Integrand size = 19, antiderivative size = 90 \[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=-\frac {2 \sqrt [4]{-1+x^2}}{x}+\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^2}}{\sqrt {2}}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{1+\sqrt {-1+x^2}}\right )}{\sqrt {2}} \] Output:
-2*(x^2-1)^(1/4)/x+1/2*arctan((-1/2*2^(1/2)+1/2*(x^2-1)^(1/2)*2^(1/2))/(x^ 2-1)^(1/4))*2^(1/2)+1/2*arctanh(2^(1/2)*(x^2-1)^(1/4)/(1+(x^2-1)^(1/2)))*2 ^(1/2)
Time = 10.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.53 \[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=-\frac {2 \sqrt [4]{-1+x^2}}{x}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{-1+x^2}\right )}{\sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} \sqrt [4]{-1+x^2}\right )}{\sqrt {2}}-\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{2 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{2 \sqrt {2}} \] Input:
Integrate[(-2 + x + x^2)/(x^2*(-1 + x^2)^(3/4)),x]
Output:
(-2*(-1 + x^2)^(1/4))/x - ArcTan[1 - Sqrt[2]*(-1 + x^2)^(1/4)]/Sqrt[2] + A rcTan[1 + Sqrt[2]*(-1 + x^2)^(1/4)]/Sqrt[2] - Log[1 - Sqrt[2]*(-1 + x^2)^( 1/4) + Sqrt[-1 + x^2]]/(2*Sqrt[2]) + Log[1 + Sqrt[2]*(-1 + x^2)^(1/4) + Sq rt[-1 + x^2]]/(2*Sqrt[2])
Time = 0.36 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.54, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {2338, 243, 73, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
\(\displaystyle \int \frac {x^2+x-2}{x^2 \left (x^2-1\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle \int \frac {1}{x \left (x^2-1\right )^{3/4}}dx-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (x^2-1\right )^{3/4}}dx^2-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 2 \int \frac {1}{x^8+1}d\sqrt [4]{x^2-1}-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle 2 \left (\frac {1}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{x^2-1}+\frac {1}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{x^2-1}\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^4-\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}+\frac {1}{2} \int \frac {1}{x^4+\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}\right )+\frac {1}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{x^2-1}\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^4-1}d\left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x^4-1}d\left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{x^2-1}\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 2 \left (\frac {1}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{x^2-1}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{\sqrt {2}}\right )\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle 2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt [4]{x^2-1}}{x^4-\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{x^4+\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{\sqrt {2}}\right )\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{x^2-1}}{x^4-\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{x^4+\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{\sqrt {2}}\right )\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{x^2-1}}{x^4-\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt [4]{x^2-1}+1}{x^4+\sqrt {2} \sqrt [4]{x^2-1}+1}d\sqrt [4]{x^2-1}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{\sqrt {2}}\right )\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x^4+\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^4-\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{2 \sqrt {2}}\right )\right )-\frac {2 \sqrt [4]{x^2-1}}{x}\) |
Input:
Int[(-2 + x + x^2)/(x^2*(-1 + x^2)^(3/4)),x]
Output:
(-2*(-1 + x^2)^(1/4))/x + 2*((-(ArcTan[1 - Sqrt[2]*(-1 + x^2)^(1/4)]/Sqrt[ 2]) + ArcTan[1 + Sqrt[2]*(-1 + x^2)^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + x^4 - Sqrt[2]*(-1 + x^2)^(1/4)]/Sqrt[2] + Log[1 + x^4 + Sqrt[2]*(-1 + x^2)^(1/ 4)]/(2*Sqrt[2]))/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.74 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {2 \left (x^{2}-1\right )^{\frac {1}{4}}}{x}+\frac {{\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} \left (\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{2}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{2 \Gamma \left (\frac {3}{4}\right ) \operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\) | \(76\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} x \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], x^{2}\right )}{\operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} \left (\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{2}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{2 \Gamma \left (\frac {3}{4}\right ) \operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}+\frac {2 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {1}{2}\right ], x^{2}\right )}{\operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}} x}\) | \(125\) |
trager | \(-\frac {2 \left (x^{2}-1\right )^{\frac {1}{4}}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{2}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \sqrt {x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-2 \left (x^{2}-1\right )^{\frac {3}{4}}}{x^{2}}\right )}{2}\) | \(170\) |
Input:
int((x^2+x-2)/x^2/(x^2-1)^(3/4),x,method=_RETURNVERBOSE)
Output:
-2*(x^2-1)^(1/4)/x+1/2/GAMMA(3/4)/signum(x^2-1)^(3/4)*(-signum(x^2-1))^(3/ 4)*(3/4*GAMMA(3/4)*x^2*hypergeom([1,1,7/4],[2,2],x^2)+(-3*ln(2)+1/2*Pi+2*l n(x)+I*Pi)*GAMMA(3/4))
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (73) = 146\).
