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

Optimal. Leaf size=48 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {442} \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[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 442

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

Rubi steps

\begin {align*} \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 48, normalized size = 1.00 \begin {gather*} -\frac {x^3 \left (1-x^2\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};x^2,\frac {x^2}{2}\right )}{6 \left (x^2-1\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(x^3*(1 - x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, x^2, x^2/2])/(-1 + x^2)^(3/4)

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IntegrateAlgebraic [A]  time = 2.02, size = 48, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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]

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fricas [B]  time = 0.46, size = 91, normalized size = 1.90 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 1.46, size = 121, normalized size = 2.52

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{{\left (x^2-1\right )}^{3/4}\,\left (x^2-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{4}} \left (x^{2} - 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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