Integrand size = 19, antiderivative size = 139 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \arctan \left (\sqrt {-1-c^2 x^2}\right )}{2 d e \sqrt {-c^2 x^2}}+\frac {b c x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{2 d \sqrt {c^2 d-e} \sqrt {e} \sqrt {-c^2 x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6435, 457, 88, 65, 211, 214} \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \arctan \left (\sqrt {-c^2 x^2-1}\right )}{2 d e \sqrt {-c^2 x^2}}+\frac {b c x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{2 d \sqrt {e} \sqrt {-c^2 x^2} \sqrt {c^2 d-e}} \]
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Rule 65
Rule 88
Rule 211
Rule 214
Rule 457
Rule 6435
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b c x) \int \frac {1}{x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e \sqrt {-c^2 x^2}} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 e \sqrt {-c^2 x^2}} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 d \sqrt {-c^2 x^2}}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{4 d e \sqrt {-c^2 x^2}} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b x) \text {Subst}\left (\int \frac {1}{d-\frac {e}{c^2}-\frac {e x^2}{c^2}} \, dx,x,\sqrt {-1-c^2 x^2}\right )}{2 c d \sqrt {-c^2 x^2}}-\frac {(b x) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {-1-c^2 x^2}\right )}{2 c d e \sqrt {-c^2 x^2}} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \arctan \left (\sqrt {-1-c^2 x^2}\right )}{2 d e \sqrt {-c^2 x^2}}+\frac {b c x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{2 d \sqrt {c^2 d-e} \sqrt {e} \sqrt {-c^2 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.95 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a}{d+e x^2}+\frac {2 b \text {csch}^{-1}(c x)}{d+e x^2}-\frac {2 b \text {arcsinh}\left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \log \left (-\frac {4 \left (i d e+c d \sqrt {e} \left (c \sqrt {d}+i \sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \sqrt {-c^2 d+e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d+e}}+\frac {b \sqrt {e} \log \left (\frac {4 i \left (d e+c d \sqrt {e} \left (i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \sqrt {-c^2 d+e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d+e}}}{4 e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(270\) vs. \(2(120)=240\).
Time = 6.22 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.95
method | result | size |
parts | \(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arccsch}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {c \sqrt {c^{2} x^{2}+1}\, \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d -e}{e}}}\right )}{c^{2}}\) | \(271\) |
derivativedivides | \(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d -e}{e}}}\right )}{c^{2}}\) | \(282\) |
default | \(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d -e}{e}}}\right )}{c^{2}}\) | \(282\) |
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Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (117) = 234\).
Time = 0.29 (sec) , antiderivative size = 615, normalized size of antiderivative = 4.42 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} - 2 \, a d e + \sqrt {-c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{2} e x^{2} - c^{2} d - 2 \, \sqrt {-c^{2} d e + e^{2}} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, e}{e x^{2} + d}\right ) - 2 \, {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 2 \, {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} - a d e + \sqrt {c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (-\frac {\sqrt {c^{2} d e - e^{2}} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{c^{2} d - e}\right ) - {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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