Integrand size = 16, antiderivative size = 60 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {x^2}{2 a}+\frac {b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \]
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Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5545, 3868, 2739, 632, 210} \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}+\frac {x^2}{2 a} \]
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Rule 210
Rule 632
Rule 2739
Rule 3868
Rule 5545
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b \text {csch}(c+d x)} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2 a}-\frac {\text {Subst}\left (\int \frac {1}{1+\frac {a \sinh (c+d x)}{b}} \, dx,x,x^2\right )}{2 a} \\ & = \frac {x^2}{2 a}+\frac {i \text {Subst}\left (\int \frac {1}{1-\frac {2 i a x}{b}+x^2} \, dx,x,i \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a d} \\ & = \frac {x^2}{2 a}-\frac {(2 i) \text {Subst}\left (\int \frac {1}{-4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,-\frac {2 i a}{b}+2 i \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a d} \\ & = \frac {x^2}{2 a}+\frac {b \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {\frac {c}{d}+x^2-\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}}{2 a} \]
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Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.48
method | result | size |
derivativedivides | \(\frac {\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a}}{2 d}\) | \(89\) |
default | \(\frac {\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a}}{2 d}\) | \(89\) |
risch | \(\frac {x^{2}}{2 a}+\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{2 \sqrt {a^{2}+b^{2}}\, d a}-\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{2 \sqrt {a^{2}+b^{2}}\, d a}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (55) = 110\).
Time = 0.27 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.55 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {{\left (a^{2} + b^{2}\right )} d x^{2} + \sqrt {a^{2} + b^{2}} b \log \left (\frac {a^{2} \cosh \left (d x^{2} + c\right )^{2} + a^{2} \sinh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x^{2} + c\right ) + a b\right )} \sinh \left (d x^{2} + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b\right )}}{a \cosh \left (d x^{2} + c\right )^{2} + a \sinh \left (d x^{2} + c\right )^{2} + 2 \, b \cosh \left (d x^{2} + c\right ) + 2 \, {\left (a \cosh \left (d x^{2} + c\right ) + b\right )} \sinh \left (d x^{2} + c\right ) - a}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \]
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\[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x}{a + b \operatorname {csch}{\left (c + d x^{2} \right )}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=-\frac {b \log \left (\frac {a e^{\left (-d x^{2} - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x^{2} - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a d} + \frac {d x^{2} + c}{2 \, a d} \]
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Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=-\frac {b \log \left (\frac {{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a d} + \frac {d x^{2} + c}{2 \, a d} \]
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Time = 2.78 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.92 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {x^2}{2\,a}-\frac {\mathrm {atan}\left (\frac {a\,d\,\sqrt {b^2}}{\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}-\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}{a^2\,d\,\sqrt {b^2}}+\frac {a^2\,b\,d\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {b^2}\,\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}{a^6\,d^2+a^4\,b^2\,d^2}\right )\,\sqrt {b^2}}{\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}} \]
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