Integrand size = 44, antiderivative size = 136 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {41 \text {arcsinh}(x)}{54}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+3 x}{2 \sqrt {2}}\right )+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )+\frac {7}{27} \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x^2}}\right )-\frac {7}{54} \log \left (3+2 x+3 x^2\right ) \]
8/9*x-1/6*x^2-41/54*arcsinh(x)+7/27*arctanh(1/2*(1-x)/(x^2+1)^(1/2))-7/54* ln(3*x^2+2*x+3)+4/27*arctan(1/4*(1+3*x)*2^(1/2))*2^(1/2)+4/27*arctan(1/2*( 1+x)*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)+8/9*(x^2+1)^(1/2)-1/6*x*(x^2+1)^(1/2)
Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\frac {1}{54} \left (48 x-9 x^2+48 \sqrt {1+x^2}-9 x \sqrt {1+x^2}+16 \sqrt {2} \arctan \left (\frac {1+x-\sqrt {1+x^2}}{\sqrt {2}}\right )+55 \log \left (-x+\sqrt {1+x^2}\right )-14 \log \left (-2-x-x^2+(1+x) \sqrt {1+x^2}\right )\right ) \]
(48*x - 9*x^2 + 48*Sqrt[1 + x^2] - 9*x*Sqrt[1 + x^2] + 16*Sqrt[2]*ArcTan[( 1 + x - Sqrt[1 + x^2])/Sqrt[2]] + 55*Log[-x + Sqrt[1 + x^2]] - 14*Log[-2 - x - x^2 + (1 + x)*Sqrt[1 + x^2]])/54
Time = 1.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, 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.
| \(\displaystyle \int \frac {x^2 \left (2-\sqrt {x^2+1}\right )}{\sqrt {x^2+1} \left (-x^3+\left (x^2+1\right )^{3/2}+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {x^2}{-x^3+\sqrt {x^2+1} x^2+\sqrt {x^2+1}+1}-\frac {2 x^2}{\sqrt {x^2+1} \left (x^3-\left (x^2+1\right )^{3/2}-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {41 \text {arcsinh}(x)}{54}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )+\frac {4}{27} \sqrt {2} \arctan \left (\frac {3 x+1}{2 \sqrt {2}}\right )+\frac {7}{27} \text {arctanh}\left (\frac {1-x}{2 \sqrt {x^2+1}}\right )-\frac {x^2}{6}-\frac {1}{6} \sqrt {x^2+1} x+\frac {8 \sqrt {x^2+1}}{9}-\frac {7}{54} \log \left (3 x^2+2 x+3\right )+\frac {8 x}{9}\) |
(8*x)/9 - x^2/6 + (8*Sqrt[1 + x^2])/9 - (x*Sqrt[1 + x^2])/6 - (41*ArcSinh[ x])/54 + (4*Sqrt[2]*ArcTan[(1 + 3*x)/(2*Sqrt[2])])/27 + (4*Sqrt[2]*ArcTan[ (1 + x)/(Sqrt[2]*Sqrt[1 + x^2])])/27 + (7*ArcTanh[(1 - x)/(2*Sqrt[1 + x^2] )])/27 - (7*Log[3 + 2*x + 3*x^2])/54
3.3.60.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(653\) vs. \(2(99)=198\).
Time = 0.04 (sec) , antiderivative size = 654, normalized size of antiderivative = 4.81
\[-\frac {x^{2}}{6}+\frac {8 x}{9}-\frac {7 \ln \left (3 x^{2}+2 x +3\right )}{54}+\frac {4 \sqrt {2}\, \arctan \left (\frac {\left (6 x +2\right ) \sqrt {2}}{8}\right )}{27}-\frac {41 \,\operatorname {arcsinh}\left (x \right )}{54}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+5 \,\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{12 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}+\frac {3 \sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{8 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}-\frac {x \sqrt {x^{2}+1}}{6}+\frac {8 \sqrt {x^{2}+1}}{9}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (13 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+43 \,\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{216 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (-11 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{36 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}\]
