Optimal. Leaf size=45 \[ -\sin ^{-1}(x)-\sqrt {1-x^2} \tan ^{-1}(x)+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5094, 399, 222,
385, 209} \begin {gather*} -\sqrt {1-x^2} \tan ^{-1}(x)+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )-\sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 222
Rule 385
Rule 399
Rule 5094
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(x)}{\sqrt {1-x^2}} \, dx &=-\sqrt {1-x^2} \tan ^{-1}(x)+\int \frac {\sqrt {1-x^2}}{1+x^2} \, dx\\ &=-\sqrt {1-x^2} \tan ^{-1}(x)+2 \int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\sin ^{-1}(x)-\sqrt {1-x^2} \tan ^{-1}(x)+2 \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=-\sin ^{-1}(x)-\sqrt {1-x^2} \tan ^{-1}(x)+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} -\sin ^{-1}(x)-\sqrt {1-x^2} \tan ^{-1}(x)+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x \arctan \left (x \right )}{\sqrt {-x^{2}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 69, normalized size = 1.53 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (3 \, x^{2} - 1\right )} \sqrt {-x^{2} + 1}}{4 \, {\left (x^{3} - x\right )}}\right ) - \sqrt {-x^{2} + 1} \arctan \left (x\right ) + \arctan \left (\frac {\sqrt {-x^{2} + 1} x}{x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \operatorname {atan}{\left (x \right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (37) = 74\).
time = 0.01, size = 133, normalized size = 2.96 \begin {gather*} -\frac {1}{2} \pi \mathrm {sign}\left (x\right )-\sqrt {2} \left (-\frac {1}{2} \pi \mathrm {sign}\left (x\right )-\arctan \left (\frac {x \left (\left (-\frac {-2 \sqrt {-x^{2}+1}+2}{2 x}\right )^{2}-1\right )}{\sqrt {2} \left (-2 \sqrt {-x^{2}+1}+2\right )}\right )\right )-\arctan \left (\frac {x \left (\left (-\frac {-2 \sqrt {-x^{2}+1}+2}{2 x}\right )^{2}-1\right )}{-2 \sqrt {-x^{2}+1}+2}\right )-\sqrt {-x^{2}+1} \arctan x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 37, normalized size = 0.82 \begin {gather*} \sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{\sqrt {1-x^2}}\right )-\mathrm {atan}\left (x\right )\,\sqrt {1-x^2}-\mathrm {asin}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________