Optimal. Leaf size=34 \[ \frac {x \log \left (\sqrt {x^2+1}+x\right )}{\sqrt {1-x^2}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {191, 2554, 275, 216} \[ \frac {x \log \left (\sqrt {x^2+1}+x\right )}{\sqrt {1-x^2}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 216
Rule 275
Rule 2554
Rubi steps
\begin {align*} \int \frac {\log \left (x+\sqrt {1+x^2}\right )}{\left (1-x^2\right )^{3/2}} \, dx &=\frac {x \log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1-x^2}}-\int \frac {x}{\sqrt {1-x^4}} \, dx\\ &=\frac {x \log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1-x^2}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{2} \sin ^{-1}\left (x^2\right )+\frac {x \log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1-x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 64, normalized size = 1.88 \[ \frac {1}{2} \sqrt {1-x^2} \left (-\frac {2 x \log \left (\sqrt {x^2+1}+x\right )}{x^2-1}-\frac {\sqrt {x^2+1} \sin ^{-1}\left (x^2\right )}{\sqrt {1-x^4}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 62, normalized size = 1.82 \[ -\frac {\sqrt {-x^{2} + 1} x \log \left (x + \sqrt {x^{2} + 1}\right ) - {\left (x^{2} - 1\right )} \arctan \left (\frac {\sqrt {x^{2} + 1} \sqrt {-x^{2} + 1} - 1}{x^{2}}\right )}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.38, size = 36, normalized size = 1.06 \[ -\frac {\sqrt {-x^{2} + 1} x \log \left (x + \sqrt {x^{2} + 1}\right )}{x^{2} - 1} - \frac {1}{2} \, \arcsin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (x +\sqrt {x^{2}+1}\right )}{\left (-x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (x + \sqrt {x^{2} + 1}\right )}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\ln \left (x+\sqrt {x^2+1}\right )}{{\left (1-x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________