3.9.68 \(\int \frac {\sqrt {1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx\)

Optimal. Leaf size=66 \[ \frac {\sqrt {x^4+6 x^2+1}}{4 (x+1)^2}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {2} x}{x^2+\sqrt {x^4+6 x^2+1}-2 x+1}\right )}{2 \sqrt {2}} \]

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Rubi [F]  time = 0.57, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(x*(3 + 2*Sqrt[2] + x^2))/(4*Sqrt[1 + 6*x^2 + x^4]) - ArcTanh[(2*Sqrt[2]*x)/Sqrt[1 + 6*x^2 + x^4]]/(2*Sqrt[2])
- (Sqrt[3 + 2*Sqrt[2]]*Sqrt[(1 + (3 - 2*Sqrt[2])*x^2)/(1 + (3 + 2*Sqrt[2])*x^2)]*(1 + (3 + 2*Sqrt[2])*x^2)*El
lipticE[ArcTan[Sqrt[3 + 2*Sqrt[2]]*x], -4*(4 - 3*Sqrt[2])])/(4*Sqrt[1 + 6*x^2 + x^4]) + (3*Sqrt[(1 + (3 - 2*Sq
rt[2])*x^2)/(1 + (3 + 2*Sqrt[2])*x^2)]*(1 + (3 + 2*Sqrt[2])*x^2)*EllipticF[ArcTan[Sqrt[3 + 2*Sqrt[2]]*x], -4*(
4 - 3*Sqrt[2])])/(4*Sqrt[3 + 2*Sqrt[2]]*Sqrt[1 + 6*x^2 + x^4]) - Defer[Int][Sqrt[1 + 6*x^2 + x^4]/(1 + x)^3, x
]/2 - Defer[Int][Sqrt[1 + 6*x^2 + x^4]/(1 + x)^2, x]/4

Rubi steps

\begin {align*} \int \frac {\sqrt {1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx &=\int \left (-\frac {\sqrt {1+6 x^2+x^4}}{2 (1+x)^3}-\frac {\sqrt {1+6 x^2+x^4}}{4 (1+x)^2}+\frac {\sqrt {1+6 x^2+x^4}}{4 \left (-1+x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx\right )+\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{-1+x^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx\\ &=-\left (\frac {1}{4} \int \frac {-7-x^2}{\sqrt {1+6 x^2+x^4}} \, dx\right )-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx+2 \int \frac {1}{\left (-1+x^2\right ) \sqrt {1+6 x^2+x^4}} \, dx\\ &=\frac {1}{4} \int \frac {x^2}{\sqrt {1+6 x^2+x^4}} \, dx-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx+\frac {7}{4} \int \frac {1}{\sqrt {1+6 x^2+x^4}} \, dx-\int \frac {1}{\sqrt {1+6 x^2+x^4}} \, dx-\int \frac {-1-x^2}{\left (-1+x^2\right ) \sqrt {1+6 x^2+x^4}} \, dx\\ &=\frac {x \left (3+2 \sqrt {2}+x^2\right )}{4 \sqrt {1+6 x^2+x^4}}-\frac {\sqrt {3+2 \sqrt {2}} \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {3+2 \sqrt {2}} x\right )|-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {1+6 x^2+x^4}}+\frac {3 \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {3+2 \sqrt {2}} x\right )|-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {3+2 \sqrt {2}} \sqrt {1+6 x^2+x^4}}-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx+\operatorname {Subst}\left (\int \frac {1}{-1+8 x^2} \, dx,x,\frac {x}{\sqrt {1+6 x^2+x^4}}\right )\\ &=\frac {x \left (3+2 \sqrt {2}+x^2\right )}{4 \sqrt {1+6 x^2+x^4}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {2} x}{\sqrt {1+6 x^2+x^4}}\right )}{2 \sqrt {2}}-\frac {\sqrt {3+2 \sqrt {2}} \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {3+2 \sqrt {2}} x\right )|-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {1+6 x^2+x^4}}+\frac {3 \sqrt {\frac {1+\left (3-2 \sqrt {2}\right ) x^2}{1+\left (3+2 \sqrt {2}\right ) x^2}} \left (1+\left (3+2 \sqrt {2}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {3+2 \sqrt {2}} x\right )|-4 \left (4-3 \sqrt {2}\right )\right )}{4 \sqrt {3+2 \sqrt {2}} \sqrt {1+6 x^2+x^4}}-\frac {1}{4} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {1+6 x^2+x^4}}{(1+x)^3} \, dx\\ \end {align*}

