Integrand size = 19, antiderivative size = 147 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{2 d e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e}} \]
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Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6434, 531, 457, 88, 65, 214} \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{2 d e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e}} \]
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Rule 65
Rule 88
Rule 214
Rule 457
Rule 531
Rule 6434
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{2 e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 d}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{4 d e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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^2 d}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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^2 d e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{2 d e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.35 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a}{d+e x^2}+\frac {2 b \text {sech}^{-1}(c x)}{d+e x^2}+\frac {2 b \log (x)}{d}-\frac {2 b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d}+\frac {b \sqrt {e} \log \left (\frac {4 \left (\frac {i d e+c^2 d^{3/2} \sqrt {e} x}{\sqrt {c^2 d+e} \left (\sqrt {d}+i \sqrt {e} x\right )}+\frac {d e \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{-i \sqrt {d} \sqrt {e}+e x}\right )}{b}\right )}{d \sqrt {c^2 d+e}}+\frac {b \sqrt {e} \log \left (\frac {4 \left (\frac {d e+i c^2 d^{3/2} \sqrt {e} x}{\sqrt {c^2 d+e} \left (i \sqrt {d}+\sqrt {e} x\right )}+\frac {d e \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{i \sqrt {d} \sqrt {e}+e x}\right )}{b}\right )}{d \sqrt {c^2 d+e}}}{4 e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(124)=248\).
Time = 5.52 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.14
method | result | size |
parts | \(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arcsech}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {c^{3} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (-2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} d +\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d +\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d -2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e +\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e +\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 \sqrt {-c^{2} x^{2}+1}\, d \left (e +\sqrt {-c^{2} d e}\right ) \left (-e +\sqrt {-c^{2} d e}\right ) \sqrt {\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(462\) |
derivativedivides | \(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} d -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d +2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 c \sqrt {-c^{2} x^{2}+1}\, d \left (e +\sqrt {-c^{2} d e}\right ) \left (-e +\sqrt {-c^{2} d e}\right ) \sqrt {\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(477\) |
default | \(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} d -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d +2 \sqrt {\frac {c^{2} d +e}{e}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e -\ln \left (\frac {2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e -\ln \left (-\frac {2 \left (\sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-c^{2} x^{2}+1}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 c \sqrt {-c^{2} x^{2}+1}\, d \left (e +\sqrt {-c^{2} d e}\right ) \left (-e +\sqrt {-c^{2} d e}\right ) \sqrt {\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(477\) |
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Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (89) = 178\).
Time = 0.31 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.10 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d e - \sqrt {c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{4} d^{2} + 4 \, c^{2} d e - {\left (c^{4} d e + 2 \, c^{2} e^{2}\right )} x^{2} + 4 \, {\left (c^{3} d e + c e^{2}\right )} x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 4 \, e^{2} + 2 \, {\left (c^{2} e x^{2} - c^{2} d - {\left (c^{3} d + 2 \, c e\right )} x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, e\right )} \sqrt {c^{2} d e + e^{2}}}{e x^{2} + d}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} + a d e + \sqrt {-c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (\frac {\sqrt {-c^{2} d e - e^{2}} c d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - \sqrt {-c^{2} d e - e^{2}} {\left (e x^{2} + d\right )}}{{\left (c^{2} d e + e^{2}\right )} x^{2}}\right ) + {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c^{2} d^{2} + b d e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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