3.15.36 \(\int \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\)
Optimal. Leaf size=101 \[ \frac {2 \sqrt {x^4+1} x^4+\left (2 x^4+3\right ) x^2}{8 x \sqrt {\sqrt {x^4+1}+x^2}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{4 \sqrt {2}} \]
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Rubi [F] time = 0.11, 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 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]
[Out]
Defer[Int][Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]], x]
Rubi steps
\begin {align*} \int \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx &=\int \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.39, size = 1439, normalized size = 14.25
result too large to display
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]
[Out]
(7*(x^2 + Sqrt[1 + x^4])^(7/2)*(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*Sq
rt[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])*HypergeometricPFQ[{1/2, 3/2, 2}, {1, 7/2}, (x^2 +
Sqrt[1 + x^4])^2]))/(2*x*(630*x^2*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 3990*x^6*Hyper
geometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 6720*x^10*Hypergeometric2F1[-1/2, 1/2, 5/2, (x^2 + Sqr
t[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]*Hype
rgeometric2F1[-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*Hypergeometr
ic2F1[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] - 448*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*Sqr
t[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]*Hypergeometric2F1[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*Hype
rgeometric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 720*x^6*Hypergeometric2F1[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*Hypergeometric2F
1[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]*Hypergeo
metric2F1[3/2, 5/2, 9/2, (x^2 + Sqrt[1 + x^4])^2] + 1440*x^8*Sqrt[1 + x^4]*Hypergeometric2F1[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] + 1
536*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*Sqr
t[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])*HypergeometricP
FQ[{3/2, 5/2, 3}, {2, 9/2}, (x^2 + Sqrt[1 + x^4])^2]))
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IntegrateAlgebraic [A] time = 0.42, size = 101, normalized size = 1.00 \begin {gather*} \frac {2 x^4 \sqrt {1+x^4}+x^2 \left (3+2 x^4\right )}{8 x \sqrt {x^2+\sqrt {1+x^4}}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]
[Out]
(2*x^4*Sqrt[1 + x^4] + x^2*(3 + 2*x^4))/(8*x*Sqrt[x^2 + Sqrt[1 + x^4]]) + (5*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqr
t[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/(4*Sqrt[2])
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fricas [A] time = 1.01, size = 90, normalized size = 0.89 \begin {gather*} -\frac {1}{8} \, {\left (x^{3} - 3 \, \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {5}{32} \, \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((x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")
[Out]
-1/8*(x^3 - 3*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 5/32*sqrt(2)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(s
qrt(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 \sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(x^4 + 1)*sqrt(x^2 + sqrt(x^4 + 1)), x)
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x)
[Out]
int((x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(x^4 + 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 \sqrt {x^4+1}\,\sqrt {\sqrt {x^4+1}+x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2),x)
[Out]
int((x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {x^{4} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**4+1)**(1/2)*(x**2+(x**4+1)**(1/2))**(1/2),x)
[Out]
Integral(sqrt(x**2 + sqrt(x**4 + 1))*sqrt(x**4 + 1), x)
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