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Mathematica [C]  time = 2.69, size = 641, normalized size = 9.71 \begin {gather*} \frac {\frac {x^4+6 x^2+1}{(x+1)^2}+\frac {8 \sqrt {\frac {\sqrt {3+2 \sqrt {2}}+i x}{\sqrt {3-2 \sqrt {2}}+i x}} \sqrt {-\frac {x+i \sqrt {3+2 \sqrt {2}}}{\sqrt {3+2 \sqrt {2}} x-\sqrt {3-2 \sqrt {2}} x-2 i \sqrt {2}+2 i}} \sqrt {\frac {-i \left (\sqrt {3-2 \sqrt {2}}-\sqrt {3+2 \sqrt {2}}\right ) x-2 \sqrt {2}+2}{\sqrt {3-2 \sqrt {2}}+i x}} \left (\sqrt {3-2 \sqrt {2}}+i x\right )^2 \left (\left (\sqrt {3-2 \sqrt {2}}-i \left (\sqrt {2}-1\right )\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt {3-2 \sqrt {2}}-\sqrt {3+2 \sqrt {2}}\right ) x+2 \left (-1+\sqrt {2}\right )}{2 \left (-2+\sqrt {2}\right )-i \left (\sqrt {3-2 \sqrt {2}}+\sqrt {3+2 \sqrt {2}}\right ) x}}\right )\right |2\right )-2 \sqrt {3-2 \sqrt {2}} \Pi \left (\frac {\left (i+\sqrt {3-2 \sqrt {2}}\right ) \left (\sqrt {3-2 \sqrt {2}}+\sqrt {3+2 \sqrt {2}}\right )}{\left (-i+\sqrt {3-2 \sqrt {2}}\right ) \left (\sqrt {3-2 \sqrt {2}}-\sqrt {3+2 \sqrt {2}}\right )};\left .\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt {3-2 \sqrt {2}}-\sqrt {3+2 \sqrt {2}}\right ) x+2 \left (-1+\sqrt {2}\right )}{2 \left (-2+\sqrt {2}\right )-i \left (\sqrt {3-2 \sqrt {2}}+\sqrt {3+2 \sqrt {2}}\right ) x}}\right )\right |2\right )\right )}{\left (\sqrt {3-2 \sqrt {2}}-i\right ) \left (\sqrt {3-2 \sqrt {2}}+i\right ) \left (\sqrt {3-2 \sqrt {2}}-\sqrt {3+2 \sqrt {2}}\right )}}{4 \sqrt {x^4+6 x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + 6*x^2 + x^4)/(1 + x)^2 + (8*(Sqrt[3 - 2*Sqrt[2]] + I*x)^2*Sqrt[(Sqrt[3 + 2*Sqrt[2]] + I*x)/(Sqrt[3 - 2*S
qrt[2]] + I*x)]*Sqrt[-((I*Sqrt[3 + 2*Sqrt[2]] + x)/(2*I - (2*I)*Sqrt[2] - Sqrt[3 - 2*Sqrt[2]]*x + Sqrt[3 + 2*S
qrt[2]]*x))]*Sqrt[(2 - 2*Sqrt[2] - I*(Sqrt[3 - 2*Sqrt[2]] - Sqrt[3 + 2*Sqrt[2]])*x)/(Sqrt[3 - 2*Sqrt[2]] + I*x
)]*((Sqrt[3 - 2*Sqrt[2]] - I*(-1 + Sqrt[2]))*EllipticF[ArcSin[Sqrt[(2*(-1 + Sqrt[2]) + I*(Sqrt[3 - 2*Sqrt[2]]
- Sqrt[3 + 2*Sqrt[2]])*x)/(2*(-2 + Sqrt[2]) - I*(Sqrt[3 - 2*Sqrt[2]] + Sqrt[3 + 2*Sqrt[2]])*x)]], 2] - 2*Sqrt[
3 - 2*Sqrt[2]]*EllipticPi[((I + Sqrt[3 - 2*Sqrt[2]])*(Sqrt[3 - 2*Sqrt[2]] + Sqrt[3 + 2*Sqrt[2]]))/((-I + Sqrt[
3 - 2*Sqrt[2]])*(Sqrt[3 - 2*Sqrt[2]] - Sqrt[3 + 2*Sqrt[2]])), ArcSin[Sqrt[(2*(-1 + Sqrt[2]) + I*(Sqrt[3 - 2*Sq
rt[2]] - Sqrt[3 + 2*Sqrt[2]])*x)/(2*(-2 + Sqrt[2]) - I*(Sqrt[3 - 2*Sqrt[2]] + Sqrt[3 + 2*Sqrt[2]])*x)]], 2]))/
((-I + Sqrt[3 - 2*Sqrt[2]])*(I + Sqrt[3 - 2*Sqrt[2]])*(Sqrt[3 - 2*Sqrt[2]] - Sqrt[3 + 2*Sqrt[2]])))/(4*Sqrt[1
+ 6*x^2 + x^4])

