Optimal. Leaf size=42 \[ \sin ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {1+4 x \sqrt {1-x^2}}{\sqrt {3} \left (1-2 x^2\right )}\right )}{\sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [C] Result contains complex when optimal does not.
time = 0.11, antiderivative size = 122, normalized size of antiderivative = 2.90, number of steps
used = 12, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6874, 1121,
632, 210, 1307, 222, 1188, 385, 211} \begin {gather*} \text {ArcSin}(x)-\frac {\text {ArcTan}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {ArcTan}\left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 211
Rule 222
Rule 385
Rule 632
Rule 1121
Rule 1188
Rule 1307
Rule 6874
Rubi steps
\begin {align*} \int \frac {1}{\frac {1}{x}+\sqrt {1-x^2}} \, dx &=\int \left (\frac {x}{1-x^2+x^4}-\frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4}\right ) \, dx\\ &=\int \frac {x}{1-x^2+x^4} \, dx-\int \frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )+\int \frac {1}{\sqrt {1-x^2}} \, dx-\int \frac {1}{\sqrt {1-x^2} \left (1-x^2+x^4\right )} \, dx\\ &=\sin ^{-1}(x)+\frac {(2 i) \int \frac {1}{\sqrt {1-x^2} \left (-1-i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}}-\frac {(2 i) \int \frac {1}{\sqrt {1-x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}}-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )\\ &=\sin ^{-1}(x)+\frac {\tan ^{-1}\left (\frac {-1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {(2 i) \text {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-1-i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-1+i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}\\ &=\sin ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {-1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 56, normalized size = 1.33 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )+\frac {2 i \tanh ^{-1}\left (\frac {(2+i)-2 i x^2+2 x \sqrt {1-x^2}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.22, size = 234, normalized size = 5.57
method | result | size |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+3 x \sqrt {-x^{2}+1}-\RootOf \left (\textit {\_Z}^{2}+3\right )}{\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}-x^{2}+2}\right )}{3}\) | \(89\) |
default | \(-2 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )+\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{6}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{6}+\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{6}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{3}\) | \(234\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 73, normalized size = 1.74 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - 1\right )} \sqrt {-x^{2} + 1}}{3 \, {\left (x^{3} - x\right )}}\right ) - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\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}{x \sqrt {1 - x^{2}} + 1}\, 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. 193 vs.
\(2 (37) = 74\).
time = 2.62, size = 193, normalized size = 4.60 \begin {gather*} \frac {1}{2} \, \pi \mathrm {sgn}\left (x\right ) - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.92, size = 549, normalized size = 13.07 \begin {gather*} \mathrm {asin}\left (x\right )-\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )-1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{\frac {\sqrt {3}}{2}-x+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}+\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )-1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )+1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}-\frac {\ln \left (x-\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}-\frac {\ln \left (x+\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}+\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )+1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}+\frac {\ln \left (x-\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}+\frac {\ln \left (x+\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________