Integrand size = 8, antiderivative size = 75 \[ \int x \text {csch}^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 b^2}-\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {a \text {arctanh}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{b^2} \]
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Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6457, 5577, 3858, 3855, 3852, 8} \[ \int x \text {csch}^{-1}(a+b x) \, dx=-\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}-\frac {a \text {arctanh}\left (\sqrt {\frac {1}{(a+b x)^2}+1}\right )}{b^2}+\frac {(a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x) \]
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Rule 8
Rule 3852
Rule 3855
Rule 3858
Rule 5577
Rule 6457
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x \coth (x) \text {csch}(x) (-a+\text {csch}(x)) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{b^2} \\ & = \frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {\text {Subst}\left (\int (-a+\text {csch}(x))^2 \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 b^2} \\ & = -\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {\text {Subst}\left (\int \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 b^2}+\frac {a \text {Subst}\left (\int \text {csch}(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{b^2} \\ & = -\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {a \text {arctanh}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{b^2}+\frac {i \text {Subst}\left (\int 1 \, dx,x,-i (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}\right )}{2 b^2} \\ & = \frac {(a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 b^2}-\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {a \text {arctanh}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{b^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int x \text {csch}^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+b^2 x^2 \text {csch}^{-1}(a+b x)-a^2 \text {arcsinh}\left (\frac {1}{a+b x}\right )-2 a \log \left ((a+b x) \left (1+\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{2 b^2} \]
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Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )+\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (2 a \,\operatorname {arcsinh}\left (b x +a \right )-\sqrt {\left (b x +a \right )^{2}+1}\right )}{2 \left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}}}{b^{2}}\) | \(97\) |
default | \(\frac {-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )+\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (2 a \,\operatorname {arcsinh}\left (b x +a \right )-\sqrt {\left (b x +a \right )^{2}+1}\right )}{2 \left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}}}{b^{2}}\) | \(97\) |
parts | \(\frac {x^{2} \operatorname {arccsch}\left (b x +a \right )}{2}+\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \left (-a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right ) \sqrt {b^{2}}-2 a \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b +\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \sqrt {b^{2}}\right )}{2 b^{2} \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\) | \(177\) |
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Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (65) = 130\).
Time = 0.26 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.80 \[ \int x \text {csch}^{-1}(a+b x) \, dx=\frac {b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - a^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) + a^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) + 2 \, a \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, b^{2}} \]
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\[ \int x \text {csch}^{-1}(a+b x) \, dx=\int x \operatorname {acsch}{\left (a + b x \right )}\, dx \]
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\[ \int x \text {csch}^{-1}(a+b x) \, dx=\int { x \operatorname {arcsch}\left (b x + a\right ) \,d x } \]
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\[ \int x \text {csch}^{-1}(a+b x) \, dx=\int { x \operatorname {arcsch}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \text {csch}^{-1}(a+b x) \, dx=\int x\,\mathrm {asinh}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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