Integrand size = 14, antiderivative size = 26 \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^2}{2}-\frac {b \text {arctanh}\left (\cosh \left (c+d x^2\right )\right )}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 5545, 3855} \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^2}{2}-\frac {b \text {arctanh}\left (\cosh \left (c+d x^2\right )\right )}{2 d} \]
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Rule 14
Rule 3855
Rule 5545
Rubi steps \begin{align*} \text {integral}& = \int \left (a x+b x \text {csch}\left (c+d x^2\right )\right ) \, dx \\ & = \frac {a x^2}{2}+b \int x \text {csch}\left (c+d x^2\right ) \, dx \\ & = \frac {a x^2}{2}+\frac {1}{2} b \text {Subst}\left (\int \text {csch}(c+d x) \, dx,x,x^2\right ) \\ & = \frac {a x^2}{2}-\frac {b \text {arctanh}\left (\cosh \left (c+d x^2\right )\right )}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(26)=52\).
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^2}{2}-\frac {b \log \left (\cosh \left (\frac {c}{2}+\frac {d x^2}{2}\right )\right )}{2 d}+\frac {b \log \left (\sinh \left (\frac {c}{2}+\frac {d x^2}{2}\right )\right )}{2 d} \]
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Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
parts | \(\frac {a \,x^{2}}{2}+\frac {b \ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(26\) |
parallelrisch | \(\frac {a d \,x^{2}+b \ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(27\) |
derivativedivides | \(\frac {\left (d \,x^{2}+c \right ) a +b \ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(30\) |
default | \(\frac {\left (d \,x^{2}+c \right ) a +b \ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(30\) |
risch | \(\frac {a \,x^{2}}{2}+\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}-1\right )}{2 d}-\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+1\right )}{2 d}\) | \(42\) |
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12 \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a d x^{2} - b \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) + b \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right )}{2 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\begin {cases} \frac {a \left (c + d x^{2}\right ) + b \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x^{2}}{2} \right )} \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x^{2} \left (a + b \operatorname {csch}{\left (c \right )}\right )}{2} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {1}{2} \, a x^{2} + \frac {b \log \left (\tanh \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {{\left (d x^{2} + c\right )} a}{2 \, d} - \frac {b \log \left (e^{\left (d x^{2} + c\right )} + 1\right )}{2 \, d} + \frac {b \log \left ({\left | e^{\left (d x^{2} + c\right )} - 1 \right |}\right )}{2 \, d} \]
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Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a\,x^2}{2}-\frac {\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \]
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