (c^{2} x^{2}+1\right )}{c^{2}}\right )}{2}-\frac {1}{2 x^{2}}-2 c^{2} \ln \left (x \right )\right )}{d^{2}}+\frac {b \,c^{2} \left (-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}+1\right )}-2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\) | \(253\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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