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IntegrateAlgebraic [A]  time = 0.63, size = 66, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+6 x^2+x^4}}{4 (1+x)^2}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {2} x}{1-2 x+x^2+\sqrt {1+6 x^2+x^4}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + 6*x^2 + x^4]/(4*(1 + x)^2) - ArcTanh[(2*Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + 6*x^2 + x^4])]/(2*Sqrt[2
])

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fricas [A]  time = 0.58, size = 106, normalized size = 1.61 \begin {gather*} \frac {\sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \log \left (\frac {3 \, x^{4} + 4 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 18 \, x^{2} + 4 \, x + 3}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + 4 \, \sqrt {x^{4} + 6 \, x^{2} + 1}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6*x^2+1)^(1/2)/(-1+x)/(1+x)^3,x, algorithm="fricas")

[Out]

1/16*(sqrt(2)*(x^2 + 2*x + 1)*log((3*x^4 + 4*x^3 - 2*sqrt(2)*sqrt(x^4 + 6*x^2 + 1)*(x^2 + 2*x + 1) + 18*x^2 +
4*x + 3)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) + 4*sqrt(x^4 + 6*x^2 + 1))/(x^2 + 2*x + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 6 \, x^{2} + 1}}{{\left (x + 1\right )}^{3} {\left (x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6*x^2+1)^(1/2)/(-1+x)/(1+x)^3,x, algorithm="giac")

[Out]

integrate(sqrt(x^4 + 6*x^2 + 1)/((x + 1)^3*(x - 1)), x)

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maple [C]  time = 0.52, size = 80, normalized size = 1.21

