Integrand size = 20, antiderivative size = 51 \[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=\left (-\frac {1}{4}-\frac {i}{4}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {x \left (-1+x^2\right )}}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \text {arctanh}\left (\frac {(1+i) x}{\sqrt {x \left (-1+x^2\right )}}\right ) \] Output:
(-1/4-1/4*I)*arctan((1+I)*x/(x*(x^2-1))^(1/2))+(1/4+1/4*I)*arctanh((1+I)*x /(x*(x^2-1))^(1/2))
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.57 \[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {x} \sqrt {-1+x^2} \left (\arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {-1+x^2}}\right )+i \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^2}}{\sqrt {x}}\right )\right )}{\sqrt {x \left (-1+x^2\right )}} \] Input:
Integrate[x/(Sqrt[x*(-1 + x^2)]*(1 + x^2)),x]
Output:
((-1/4 - I/4)*Sqrt[x]*Sqrt[-1 + x^2]*(ArcTan[((1 + I)*Sqrt[x])/Sqrt[-1 + x ^2]] + I*ArcTan[((1/2 + I/2)*Sqrt[-1 + x^2])/Sqrt[x]]))/Sqrt[x*(-1 + x^2)]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.81 (sec) , antiderivative size = 209, normalized size of antiderivative = 4.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2467, 368, 993, 1535, 763, 2213, 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.
| \(\displaystyle \int \frac {x}{\sqrt {x \left (x^2-1\right )} \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-1} \int \frac {\sqrt {x}}{\sqrt {x^2-1} \left (x^2+1\right )}dx}{\sqrt {-x \left (1-x^2\right )}}\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \frac {x}{\sqrt {x^2-1} \left (x^2+1\right )}d\sqrt {x}}{\sqrt {-x \left (1-x^2\right )}}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {1}{2} \int \frac {1}{(x+i) \sqrt {x^2-1}}d\sqrt {x}-\frac {1}{2} \int \frac {1}{(i-x) \sqrt {x^2-1}}d\sqrt {x}\right )}{\sqrt {-x \left (1-x^2\right )}}\) |
| \(\Big \downarrow \) 1535 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {1}{2} \left (-\frac {1}{2} i \int \frac {1}{\sqrt {x^2-1}}d\sqrt {x}-\frac {1}{2} i \int \frac {i-x}{(x+i) \sqrt {x^2-1}}d\sqrt {x}\right )+\frac {1}{2} \left (\frac {1}{2} i \int \frac {1}{\sqrt {x^2-1}}d\sqrt {x}+\frac {1}{2} i \int \frac {x+i}{(i-x) \sqrt {x^2-1}}d\sqrt {x}\right )\right )}{\sqrt {-x \left (1-x^2\right )}}\) |
| \(\Big \downarrow \) 763 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {1}{2} \left (-\frac {1}{2} i \int \frac {i-x}{(x+i) \sqrt {x^2-1}}d\sqrt {x}-\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}\right )+\frac {1}{2} \left (\frac {1}{2} i \int \frac {x+i}{(i-x) \sqrt {x^2-1}}d\sqrt {x}+\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}\right )\right )}{\sqrt {-x \left (1-x^2\right )}}\) |
| \(\Big \downarrow \) 2213 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{i-2 x}d\frac {\sqrt {x}}{\sqrt {x^2-1}}-\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}\right )+\frac {1}{2} \left (\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}-\frac {1}{2} \int \frac {1}{2 x+i}d\frac {\sqrt {x}}{\sqrt {x^2-1}}\right )\right )}{\sqrt {-x \left (1-x^2\right )}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{i-2 x}d\frac {\sqrt {x}}{\sqrt {x^2-1}}-\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}\right )+\frac {1}{2} \left (\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}+\left (\frac {1}{4}+\frac {i}{4}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )\right )\right )}{\sqrt {-x \left (1-x^2\right )}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {1}{2} \left (\left (-\frac {1}{4}-\frac {i}{4}\right ) \arctan \left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )-\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}\right )+\frac {1}{2} \left (\frac {i \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^2-1}}+\left (\frac {1}{4}+\frac {i}{4}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {x}}{\sqrt {x^2-1}}\right )\right )\right )}{\sqrt {-x \left (1-x^2\right )}}\) |
Input:
Int[x/(Sqrt[x*(-1 + x^2)]*(1 + x^2)),x]
Output:
(2*Sqrt[x]*Sqrt[-1 + x^2]*(((-1/4 - I/4)*ArcTan[((1 + I)*Sqrt[x])/Sqrt[-1 + x^2]] - ((I/2)*Sqrt[-1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x ])/Sqrt[-1 + x]], 1/2])/(Sqrt[2]*Sqrt[-1 + x^2]))/2 + ((1/4 + I/4)*ArcTanh [((1 + I)*Sqrt[x])/Sqrt[-1 + x^2]] + ((I/2)*Sqrt[-1 + x]*Sqrt[1 + x]*Ellip ticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(Sqrt[2]*Sqrt[-1 + x^2] ))/2))/Sqrt[-(x*(1 - x^2))]
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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Sim p[Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2)/q]/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4])) *EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; IntegerQ[q]] /; F reeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 1/(2*d) Int[1/Sqrt[a + c*x^4], x], x] + Simp[1/(2*d) Int[(d - e*x^2)/(( d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> Simp[A Subst[Int[1/(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^ 4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(39)=78\).
Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.80
| method | result | size |
| default | \(\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )}{4}-\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )}{4}\) | \(92\) |
| pseudoelliptic | \(\frac {\ln \left (\frac {x^{2}+2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}+x}{x}\right )}{4}-\frac {\ln \left (\frac {x^{2}-2 \sqrt {x^{3}-x}+2 x -1}{x}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{3}-x}-x}{x}\right )}{4}\) | \(92\) |
| elliptic | \(-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}+\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {i \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\) | \(172\) |
| trager | \(-\frac {\ln \left (-\frac {464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{2}-696 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x +157 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{2}-464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2}+400 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-598 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +8 x^{2}-157 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+125 \sqrt {x^{3}-x}-112 x -8}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+3 x -1\right )}^{2}}\right )}{4}-\frac {\ln \left (-\frac {464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{2}-696 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x +157 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{2}-464 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2}+400 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-598 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +8 x^{2}-157 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+125 \sqrt {x^{3}-x}-112 x -8}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+3 x -1\right )}^{2}}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{2}+\frac {\operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \ln \left (-\frac {928 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{2}-1392 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x +614 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{2}-928 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2}-800 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \sqrt {x^{3}-x}-196 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +91 x^{2}-614 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )-150 \sqrt {x^{3}-x}+26 x -91}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )-x -3\right )}^{2}}\right )}{2}\) | \(546\) |
Input:
int(x/(x*(x^2-1))^(1/2)/(x^2+1),x,method=_RETURNVERBOSE)
Output:
1/8*ln((x^2+2*(x^3-x)^(1/2)+2*x-1)/x)+1/4*arctan(((x^3-x)^(1/2)+x)/x)-1/8* ln((x^2-2*(x^3-x)^(1/2)+2*x-1)/x)+1/4*arctan(((x^3-x)^(1/2)-x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (31) = 62\).
Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.49 \[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=\frac {1}{4} \, \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x}}\right ) + \frac {1}{8} \, \log \left (\frac {x^{4} + 8 \, x^{3} + 2 \, x^{2} + 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} - 8 \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) \] Input:
integrate(x/(x*(x^2-1))^(1/2)/(x^2+1),x, algorithm="fricas")
Output:
1/4*arctan(1/2*(x^2 - 2*x - 1)/sqrt(x^3 - x)) + 1/8*log((x^4 + 8*x^3 + 2*x ^2 + 4*sqrt(x^3 - x)*(x^2 + 2*x - 1) - 8*x + 1)/(x^4 + 2*x^2 + 1))
\[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=\int \frac {x}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:
integrate(x/(x*(x**2-1))**(1/2)/(x**2+1),x)
Output:
Integral(x/(sqrt(x*(x - 1)*(x + 1))*(x**2 + 1)), x)
\[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=\int { \frac {x}{\sqrt {{\left (x^{2} - 1\right )} x} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(x/(x*(x^2-1))^(1/2)/(x^2+1),x, algorithm="maxima")
Output:
integrate(x/(sqrt((x^2 - 1)*x)*(x^2 + 1)), x)
\[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=\int { \frac {x}{\sqrt {{\left (x^{2} - 1\right )} x} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(x/(x*(x^2-1))^(1/2)/(x^2+1),x, algorithm="giac")
Output:
integrate(x/(sqrt((x^2 - 1)*x)*(x^2 + 1)), x)
Time = 22.53 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.63 \[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=\frac {\sqrt {-x}\,\sqrt {\frac {1}{2}-\frac {x}{2}}\,\sqrt {x+1}\,\Pi \left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {x+1}\right )\middle |\frac {1}{2}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {\frac {1}{2}-\frac {x}{2}}\,\sqrt {x+1}\,\Pi \left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {x+1}\right )\middle |\frac {1}{2}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {x^3-x}} \] Input:
int(x/((x*(x^2 - 1))^(1/2)*(x^2 + 1)),x)
Output:
- ((-x)^(1/2)*(1/2 - x/2)^(1/2)*(x + 1)^(1/2)*ellipticPi(1/2 - 1i/2, asin( (x + 1)^(1/2)), 1/2)*(1/2 - 1i/2))/(x^3 - x)^(1/2) - ((-x)^(1/2)*(1/2 - x/ 2)^(1/2)*(x + 1)^(1/2)*ellipticPi(1/2 + 1i/2, asin((x + 1)^(1/2)), 1/2)*(1 /2 + 1i/2))/(x^3 - x)^(1/2)
\[ \int \frac {x}{\sqrt {x \left (-1+x^2\right )} \left (1+x^2\right )} \, dx=\int \frac {\sqrt {x}\, \sqrt {x^{2}-1}}{x^{4}-1}d x \] Input:
int(x/(x*(x^2-1))^(1/2)/(x^2+1),x)
Output:
int((sqrt(x)*sqrt(x**2 - 1))/(x**4 - 1),x)