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

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

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Rubi [F]  time = 0.03, 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 {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/Sqrt[x^2 + Sqrt[1 + x^4]],x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 11.73, size = 1665, normalized size = 23.79

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x^2 + Sqrt[1 + x^4]]*((Sqrt[2]*Sqrt[1 + x^4]*Sqrt[x^2*(x^2 + Sqrt[1 + x^4])]*(-(x^4*(1 + 2*x^4 + 2*x^2*S
qrt[1 + x^4])))^(3/2)*(Sqrt[2]*Sqrt[-(x^2*(x^2 + Sqrt[1 + x^4]))]*(1 - 2*x^4 - 2*x^2*Sqrt[1 + x^4]) + (x^2 + S
qrt[1 + x^4])*ArcSin[x^2 + Sqrt[1 + x^4]]))/(1 + 13*x^4 + 28*x^8 + 16*x^12 + 5*x^2*Sqrt[1 + x^4] + 20*x^6*Sqrt
[1 + x^4] + 16*x^10*Sqrt[1 + x^4]) + (56*x^6*(x^2 + Sqrt[1 + x^4])^3*(1 + x^4 + x^2*Sqrt[1 + x^4])*(5*(1 + 5*x
^4 + 2*x^8 + 4*x^2*Sqrt[1 + x^4] + 2*x^6*Sqrt[1 + x^4])*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 + x^4]
)^2] + 2*(1 + 5*x^4 + 4*x^8 + 3*x^2*Sqrt[1 + x^4] + 4*x^6*Sqrt[1 + x^4])*Hypergeometric2F1[1/2, 3/2, 7/2, (x^2
+ Sqrt[1 + x^4])^2] + 2*(1 + x^4)*(1 + 8*x^4 + 8*x^8 + 4*x^2*Sqrt[1 + x^4] + 8*x^6*Sqrt[1 + x^4])*Hypergeomet
ricPFQ[{1/2, 3/2, 2}, {1, 7/2}, (x^2 + Sqrt[1 + x^4])^2]))/(630*x^2*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + S
qrt[1 + x^4])^2] + 3990*x^6*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 6720*x^10*Hypergeomet
ric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 3360*x^14*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 +
x^4])^2] + 105*Sqrt[1 + x^4]*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 1890*x^4*Sqrt[1 + x^
4]*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 5040*x^8*Sqrt[1 + x^4]*Hypergeometric2F1[-1/2,
1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 3360*x^12*Sqrt[1 + x^4]*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1
+ x^4])^2] + 140*x^2*Hypergeometric2F1[1/2, 3/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 952*x^6*Hypergeometric2F1[1/2
, 3/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 1232*x^10*Hypergeometric2F1[1/2, 3/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] - 4
48*x^14*Hypergeometric2F1[1/2, 3/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] - 896*x^18*Hypergeometric2F1[1/2, 3/2, 7/2,
(x^2 + Sqrt[1 + x^4])^2] + 21*Sqrt[1 + x^4]*Hypergeometric2F1[1/2, 3/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 448*x^
4*Sqrt[1 + x^4]*Hypergeometric2F1[1/2, 3/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 1120*x^8*Sqrt[1 + x^4]*Hypergeomet
ric2F1[1/2, 3/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] - 896*x^16*Sqrt[1 + x^4]*Hypergeometric2F1[1/2, 3/2, 7/2, (x^2
+ Sqrt[1 + x^4])^2] + 54*x^2*Hypergeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 720*x^6*Hypergeometri
c2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 2592*x^10*Hypergeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4]
)^2] + 3456*x^14*Hypergeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 1536*x^18*Hypergeometric2F1[3/2,
5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 6*Sqrt[1 + x^4]*Hypergeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2]
+ 240*x^4*Sqrt[1 + x^4]*Hypergeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 1440*x^8*Sqrt[1 + x^4]*Hy
pergeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 2688*x^12*Sqrt[1 + x^4]*Hypergeometric2F1[3/2, 5/2,
9/2, (x^2 + Sqrt[1 + x^4])^2] + 1536*x^16*Sqrt[1 + x^4]*Hypergeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])
^2] + 14*(26*x^2 + 328*x^6 + 1136*x^10 + 1472*x^14 + 640*x^18 + 3*Sqrt[1 + x^4] + 112*x^4*Sqrt[1 + x^4] + 640*
x^8*Sqrt[1 + x^4] + 1152*x^12*Sqrt[1 + x^4] + 640*x^16*Sqrt[1 + x^4])*HypergeometricPFQ[{1/2, 3/2, 2}, {1, 7/2
}, (x^2 + Sqrt[1 + x^4])^2] + 12*(10*x^2 + 170*x^6 + 832*x^10 + 1696*x^14 + 1536*x^18 + 512*x^22 + Sqrt[1 + x^
4] + 50*x^4*Sqrt[1 + x^4] + 400*x^8*Sqrt[1 + x^4] + 1120*x^12*Sqrt[1 + x^4] + 1280*x^16*Sqrt[1 + x^4] + 512*x^
20*Sqrt[1 + x^4])*HypergeometricPFQ[{3/2, 5/2, 3}, {2, 9/2}, (x^2 + Sqrt[1 + x^4])^2])))/(16*x^7)

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IntegrateAlgebraic [A]  time = 0.38, size = 70, normalized size = 1.00 \begin {gather*} \frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 90, normalized size = 1.29 \begin {gather*} -\frac {1}{2} \, {\left (x^{3} - \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {1}{8} \, \sqrt {2} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x^3 - sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1/8*sqrt(2)*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.06, size = 51, normalized size = 0.73

method	result	size
meijerg	\(-\frac {-\frac {\sqrt {\pi }\, \sqrt {2}\, \hypergeom \left (\left [1, 1, \frac {5}{4}, \frac {7}{4}\right ], \left [2, 2, \frac {5}{2}\right ], -\frac {1}{x^{4}}\right )}{4 x^{4}}+2 \left (-4 \ln \relax (2)-2-4 \ln \relax (x )\right ) \sqrt {\pi }\, \sqrt {2}}{16 \sqrt {\pi }}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/Pi^(1/2)*(-1/4*Pi^(1/2)*2^(1/2)/x^4*hypergeom([1,1,5/4,7/4],[2,2,5/2],-1/x^4)+2*(-4*ln(2)-2-4*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 {1}{\sqrt {x^{2} + \sqrt {x^{4} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.98, size = 15, normalized size = 0.21 \begin {gather*} \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 1, 1 & \frac {3}{2} \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{16 \sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

meijerg(((1, 1), (3/2,)), ((1/4, 3/4), (0,)), x**4)/(16*sqrt(pi))

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