3.23.47 \(\int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\)
Optimal. Leaf size=167 \[ \frac {x \left (192 x^{12}+360 x^8+212 x^4+39\right ) \sqrt {\sqrt {x^4+1}+x^2}+x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \left (192 x^{10}+264 x^6+104 x^2\right )}{384 \sqrt {x^4+1} \left (4 x^4+1\right )+384 \left (4 x^6+3 x^2\right )}-\frac {13 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{64 \sqrt {2}} \]
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Rubi [F] time = 0.35, 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 x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[x^4*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]
[Out]
Defer[Int][x^4*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]], x]
Rubi steps
\begin {align*} \int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx &=\int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 7.29, size = 1550, normalized size = 9.28
result too large to display
Warning: Unable to verify antiderivative.
[In]
Integrate[x^4*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]
[Out]
(5*x^3*(x^2 + Sqrt[1 + x^4])^(7/2)*(1 + x^4 + x^2*Sqrt[1 + x^4])*(-((-1 + 3*x^4 + 6*x^8 + 6*x^6*Sqrt[1 + x^4])
*Hypergeometric2F1[-3/2, -3/2, 3/2, (x^2 + Sqrt[1 + x^4])^2]) + 6*(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])*Hypergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1 + x^
4])^2] - 6*(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,
-1/2, 2}, {1, 5/2}, (x^2 + Sqrt[1 + x^4])^2]))/(6*(-30*x^2*Hypergeometric2F1[-3/2, -3/2, 3/2, (x^2 + Sqrt[1 +
x^4])^2] - 190*x^6*Hypergeometric2F1[-3/2, -3/2, 3/2, (x^2 + Sqrt[1 + x^4])^2] - 320*x^10*Hypergeometric2F1[-3
/2, -3/2, 3/2, (x^2 + Sqrt[1 + x^4])^2] - 160*x^14*Hypergeometric2F1[-3/2, -3/2, 3/2, (x^2 + Sqrt[1 + x^4])^2]
- 5*Sqrt[1 + x^4]*Hypergeometric2F1[-3/2, -3/2, 3/2, (x^2 + Sqrt[1 + x^4])^2] - 90*x^4*Sqrt[1 + x^4]*Hypergeo
metric2F1[-3/2, -3/2, 3/2, (x^2 + Sqrt[1 + x^4])^2] - 240*x^8*Sqrt[1 + x^4]*Hypergeometric2F1[-3/2, -3/2, 3/2,
(x^2 + Sqrt[1 + x^4])^2] - 160*x^12*Sqrt[1 + x^4]*Hypergeometric2F1[-3/2, -3/2, 3/2, (x^2 + Sqrt[1 + x^4])^2]
+ 220*x^2*Hypergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 2840*x^6*Hypergeometric2F1[-1/2, -1/2
, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 10000*x^10*Hypergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 131
20*x^14*Hypergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 5760*x^18*Hypergeometric2F1[-1/2, -1/2,
5/2, (x^2 + Sqrt[1 + x^4])^2] + 25*Sqrt[1 + x^4]*Hypergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] +
960*x^4*Sqrt[1 + x^4]*Hypergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 5600*x^8*Sqrt[1 + x^4]*Hy
pergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 10240*x^12*Sqrt[1 + x^4]*Hypergeometric2F1[-1/2, -
1/2, 5/2, (x^2 + Sqrt[1 + x^4])^2] + 5760*x^16*Sqrt[1 + x^4]*Hypergeometric2F1[-1/2, -1/2, 5/2, (x^2 + Sqrt[1
+ x^4])^2] + 22*x^2*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 440*x^6*Hypergeometric2F1[1/2,
1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 2464*x^10*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 56
32*x^14*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 5632*x^18*Hypergeometric2F1[1/2, 1/2, 7/2,
(x^2 + Sqrt[1 + x^4])^2] + 2048*x^22*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 2*Sqrt[1 + x
^4]*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 120*x^4*Sqrt[1 + x^4]*Hypergeometric2F1[1/2, 1
/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 1120*x^8*Sqrt[1 + x^4]*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^
4])^2] + 3584*x^12*Sqrt[1 + x^4]*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 4608*x^16*Sqrt[1
+ x^4]*Hypergeometric2F1[1/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] + 2048*x^20*Sqrt[1 + x^4]*Hypergeometric2F1[1
/2, 1/2, 7/2, (x^2 + Sqrt[1 + x^4])^2] - 10*(10*x^2 + 152*x^6 + 592*x^10 + 832*x^14 + 384*x^18 + Sqrt[1 + x^4]
+ 48*x^4*Sqrt[1 + x^4] + 320*x^8*Sqrt[1 + x^4] + 640*x^12*Sqrt[1 + x^4] + 384*x^16*Sqrt[1 + x^4])*Hypergeomet
ricPFQ[{-1/2, -1/2, 2}, {1, 5/2}, (x^2 + Sqrt[1 + x^4])^2] - 4*(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] + 12
80*x^16*Sqrt[1 + x^4] + 512*x^20*Sqrt[1 + x^4])*HypergeometricPFQ[{1/2, 1/2, 3}, {2, 7/2}, (x^2 + Sqrt[1 + x^4
])^2]))
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IntegrateAlgebraic [A] time = 0.59, size = 167, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {1+x^4} \left (104 x^2+264 x^6+192 x^{10}\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (39+212 x^4+360 x^8+192 x^{12}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{384 \sqrt {1+x^4} \left (1+4 x^4\right )+384 \left (3 x^2+4 x^6\right )}-\frac {13 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{64 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^4*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]
[Out]
(x*Sqrt[1 + x^4]*(104*x^2 + 264*x^6 + 192*x^10)*Sqrt[x^2 + Sqrt[1 + x^4]] + x*(39 + 212*x^4 + 360*x^8 + 192*x^
12)*Sqrt[x^2 + Sqrt[1 + x^4]])/(384*Sqrt[1 + x^4]*(1 + 4*x^4) + 384*(3*x^2 + 4*x^6)) - (13*ArcTanh[(Sqrt[2]*x*
Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/(64*Sqrt[2])
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fricas [A] time = 1.08, size = 105, normalized size = 0.63 \begin {gather*} -\frac {1}{384} \, {\left (8 \, x^{7} + 13 \, x^{3} - {\left (56 \, x^{5} + 39 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {13}{512} \, \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*(x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")
[Out]
-1/384*(8*x^7 + 13*x^3 - (56*x^5 + 39*x)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1)) + 13/512*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 \sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(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^4, x)
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int x^{4} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x)
[Out]
int(x^4*(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}} x^{4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(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^4, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,\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*(x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2),x)
[Out]
int(x^4*(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 x^{4} \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*(x**4+1)**(1/2)*(x**2+(x**4+1)**(1/2))**(1/2),x)
[Out]
Integral(x**4*sqrt(x**2 + sqrt(x**4 + 1))*sqrt(x**4 + 1), x)
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