\(\int \frac {x^2}{(-2+x^2) (-1+x^2)^{3/4}} \, dx\) [606]

Optimal result

Integrand size = 20, antiderivative size = 48 \[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}} \]

[Out]

1/2*arctan(1/2*x*2^(1/2)/(x^2-1)^(1/4))*2^(1/2)-1/2*arctanh(1/2*x*2^(1/2)/(x^2-1)^(1/4))*2^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {453} \[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}} \]

[In]

Int[x^2/((-2 + x^2)*(-1 + x^2)^(3/4)),x]

[Out]

ArcTan[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/Sqrt[2] - ArcTanh[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/Sqrt[2]

Rule 453

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-b/(Sqrt[2]*a*d*Rt[-b^2/a,
4]^3))*ArcTan[(Rt[-b^2/a, 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))], x] + Simp[(b/(Sqrt[2]*a*d*Rt[-b^2/a, 4]^3))*ArcT
anh[(Rt[-b^2/a, 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && Neg
Q[b^2/a]

Rubi steps \begin{align*} \text {integral}& = \frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.45 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )-\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}} \]

[In]

Integrate[x^2/((-2 + x^2)*(-1 + x^2)^(3/4)),x]

[Out]

(ArcTan[x/(Sqrt[2]*(-1 + x^2)^(1/4))] - ArcTanh[x/(Sqrt[2]*(-1 + x^2)^(1/4))])/Sqrt[2]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.70 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.52

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-x \sqrt {x^{2}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+x \sqrt {x^{2}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{2}\)	\(121\)

[In]

int(x^2/(x^2-2)/(x^2-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^2-2)*ln(-((x^2-1)^(3/4)*RootOf(_Z^2-2)-x*(x^2-1)^(1/2)+RootOf(_Z^2-2)*(x^2-1)^(1/4)-x)/(x^2-2))+
1/2*RootOf(_Z^2+2)*ln(((x^2-1)^(3/4)*RootOf(_Z^2+2)+x*(x^2-1)^(1/2)-RootOf(_Z^2+2)*(x^2-1)^(1/4)-x)/(x^2-2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (39) = 78\).

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.90 \[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{2} - 1} x^{2} - 4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} x + 4 \, x^{2} - 4}{x^{4} - 4 \, x^{2} + 4}\right ) \]

[In]

integrate(x^2/(x^2-2)/(x^2-1)^(3/4),x, algorithm="fricas")

[Out]

-1/2*sqrt(2)*arctan(sqrt(2)*(x^2 - 1)^(1/4)/x) + 1/4*sqrt(2)*log(-(x^4 - 2*sqrt(2)*(x^2 - 1)^(1/4)*x^3 + 4*sqr
t(x^2 - 1)*x^2 - 4*sqrt(2)*(x^2 - 1)^(3/4)*x + 4*x^2 - 4)/(x^4 - 4*x^2 + 4))

Sympy [F]

\[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=\int \frac {x^{2}}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{4}} \left (x^{2} - 2\right )}\, dx \]

[In]

integrate(x**2/(x**2-2)/(x**2-1)**(3/4),x)

[Out]

Integral(x**2/(((x - 1)*(x + 1))**(3/4)*(x**2 - 2)), x)

Maxima [F]

\[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 2\right )}} \,d x } \]

[In]

integrate(x^2/(x^2-2)/(x^2-1)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^2/((x^2 - 1)^(3/4)*(x^2 - 2)), x)

Giac [F]

\[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 2\right )}} \,d x } \]

[In]

integrate(x^2/(x^2-2)/(x^2-1)^(3/4),x, algorithm="giac")

[Out]

integrate(x^2/((x^2 - 1)^(3/4)*(x^2 - 2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx=\int \frac {x^2}{{\left (x^2-1\right )}^{3/4}\,\left (x^2-2\right )} \,d x \]

[In]

int(x^2/((x^2 - 1)^(3/4)*(x^2 - 2)),x)

[Out]

int(x^2/((x^2 - 1)^(3/4)*(x^2 - 2)), x)