-1/6*x^2+8/9*x-7/54*ln(3*x^2+2*x+3)+4/27*2^(1/2)*arctan(1/8*(6*x+2)*2^(1/2 ))-41/54*arcsinh(x)-1/12*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)*(-2^(1/2)*arc tan(1/2*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)/((1+x)^2/(1-x)^2+1)*(1+x)/(1-x ))+5*arctanh((2*(1+x)^2/(1-x)^2+2)^(1/2)))/(((1+x)^2/(1-x)^2+1)/((1+x)/(1- x)+1)^2)^(1/2)/((1+x)/(1-x)+1)+3/8*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)*(-2 ^(1/2)*arctan(1/2*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)/((1+x)^2/(1-x)^2+1)* (1+x)/(1-x))+arctanh((2*(1+x)^2/(1-x)^2+2)^(1/2)))/(((1+x)^2/(1-x)^2+1)/(( 1+x)/(1-x)+1)^2)^(1/2)/((1+x)/(1-x)+1)-1/6*x*(x^2+1)^(1/2)+8/9*(x^2+1)^(1/ 2)+1/216*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)*(13*2^(1/2)*arctan(1/2*2^(1/2 )*(2*(1+x)^2/(1-x)^2+2)^(1/2)/((1+x)^2/(1-x)^2+1)*(1+x)/(1-x))+43*arctanh( (2*(1+x)^2/(1-x)^2+2)^(1/2)))/(((1+x)^2/(1-x)^2+1)/((1+x)/(1-x)+1)^2)^(1/2 )/((1+x)/(1-x)+1)-1/36*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)*(-11*2^(1/2)*ar ctan(1/2*2^(1/2)*(2*(1+x)^2/(1-x)^2+2)^(1/2)/((1+x)^2/(1-x)^2+1)*(1+x)/(1- x))+arctanh((2*(1+x)^2/(1-x)^2+2)^(1/2)))/(((1+x)^2/(1-x)^2+1)/((1+x)/(1-x )+1)^2)^(1/2)/((1+x)/(1-x)+1)
Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.25 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=-\frac {1}{6} \, x^{2} - \frac {1}{18} \, \sqrt {x^{2} + 1} {\left (3 \, x - 16\right )} + \frac {4}{27} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) + \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 1\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 1}\right ) - \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )} + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 1}\right ) + \frac {8}{9} \, x + \frac {7}{54} \, \log \left (3 \, x^{2} - \sqrt {x^{2} + 1} {\left (3 \, x - 1\right )} - x + 2\right ) - \frac {7}{54} \, \log \left (3 \, x^{2} + 2 \, x + 3\right ) - \frac {7}{54} \, \log \left (x^{2} - \sqrt {x^{2} + 1} {\left (x + 1\right )} + x + 2\right ) + \frac {41}{54} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
-1/6*x^2 - 1/18*sqrt(x^2 + 1)*(3*x - 16) + 4/27*sqrt(2)*arctan(1/4*sqrt(2) *(3*x + 1)) + 4/27*sqrt(2)*arctan(-1/2*sqrt(2)*(3*x - 1) + 3/2*sqrt(2)*sqr t(x^2 + 1)) - 4/27*sqrt(2)*arctan(-1/2*sqrt(2)*(x + 1) + 1/2*sqrt(2)*sqrt( x^2 + 1)) + 8/9*x + 7/54*log(3*x^2 - sqrt(x^2 + 1)*(3*x - 1) - x + 2) - 7/ 54*log(3*x^2 + 2*x + 3) - 7/54*log(x^2 - sqrt(x^2 + 1)*(x + 1) + x + 2) + 41/54*log(-x + sqrt(x^2 + 1))
Timed out. \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\int { \frac {x^{2} {\left (\sqrt {x^{2} + 1} - 2\right )}}{{\left (x^{3} - {\left (x^{2} + 1\right )}^{\frac {3}{2}} - 1\right )} \sqrt {x^{2} + 1}} \,d x } \]
-1/2*x/(x^2 + 1) + 1/2*arctan(x) + integrate(-1/2*(3*x^10 - 4*x^9 + 5*x^8 - 2*x^7 + 15*x^6 + 6*x^5 + 9*x^4)/(2*x^13 + 7*x^11 - 4*x^10 + 11*x^9 - 11* x^8 + 13*x^7 - 13*x^6 + 11*x^5 - 11*x^4 + 4*x^3 - 7*x^2 - 2*(x^12 + 3*x^10 - 2*x^9 + 3*x^8 - 6*x^7 + 2*x^6 - 6*x^5 + 3*x^4 - 2*x^3 + 3*x^2 + 1)*sqrt (x^2 + 1) - 2), x) + 1/6*log(x^2 + x + 1) + 1/6*log(x - 1)
Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.29 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=-\frac {1}{6} \, x^{2} - \frac {1}{18} \, \sqrt {x^{2} + 1} {\left (3 \, x - 16\right )} + \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 3 \, \sqrt {x^{2} + 1} - 1\right )}\right ) + \frac {4}{27} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) - \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} + 1} + 1\right )}\right ) + \frac {8}{9} \, x + \frac {7}{54} \, \log \left (3 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 2 \, x + 2 \, \sqrt {x^{2} + 1} + 1\right ) - \frac {7}{54} \, \log \left ({\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 2 \, x - 2 \, \sqrt {x^{2} + 1} + 3\right ) - \frac {7}{54} \, \log \left (3 \, x^{2} + 2 \, x + 3\right ) + \frac {41}{54} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
