Integrand size = 10, antiderivative size = 63 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=-\frac {b \text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x)}{x}+\frac {2 b \text {arctanh}\left (\frac {a+\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{a \sqrt {1+a^2}} \]
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Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6457, 5577, 3868, 2739, 632, 212} \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\frac {2 b \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )+a}{\sqrt {a^2+1}}\right )}{a \sqrt {a^2+1}}-\frac {b \text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x)}{x} \]
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Rule 212
Rule 632
Rule 2739
Rule 3868
Rule 5577
Rule 6457
Rubi steps \begin{align*} \text {integral}& = -\left (b \text {Subst}\left (\int \frac {x \coth (x) \text {csch}(x)}{(-a+\text {csch}(x))^2} \, dx,x,\text {csch}^{-1}(a+b x)\right )\right ) \\ & = -\frac {\text {csch}^{-1}(a+b x)}{x}+b \text {Subst}\left (\int \frac {1}{-a+\text {csch}(x)} \, dx,x,\text {csch}^{-1}(a+b x)\right ) \\ & = -\frac {b \text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x)}{x}+\frac {b \text {Subst}\left (\int \frac {1}{1-a \sinh (x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )}{a} \\ & = -\frac {b \text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x)}{x}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{1-2 a x-x^2} \, dx,x,\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )}{a} \\ & = -\frac {b \text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x)}{x}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{4 \left (1+a^2\right )-x^2} \, dx,x,-2 a-2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )}{a} \\ & = -\frac {b \text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x)}{x}+\frac {2 b \text {arctanh}\left (\frac {a+\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{a \sqrt {1+a^2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(63)=126\).
Time = 0.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.24 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=-\frac {\text {csch}^{-1}(a+b x)}{x}-\frac {b \left (\sqrt {1+a^2} \text {arcsinh}\left (\frac {1}{a+b x}\right )+\log (x)-\log \left (1+a^2+a b x+a \sqrt {1+a^2} \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+\sqrt {1+a^2} b x \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{a \sqrt {1+a^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs. \(2(57)=114\).
Time = 0.64 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.02
method | result | size |
derivativedivides | \(b \left (-\frac {\operatorname {arccsch}\left (b x +a \right )}{b x}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \sqrt {a^{2}+1}-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right )\right )}{\sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}+1}}\right )\) | \(127\) |
default | \(b \left (-\frac {\operatorname {arccsch}\left (b x +a \right )}{b x}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \sqrt {a^{2}+1}-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 \left (b x +a \right ) a +2}{b x}\right )\right )}{\sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}+1}}\right )\) | \(127\) |
parts | \(-\frac {\operatorname {arccsch}\left (b x +a \right )}{x}-\frac {b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right ) \sqrt {a^{2}+1}-\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{\sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}+1}}\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 343, normalized size of antiderivative = 5.44 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=-\frac {{\left (a^{2} + 1\right )} b x \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) - {\left (a^{2} + 1\right )} b x \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) - \sqrt {a^{2} + 1} b x \log \left (-\frac {a^{2} b x + a^{3} + {\left (a b x + a^{2} + {\left (a b x + a^{2}\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1\right )} \sqrt {a^{2} + 1} + {\left (a^{3} + {\left (a^{2} + 1\right )} b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + a}{x}\right ) + {\left (a^{3} + a\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right )}{{\left (a^{3} + a\right )} x} \]
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\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {csch}^{-1}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x^2} \,d x \]
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