Integrand size = 17, antiderivative size = 43 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {2+x^2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {2+x^2}}\right )}{2 \sqrt {3}} \] Output:
-1/2*arctan(x/(x^2+2)^(1/2))-1/6*arctanh(x*3^(1/2)/(x^2+2)^(1/2))*3^(1/2)
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\frac {1}{6} \left (3 \arctan \left (1+x^2-x \sqrt {2+x^2}\right )-\sqrt {3} \text {arctanh}\left (\frac {1-x^2+x \sqrt {2+x^2}}{\sqrt {3}}\right )\right ) \] Input:
Integrate[1/(Sqrt[2 + x^2]*(-1 + x^4)),x]
Output:
(3*ArcTan[1 + x^2 - x*Sqrt[2 + x^2]] - Sqrt[3]*ArcTanh[(1 - x^2 + x*Sqrt[2 + x^2])/Sqrt[3]])/6
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1489, 291, 216, 219}
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.
\(\displaystyle \int \frac {1}{\sqrt {x^2+2} \left (x^4-1\right )} \, dx\) |
\(\Big \downarrow \) 1489 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{\left (1-x^2\right ) \sqrt {x^2+2}}dx-\frac {1}{2} \int \frac {1}{\left (x^2+1\right ) \sqrt {x^2+2}}dx\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{1-\frac {3 x^2}{x^2+2}}d\frac {x}{\sqrt {x^2+2}}-\frac {1}{2} \int \frac {1}{\frac {x^2}{x^2+2}+1}d\frac {x}{\sqrt {x^2+2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{1-\frac {3 x^2}{x^2+2}}d\frac {x}{\sqrt {x^2+2}}-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^2+2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^2+2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {x^2+2}}\right )}{2 \sqrt {3}}\) |
Input:
Int[1/(Sqrt[2 + x^2]*(-1 + x^4)),x]
Output:
-1/2*ArcTan[x/Sqrt[2 + x^2]] - ArcTanh[(Sqrt[3]*x)/Sqrt[2 + x^2]]/(2*Sqrt[ 3])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r) Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S imp[c/(2*r) Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q]
Time = 0.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {\sqrt {x^{2}+2}}{x}\right )}{2}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {x^{2}+2}}{3 x}\right )}{6}\) | \(37\) |
default | \(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (4+2 x \right ) \sqrt {3}}{6 \sqrt {\left (-1+x \right )^{2}+1+2 x}}\right )}{12}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-2 x +4\right ) \sqrt {3}}{6 \sqrt {\left (1+x \right )^{2}+1-2 x}}\right )}{12}-\frac {\arctan \left (\frac {x}{\sqrt {x^{2}+2}}\right )}{2}\) | \(70\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+3 \sqrt {x^{2}+2}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\left (-1+x \right ) \left (1+x \right )}\right )}{12}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {x^{2}+2}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{2}+1}\right )}{4}\) | \(84\) |
Input:
int(1/(x^4-1)/(x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*arctan((x^2+2)^(1/2)/x)-1/6*3^(1/2)*arctanh(1/3*3^(1/2)*(x^2+2)^(1/2)/ x)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (31) = 62\).
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (\frac {4 \, x^{2} - \sqrt {3} {\left (2 \, x^{2} + 1\right )} - \sqrt {x^{2} + 2} {\left (2 \, \sqrt {3} x - 3 \, x\right )} + 2}{x^{2} - 1}\right ) - \frac {1}{2} \, \arctan \left (-x^{2} + \sqrt {x^{2} + 2} x - 1\right ) \] Input:
integrate(1/(x^4-1)/(x^2+2)^(1/2),x, algorithm="fricas")
Output:
1/12*sqrt(3)*log((4*x^2 - sqrt(3)*(2*x^2 + 1) - sqrt(x^2 + 2)*(2*sqrt(3)*x - 3*x) + 2)/(x^2 - 1)) - 1/2*arctan(-x^2 + sqrt(x^2 + 2)*x - 1)
\[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\int \frac {1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{2} + 2}}\, dx \] Input:
integrate(1/(x**4-1)/(x**2+2)**(1/2),x)
Output:
Integral(1/((x - 1)*(x + 1)*(x**2 + 1)*sqrt(x**2 + 2)), x)
\[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\int { \frac {1}{{\left (x^{4} - 1\right )} \sqrt {x^{2} + 2}} \,d x } \] Input:
integrate(1/(x^4-1)/(x^2+2)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((x^4 - 1)*sqrt(x^2 + 2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (31) = 62\).
