Integrand size = 19, antiderivative size = 42 \[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx=\arcsin (x)-\frac {\arctan \left (\frac {1+4 x \sqrt {1-x^2}}{\sqrt {3} \left (1-2 x^2\right )}\right )}{\sqrt {3}} \] Output:
arcsin(x)-1/3*arctan(1/3*(1+4*x*(-x^2+1)^(1/2))*3^(1/2)/(-2*x^2+1))*3^(1/2 )
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.07 \[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx=-2 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right )+\text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log (-1+x)+\log \left (\sqrt {1-x^2}+\text {$\#$1}-x \text {$\#$1}\right )-\log (-1+x) \text {$\#$1}^2+\log \left (\sqrt {1-x^2}+\text {$\#$1}-x \text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[x/(1 + x*Sqrt[1 - x^2]),x]
Output:
-2*ArcTan[Sqrt[1 - x^2]/(1 + x)] + RootSum[1 + 2*#1 + 2*#1^2 - 2*#1^3 + #1 ^4 & , (-Log[-1 + x] + Log[Sqrt[1 - x^2] + #1 - x*#1] - Log[-1 + x]*#1^2 + Log[Sqrt[1 - x^2] + #1 - x*#1]*#1^2)/(1 + 2*#1 - 3*#1^2 + 2*#1^3) & ]
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x}{\sqrt {1-x^2} x+1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x}{x^4-x^2+1}-\frac {x^2 \sqrt {1-x^2}}{x^4-x^2+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \arcsin (x)-\frac {\arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}\) |
Input:
Int[x/(1 + x*Sqrt[1 - x^2]),x]
Output:
ArcSin[x] - ArcTan[(1 - 2*x^2)/Sqrt[3]]/Sqrt[3] - ArcTan[x/(Sqrt[-((I - Sq rt[3])/(I + Sqrt[3]))]*Sqrt[1 - x^2])]/Sqrt[3] - ArcTan[(Sqrt[-((I - Sqrt[ 3])/(I + Sqrt[3]))]*x)/Sqrt[1 - x^2]]/Sqrt[3]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.02
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-2 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+3 x \sqrt {-x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+x^{2}-2}\right )}{3}\) | \(85\) |
default | \(\frac {i \sqrt {3}\, \ln \left (\frac {\left (-1+\sqrt {-x^{2}+1}\right )^{2}}{x^{2}}+\frac {\left (i \sqrt {3}-1\right ) \left (-1+\sqrt {-x^{2}+1}\right )}{x}-1\right )}{6}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (-1+\sqrt {-x^{2}+1}\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (-1+\sqrt {-x^{2}+1}\right )}{x}-1\right )}{6}-2 \arctan \left (\frac {-1+\sqrt {-x^{2}+1}}{x}\right )+\frac {i \sqrt {3}\, \ln \left (\frac {\left (-1+\sqrt {-x^{2}+1}\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-1+\sqrt {-x^{2}+1}\right )}{x}-1\right )}{6}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (-1+\sqrt {-x^{2}+1}\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-1+\sqrt {-x^{2}+1}\right )}{x}-1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{3}\) | \(234\) |
Input:
int(x/(1+x*(-x^2+1)^(1/2)),x,method=_RETURNVERBOSE)
Output:
RootOf(_Z^2+1)*ln(RootOf(_Z^2+1)*(-x^2+1)^(1/2)+x)-1/3*RootOf(_Z^2+3)*ln(( -2*x^2*RootOf(_Z^2+3)+3*x*(-x^2+1)^(1/2)+RootOf(_Z^2+3))/(x^2*RootOf(_Z^2+ 3)+x^2-2))
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74 \[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - 1\right )} \sqrt {-x^{2} + 1}}{3 \, {\left (x^{3} - x\right )}}\right ) - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \] Input:
integrate(x/(1+x*(-x^2+1)^(1/2)),x, algorithm="fricas")
Output:
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) + 1/3*sqrt(3)*arctan(1/3*sqrt( 3)*(2*x^2 - 1)*sqrt(-x^2 + 1)/(x^3 - x)) - 2*arctan((sqrt(-x^2 + 1) - 1)/x )
\[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx=\int \frac {x}{x \sqrt {1 - x^{2}} + 1}\, dx \] Input:
integrate(x/(1+x*(-x**2+1)**(1/2)),x)
Output:
Integral(x/(x*sqrt(1 - x**2) + 1), x)
\[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx=\int { \frac {x}{\sqrt {-x^{2} + 1} x + 1} \,d x } \] Input:
integrate(x/(1+x*(-x^2+1)^(1/2)),x, algorithm="maxima")
Output:
integrate(x/(sqrt(-x^2 + 1)*x + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (37) = 74\).
