Integrand size = 15, antiderivative size = 15 \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {2 x}{\sqrt {1+x^2}}+\text {arcsinh}(x) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {393, 221} \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=\text {arcsinh}(x)-\frac {2 x}{\sqrt {x^2+1}} \]
[In]
[Out]
Rule 221
Rule 393
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{\sqrt {1+x^2}}+\int \frac {1}{\sqrt {1+x^2}} \, dx \\ & = -\frac {2 x}{\sqrt {1+x^2}}+\sinh ^{-1}(x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93 \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {2 x}{\sqrt {1+x^2}}-\log \left (-x+\sqrt {1+x^2}\right ) \]
[In]
[Out]
Time = 2.38 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(\operatorname {arcsinh}\left (x \right )-\frac {2 x}{\sqrt {x^{2}+1}}\) | \(14\) |
risch | \(\operatorname {arcsinh}\left (x \right )-\frac {2 x}{\sqrt {x^{2}+1}}\) | \(14\) |
trager | \(-\frac {2 x}{\sqrt {x^{2}+1}}+\ln \left (x +\sqrt {x^{2}+1}\right )\) | \(22\) |
meijerg | \(-\frac {x}{\sqrt {x^{2}+1}}+\frac {-\frac {\sqrt {\pi }\, x}{\sqrt {x^{2}+1}}+\sqrt {\pi }\, \operatorname {arcsinh}\left (x \right )}{\sqrt {\pi }}\) | \(36\) |
pseudoelliptic | \(\frac {-\ln \left (\frac {-x +\sqrt {x^{2}+1}}{x}\right ) \sqrt {x^{2}+1}+\ln \left (\frac {x +\sqrt {x^{2}+1}}{x}\right ) \sqrt {x^{2}+1}-4 x}{2 \sqrt {x^{2}+1}}\) | \(61\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.93 \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {2 \, x^{2} + {\left (x^{2} + 1\right )} \log \left (-x + \sqrt {x^{2} + 1}\right ) + 2 \, \sqrt {x^{2} + 1} x + 2}{x^{2} + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
Time = 2.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=\frac {x^{2} \operatorname {asinh}{\left (x \right )}}{x^{2} + 1} - \frac {2 x}{\sqrt {x^{2} + 1}} + \frac {\operatorname {asinh}{\left (x \right )}}{x^{2} + 1} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {2 \, x}{\sqrt {x^{2} + 1}} + \operatorname {arsinh}\left (x\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {2 \, x}{\sqrt {x^{2} + 1}} - \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80 \[ \int \frac {-1+x^2}{\left (1+x^2\right )^{3/2}} \, dx=\frac {\mathrm {asinh}\left (x\right )+x^2\,\mathrm {asinh}\left (x\right )-2\,x\,\sqrt {x^2+1}}{x^2+1} \]
[In]
[Out]