3.10.30 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx\)

Optimal. Leaf size=70 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {\sqrt {\sqrt {x^4+1}+x^2}}{x} \]

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Rubi [F]  time = 0.14, 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 {x^2+\sqrt {1+x^4}}}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[x^2 + Sqrt[1 + x^4]]/x^2,x]

[Out]

Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/x^2, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx &=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 82, normalized size = 1.17 \begin {gather*} \frac {\sqrt {2} \sqrt {-x^2 \left (\sqrt {x^4+1}+x^2\right )} \sin ^{-1}\left (\sqrt {x^4+1}+x^2\right )-2 \left (\sqrt {x^4+1}+x^2\right )}{2 x \sqrt {\sqrt {x^4+1}+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x^2 + Sqrt[1 + x^4]]/x^2,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.37, size = 70, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^2+\sqrt {1+x^4}}}{x}+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/x^2,x]

[Out]

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

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fricas [A]  time = 0.85, size = 81, normalized size = 1.16 \begin {gather*} \frac {\sqrt {2} x \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - 4 \, \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + sqrt(x^4 + 1))/x^2, x)

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maple [C]  time = 0.05, size = 51, normalized size = 0.73

method	result	size
meijerg	\(\frac {-\frac {\sqrt {\pi }\, \sqrt {2}\, \hypergeom \left (\left [\frac {3}{4}, 1, 1, \frac {5}{4}\right ], \left [\frac {3}{2}, 2, 2\right ], -\frac {1}{x^{4}}\right )}{2 x^{4}}-4 \left (-4 \ln \relax (2)+4-4 \ln \relax (x )\right ) \sqrt {\pi }\, \sqrt {2}}{16 \sqrt {\pi }}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/Pi^(1/2)*(-1/2*Pi^(1/2)*2^(1/2)/x^4*hypergeom([3/4,1,1,5/4],[3/2,2,2],-1/x^4)-4*(-4*ln(2)+4-4*ln(x))*Pi^(
1/2)*2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + sqrt(x^4 + 1))/x^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 1)^(1/2) + x^2)^(1/2)/x^2,x)

[Out]

int(((x^4 + 1)^(1/2) + x^2)^(1/2)/x^2, x)

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sympy [C]  time = 1.11, size = 53, normalized size = 0.76 \begin {gather*} - \frac {\log {\left (\frac {1}{x^{4}} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{4 \pi } - \frac {\Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{4}F_{3}\left (\begin {matrix} \frac {3}{4}, 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2 \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{8 \pi x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+(x**4+1)**(1/2))**(1/2)/x**2,x)

[Out]

-log(x**(-4))*gamma(1/4)*gamma(3/4)/(4*pi) - gamma(3/4)*gamma(5/4)*hyper((3/4, 1, 1, 5/4), (3/2, 2, 2), exp_po
lar(I*pi)/x**4)/(8*pi*x**4)

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