Time = 0.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.60 \[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx=\frac {1}{2} \, \pi \mathrm {sgn}\left (x\right ) - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \] Input:
integrate(x/(1+x*(-x^2+1)^(1/2)),x, algorithm="giac")
Output:
1/2*pi*sgn(x) - 1/6*sqrt(3)*(pi*sgn(x) + 2*arctan(-1/3*sqrt(3)*x*((sqrt(-x ^2 + 1) - 1)/x + (sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))) - 1/6*sqrt(3)*(pi*sgn(x) + 2*arctan(1/3*sqrt(3)*x*((sqrt(-x^2 + 1) - 1)/x - (sqrt(-x^2 + 1) - 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1))) + 1/3*sqrt(3)*arcta n(1/3*sqrt(3)*(2*x^2 - 1)) + arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1 )/(sqrt(-x^2 + 1) - 1))
Time = 0.70 (sec) , antiderivative size = 549, normalized size of antiderivative = 13.07 \[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx =\text {Too large to display} \] Input:
int(x/(x*(1 - x^2)^(1/2) + 1),x)
Output:
asin(x) - log((((x*(3^(1/2)/2 + 1i/2) - 1)*1i)/(1 - (3^(1/2)/2 + 1i/2)^2)^ (1/2) - (1 - x^2)^(1/2)*1i)/(3^(1/2)/2 - x + 1i/2))/((1 - (3^(1/2)/2 + 1i/ 2)^2)^(1/2)*(3^(1/2) - 4*(3^(1/2)/2 + 1i/2)^3 + 1i)) + log((((x*(3^(1/2)/2 - 1i/2) - 1)*1i)/(1 - (3^(1/2)/2 - 1i/2)^2)^(1/2) - (1 - x^2)^(1/2)*1i)/( x - 3^(1/2)/2 + 1i/2))/((1 - (3^(1/2)/2 - 1i/2)^2)^(1/2)*(4*(3^(1/2)/2 - 1 i/2)^3 - 3^(1/2) + 1i)) - log((((x*(3^(1/2)/2 - 1i/2) + 1)*1i)/(1 - (3^(1/ 2)/2 - 1i/2)^2)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + 3^(1/2)/2 - 1i/2))/((1 - (3^(1/2)/2 - 1i/2)^2)^(1/2)*(4*(3^(1/2)/2 - 1i/2)^3 - 3^(1/2) + 1i)) - (lo g(x - 3^(1/2)/2 - 1i/2)*(3^(1/2)/2 + 1i/2))/(3^(1/2) - 4*(3^(1/2)/2 + 1i/2 )^3 + 1i) - (log(x + 3^(1/2)/2 + 1i/2)*(3^(1/2)/2 + 1i/2))/(3^(1/2) - 4*(3 ^(1/2)/2 + 1i/2)^3 + 1i) + log((((x*(3^(1/2)/2 + 1i/2) + 1)*1i)/(1 - (3^(1 /2)/2 + 1i/2)^2)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + 3^(1/2)/2 + 1i/2))/((1 - (3^(1/2)/2 + 1i/2)^2)^(1/2)*(3^(1/2) - 4*(3^(1/2)/2 + 1i/2)^3 + 1i)) + (l og(x - 3^(1/2)/2 + 1i/2)*(3^(1/2)/2 - 1i/2))/(4*(3^(1/2)/2 - 1i/2)^3 - 3^( 1/2) + 1i) + (log(x + 3^(1/2)/2 - 1i/2)*(3^(1/2)/2 - 1i/2))/(4*(3^(1/2)/2 - 1i/2)^3 - 3^(1/2) + 1i)
\[ \int \frac {x}{1+x \sqrt {1-x^2}} \, dx=\mathit {asin} \left (x \right )-2 \left (\int \frac {x^{2}}{\sqrt {-x^{2}+1}-x^{3}+x}d x \right )-3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}-2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{3}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )+1\right )+6 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1\right )+2 \,\mathrm {log}\left (\sqrt {-x^{2}+1}\, x +1\right ) \] Input:
int(x/(1+x*(-x^2+1)^(1/2)),x)
Output:
asin(x) - 2*int(x**2/(sqrt( - x**2 + 1) - x**3 + x),x) - 3*log(tan(asin(x) /2)**4 - 2*tan(asin(x)/2)**3 + 2*tan(asin(x)/2)**2 + 2*tan(asin(x)/2) + 1) + 6*log(tan(asin(x)/2)**2 + 1) + 2*log(sqrt( - x**2 + 1)*x + 1)