Integrand size = 26, antiderivative size = 194 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {a+b \text {arcsinh}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{d \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5811, 5816, 4267, 2317, 2438, 209} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {c^2 x^2+1} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}+\frac {a+b \text {arcsinh}(c x)}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \arctan (c x)}{d \sqrt {c^2 d x^2+d}} \]
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Rule 209
Rule 2317
Rule 2438
Rule 4267
Rule 5811
Rule 5816
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arcsinh}(c x)}{d \sqrt {d+c^2 d x^2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {d+c^2 d x^2}} \, dx}{d}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d \sqrt {d+c^2 d x^2}} \\ & = \frac {a+b \text {arcsinh}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{d \sqrt {d+c^2 d x^2}} \\ & = \frac {a+b \text {arcsinh}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{d \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d \sqrt {d+c^2 d x^2}} \\ & = \frac {a+b \text {arcsinh}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{d \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \\ & = \frac {a+b \text {arcsinh}(c x)}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{d \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\frac {a \sqrt {d+c^2 d x^2}}{1+c^2 x^2}+a \sqrt {d} \log (x)-a \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b d \left (\text {arcsinh}(c x)-2 \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {d+c^2 d x^2}}}{d^2} \]
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Time = 0.24 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.40
method | result | size |
default | \(\frac {a}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+2 \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2}}\) | \(271\) |
parts | \(\frac {a}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+2 \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2}}\) | \(271\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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