-1/6*x^2 - 1/18*sqrt(x^2 + 1)*(3*x - 16) + 4/27*sqrt(2)*arctan(-1/2*sqrt(2 )*(3*x - 3*sqrt(x^2 + 1) - 1)) + 4/27*sqrt(2)*arctan(1/4*sqrt(2)*(3*x + 1) ) - 4/27*sqrt(2)*arctan(-1/2*sqrt(2)*(x - sqrt(x^2 + 1) + 1)) + 8/9*x + 7/ 54*log(3*(x - sqrt(x^2 + 1))^2 - 2*x + 2*sqrt(x^2 + 1) + 1) - 7/54*log((x - sqrt(x^2 + 1))^2 + 2*x - 2*sqrt(x^2 + 1) + 3) - 7/54*log(3*x^2 + 2*x + 3 ) + 41/54*log(-x + sqrt(x^2 + 1))
Time = 0.64 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.59 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\frac {8\,x}{9}-\frac {41\,\mathrm {asinh}\left (x\right )}{54}-\left (\frac {x}{6}-\frac {8}{9}\right )\,\sqrt {x^2+1}-\frac {x^2}{6}+\frac {\sqrt {2}\,\ln \left (x+\frac {1}{3}-\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )\,\left (-\frac {16}{27}+\frac {\sqrt {2}\,14{}\mathrm {i}}{27}\right )\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\ln \left (x+\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )\,\left (\frac {16}{27}+\frac {\sqrt {2}\,14{}\mathrm {i}}{27}\right )\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\left (\frac {4}{81}+\frac {\sqrt {2}\,44{}\mathrm {i}}{81}\right )\,\left (\ln \left (x+\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )-\ln \left (1+\left (\frac {2}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}\right )\,\sqrt {x^2+1}-\frac {x}{3}-\frac {\sqrt {2}\,x\,2{}\mathrm {i}}{3}\right )\right )\,1{}\mathrm {i}}{8\,\sqrt {{\left (\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )}^2+1}}+\frac {\sqrt {2}\,\left (-\frac {4}{81}+\frac {\sqrt {2}\,44{}\mathrm {i}}{81}\right )\,\left (\ln \left (x+\frac {1}{3}-\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )-\ln \left (1-\left (-\frac {2}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}\right )\,\sqrt {x^2+1}-\frac {x}{3}+\frac {\sqrt {2}\,x\,2{}\mathrm {i}}{3}\right )\right )\,1{}\mathrm {i}}{8\,\sqrt {{\left (-\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )}^2+1}} \]
(8*x)/9 - (41*asinh(x))/54 - (x/6 - 8/9)*(x^2 + 1)^(1/2) - x^2/6 + (2^(1/2 )*log(x - (2^(1/2)*2i)/3 + 1/3)*((2^(1/2)*14i)/27 - 16/27)*1i)/8 + (2^(1/2 )*log(x + (2^(1/2)*2i)/3 + 1/3)*((2^(1/2)*14i)/27 + 16/27)*1i)/8 + (2^(1/2 )*((2^(1/2)*44i)/81 + 4/81)*(log(x + (2^(1/2)*2i)/3 + 1/3) - log(((2^(1/2) *1i)/3 + 2/3)*(x^2 + 1)^(1/2) - x/3 - (2^(1/2)*x*2i)/3 + 1))*1i)/(8*(((2^( 1/2)*2i)/3 + 1/3)^2 + 1)^(1/2)) + (2^(1/2)*((2^(1/2)*44i)/81 - 4/81)*(log( x - (2^(1/2)*2i)/3 + 1/3) - log((2^(1/2)*x*2i)/3 - ((2^(1/2)*1i)/3 - 2/3)* (x^2 + 1)^(1/2) - x/3 + 1))*1i)/(8*(((2^(1/2)*2i)/3 - 1/3)^2 + 1)^(1/2))
Time = 0.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\frac {8 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+1}+3 x -1}{\sqrt {2}}\right )}{27}-\frac {\sqrt {x^{2}+1}\, x}{6}+\frac {8 \sqrt {x^{2}+1}}{9}-\frac {\mathrm {log}\left (\sqrt {x^{2}+1}+x \right )}{2}-\frac {7 \,\mathrm {log}\left (6 \sqrt {x^{2}+1}\, x -2 \sqrt {x^{2}+1}+6 x^{2}-2 x +4\right )}{27}-\frac {x^{2}}{6}+\frac {8 x}{9}-\frac {1}{12} \]
int((x**2*(sqrt(x**2 + 1) - 2))/(sqrt(x**2 + 1)*x**3 - sqrt(x**2 + 1) - x* *4 - 2*x**2 - 1),x)