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.72 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | 2 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} - 4 \, \sqrt {3} - 8 \right |}}{{\left | 2 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} + 4 \, \sqrt {3} - 8 \right |}}\right ) + \frac {1}{2} \, \arctan \left (\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2}\right ) \] Input:
integrate(1/(x^4-1)/(x^2+2)^(1/2),x, algorithm="giac")
Output:
-1/12*sqrt(3)*log(abs(2*(x - sqrt(x^2 + 2))^2 - 4*sqrt(3) - 8)/abs(2*(x - sqrt(x^2 + 2))^2 + 4*sqrt(3) - 8)) + 1/2*arctan(1/2*(x - sqrt(x^2 + 2))^2)
Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.49 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\frac {\sqrt {3}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {3}\,\sqrt {x^2+2}+2\right )\right )}{12}-\frac {\sqrt {3}\,\left (\ln \left (x+1\right )-\ln \left (\sqrt {3}\,\sqrt {x^2+2}-x+2\right )\right )}{12}+\frac {\ln \left (\sqrt {x^2+2}+2-x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\ln \left (\sqrt {x^2+2}+2+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {\ln \left (x-\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\ln \left (x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \] Input:
int(1/((x^2 + 2)^(1/2)*(x^4 - 1)),x)
Output:
(log((x^2 + 2)^(1/2) - x*1i + 2)*1i)/4 - (log(x*1i + (x^2 + 2)^(1/2) + 2)* 1i)/4 + (log(x - 1i)*1i)/4 - (log(x + 1i)*1i)/4 + (3^(1/2)*(log(x - 1) - l og(x + 3^(1/2)*(x^2 + 2)^(1/2) + 2)))/12 - (3^(1/2)*(log(x + 1) - log(3^(1 /2)*(x^2 + 2)^(1/2) - x + 2)))/12
Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.09 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=-\frac {\mathit {atan} \left (\sqrt {x^{2}+2}+x -1\right )}{2}+\frac {\mathit {atan} \left (\sqrt {x^{2}+2}+x +1\right )}{2}+\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x^{2}+2}-2 \sqrt {3}+2 x -2}{\sqrt {2}}\right )}{12}-\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x^{2}+2}-2 \sqrt {3}+2 x +2}{\sqrt {2}}\right )}{12}-\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x^{2}+2}+2 \sqrt {3}+2 x -2}{\sqrt {2}}\right )}{12}+\frac {\sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {x^{2}+2}+2 \sqrt {3}+2 x +2}{\sqrt {2}}\right )}{12} \] Input:
int(1/(x^4-1)/(x^2+2)^(1/2),x)
Output:
( - 6*atan(sqrt(x**2 + 2) + x - 1) + 6*atan(sqrt(x**2 + 2) + x + 1) + sqrt (3)*log((2*sqrt(x**2 + 2) - 2*sqrt(3) + 2*x - 2)/sqrt(2)) - sqrt(3)*log((2 *sqrt(x**2 + 2) - 2*sqrt(3) + 2*x + 2)/sqrt(2)) - sqrt(3)*log((2*sqrt(x**2 + 2) + 2*sqrt(3) + 2*x - 2)/sqrt(2)) + sqrt(3)*log((2*sqrt(x**2 + 2) + 2* sqrt(3) + 2*x + 2)/sqrt(2)))/12