method	result	size
trager	\(\frac {\sqrt {x^{4}+6 x^{2}+1}}{4 \left (1+x \right )^{2}}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +2 \sqrt {x^{4}+6 x^{2}+1}-\RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (-1+x \right )^{2}}\right )}{8}\)	\(80\)
default	\(\frac {\sqrt {x^{4}+6 x^{2}+1}}{4 \left (1+x \right )^{2}}-\frac {\sqrt {8}\, \arctanh \left (\frac {\left (8 x^{2}+8\right ) \sqrt {8}}{16 \sqrt {x^{4}+6 x^{2}+1}}\right )}{16}+\frac {\sqrt {1-\left (-3+2 \sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (-3-2 \sqrt {2}\right ) x^{2}}\, \EllipticF \left (x \left (i \sqrt {2}-i\right ), 3+2 \sqrt {2}\right )}{2 \left (i \sqrt {2}-i\right ) \sqrt {x^{4}+6 x^{2}+1}}-\frac {\sqrt {1-\left (-3+2 \sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (-3-2 \sqrt {2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-3+2 \sqrt {2}}\, x , \frac {1}{-3+2 \sqrt {2}}, \frac {\sqrt {-3-2 \sqrt {2}}}{\sqrt {-3+2 \sqrt {2}}}\right )}{\sqrt {-3+2 \sqrt {2}}\, \sqrt {x^{4}+6 x^{2}+1}}\)	\(222\)
risch	\(\frac {\sqrt {x^{4}+6 x^{2}+1}}{4 \left (1+x \right )^{2}}-\frac {\sqrt {8}\, \arctanh \left (\frac {\left (8 x^{2}+8\right ) \sqrt {8}}{16 \sqrt {x^{4}+6 x^{2}+1}}\right )}{16}+\frac {\sqrt {1-\left (-3+2 \sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (-3-2 \sqrt {2}\right ) x^{2}}\, \EllipticF \left (x \left (i \sqrt {2}-i\right ), 3+2 \sqrt {2}\right )}{2 \left (i \sqrt {2}-i\right ) \sqrt {x^{4}+6 x^{2}+1}}-\frac {\sqrt {1-\left (-3+2 \sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (-3-2 \sqrt {2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-3+2 \sqrt {2}}\, x , \frac {1}{-3+2 \sqrt {2}}, \frac {\sqrt {-3-2 \sqrt {2}}}{\sqrt {-3+2 \sqrt {2}}}\right )}{\sqrt {-3+2 \sqrt {2}}\, \sqrt {x^{4}+6 x^{2}+1}}\)	\(222\)
elliptic	\(\frac {\left (\left (x^{2}-1\right )^{2}+8 x^{2}\right )^{\frac {3}{2}}}{16 \left (x^{2}-1\right )^{2}}-\frac {\left (\left (x^{2}-1\right )^{2}+8 x^{2}\right )^{\frac {3}{2}}}{32 \left (x^{2}-1\right )}+\frac {\sqrt {\left (x^{2}-1\right )^{2}+8 x^{2}}}{16}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (8 x^{2}+8\right ) \sqrt {2}}{8 \sqrt {\left (x^{2}-1\right )^{2}+8 x^{2}}}\right )}{8}+\frac {\left (2 x^{2}+6\right ) \sqrt {\left (x^{2}-1\right )^{2}+8 x^{2}}}{64}+\frac {\left (-\frac {1}{4 \left (2+\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}\right )}-\frac {\ln \left (2+\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}-\frac {1}{4 \left (-2+\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}\right )}+\frac {\ln \left (-2+\frac {\sqrt {x^{4}+6 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}\right ) \sqrt {2}}{2}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+6*x^2+1)^(1/2)/(-1+x)/(1+x)^3,x,method=_RETURNVERBOSE)

[Out]

1/4*(x^4+6*x^2+1)^(1/2)/(1+x)^2+1/8*RootOf(_Z^2-2)*ln(-(-RootOf(_Z^2-2)*x^2-2*RootOf(_Z^2-2)*x+2*(x^4+6*x^2+1)
^(1/2)-RootOf(_Z^2-2))/(-1+x)^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 6 \, x^{2} + 1}}{{\left (x + 1\right )}^{3} {\left (x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6*x^2+1)^(1/2)/(-1+x)/(1+x)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + 6*x^2 + 1)/((x + 1)^3*(x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x^4+6\,x^2+1}}{\left (x-1\right )\,{\left (x+1\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^2 + x^4 + 1)^(1/2)/((x - 1)*(x + 1)^3),x)

[Out]

int((6*x^2 + x^4 + 1)^(1/2)/((x - 1)*(x + 1)^3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 6 x^{2} + 1}}{\left (x - 1\right ) \left (x + 1\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+6*x**2+1)**(1/2)/(-1+x)/(1+x)**3,x)

[Out]

Integral(sqrt(x**4 + 6*x**2 + 1)/((x - 1)*(x + 1)**3), x)

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