Time = 0.72 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.83 \[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=-\frac {2 \, \sqrt {2} x \arctan \left (\frac {x^{4} + 4 \, \sqrt {x^{2} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 4\right )} + 2 \, \sqrt {2} {\left (3 \, x^{2} - 4\right )} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{x^{4} - 16 \, x^{2} + 16}\right ) + 2 \, \sqrt {2} x \arctan \left (-\frac {x^{4} + 4 \, \sqrt {x^{2} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 4\right )} - 2 \, \sqrt {2} {\left (3 \, x^{2} - 4\right )} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{x^{4} - 16 \, x^{2} + 16}\right ) - \sqrt {2} x \log \left (\frac {x^{2} + 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{2} - 1}}{x^{2}}\right ) + \sqrt {2} x \log \left (\frac {x^{2} - 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{2} - 1}}{x^{2}}\right ) + 16 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{8 \, x} \] Input:
integrate((x^2+x-2)/x^2/(x^2-1)^(3/4),x, algorithm="fricas")
Output:
-1/8*(2*sqrt(2)*x*arctan((x^4 + 4*sqrt(x^2 - 1)*x^2 + 2*sqrt(2)*(x^2 - 1)^ (3/4)*(x^2 - 4) + 2*sqrt(2)*(3*x^2 - 4)*(x^2 - 1)^(1/4))/(x^4 - 16*x^2 + 1 6)) + 2*sqrt(2)*x*arctan(-(x^4 + 4*sqrt(x^2 - 1)*x^2 - 2*sqrt(2)*(x^2 - 1) ^(3/4)*(x^2 - 4) - 2*sqrt(2)*(3*x^2 - 4)*(x^2 - 1)^(1/4))/(x^4 - 16*x^2 + 16)) - sqrt(2)*x*log((x^2 + 2*sqrt(2)*(x^2 - 1)^(3/4) + 2*sqrt(2)*(x^2 - 1 )^(1/4) + 4*sqrt(x^2 - 1))/x^2) + sqrt(2)*x*log((x^2 - 2*sqrt(2)*(x^2 - 1) ^(3/4) - 2*sqrt(2)*(x^2 - 1)^(1/4) + 4*sqrt(x^2 - 1))/x^2) + 16*(x^2 - 1)^ (1/4))/x
Result contains complex when optimal does not.
Time = 2.43 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.83 \[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=x e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {x^{2}} \right )} - \frac {2 e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{2}} \right )}}{x} - \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 x^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((x**2+x-2)/x**2/(x**2-1)**(3/4),x)
Output:
x*exp(-3*I*pi/4)*hyper((1/2, 3/4), (3/2,), x**2) - 2*exp(I*pi/4)*hyper((-1 /2, 3/4), (1/2,), x**2)/x - gamma(3/4)*hyper((3/4, 3/4), (7/4,), exp_polar (2*I*pi)/x**2)/(2*x**(3/2)*gamma(7/4))
\[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=\int { \frac {x^{2} + x - 2}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \] Input:
integrate((x^2+x-2)/x^2/(x^2-1)^(3/4),x, algorithm="maxima")
Output:
integrate((x^2 + x - 2)/((x^2 - 1)^(3/4)*x^2), x)
\[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=\int { \frac {x^{2} + x - 2}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \] Input:
integrate((x^2+x-2)/x^2/(x^2-1)^(3/4),x, algorithm="giac")
Output:
integrate((x^2 + x - 2)/((x^2 - 1)^(3/4)*x^2), x)
Time = 9.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99 \[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=\frac {4\,{\left (\frac {1}{x^2}\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {5}{4};\ \frac {9}{4};\ \frac {1}{x^2}\right )}{5\,x}+\frac {x\,{\left (1-x^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {3}{2};\ x^2\right )}{{\left (x^2-1\right )}^{3/4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^2-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^2-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right ) \] Input:
int((x + x^2 - 2)/(x^2*(x^2 - 1)^(3/4)),x)
Output:
2^(1/2)*atan(2^(1/2)*(x^2 - 1)^(1/4)*(1/2 - 1i/2))*(1/2 + 1i/2) + 2^(1/2)* atan(2^(1/2)*(x^2 - 1)^(1/4)*(1/2 + 1i/2))*(1/2 - 1i/2) + (4*(1/x^2)^(3/4) *hypergeom([3/4, 5/4], 9/4, 1/x^2))/(5*x) + (x*(1 - x^2)^(3/4)*hypergeom([ 1/2, 3/4], 3/2, x^2))/(x^2 - 1)^(3/4)
\[ \int \frac {-2+x+x^2}{x^2 \left (-1+x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (x^{2}-1\right )^{\frac {3}{4}}}d x -2 \left (\int \frac {1}{\left (x^{2}-1\right )^{\frac {3}{4}} x^{2}}d x \right )+\int \frac {1}{\left (x^{2}-1\right )^{\frac {3}{4}} x}d x \] Input:
int((x^2+x-2)/x^2/(x^2-1)^(3/4),x)
Output:
int(1/(x**2 - 1)**(3/4),x) - 2*int(1/((x**2 - 1)**(3/4)*x**2),x) + int(1/( (x**2 - 1)**(3/4)*x),x)