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

Optimal. Leaf size=604 \[ -\frac {5 \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {2}-1}}\right )}{2 \sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {2}-1}}\right )}{\sqrt {\sqrt {2}-1}}-3 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (73+53 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {1+\sqrt {2}}}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (53 \sqrt {2}-73\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {x \sqrt {\sqrt {x^4+1}+x^2} \left (-48 x^{12}+72 x^{10}-96 x^8+72 x^6-52 x^4+9 x^2-5\right )+\sqrt {x^4+1} \left (x \sqrt {\sqrt {x^4+1}+x^2} \left (-48 x^{10}+72 x^8-72 x^6+36 x^4-22 x^2\right )+\left (16 x^{12}-16 x^{10}+28 x^8-4 x^6+9 x^4+x^2\right ) \sqrt {\sqrt {x^4+1}+x^2}\right )+\left (16 x^{14}-16 x^{12}+36 x^{10}-12 x^8+21 x^6+x^4+2 x^2\right ) \sqrt {\sqrt {x^4+1}+x^2}}{2 x \left (x^2-1\right )^2 \left (\sqrt {x^4+1}+x^2\right )^5} \]

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Rubi [F]  time = 0.34, 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+x^4}}{(1+x)^3 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x^4}}{(1+x)^3 \sqrt {x^2+\sqrt {1+x^4}}} \, dx &=\int \frac {\sqrt {1+x^4}}{(1+x)^3 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^4}}{(1+x)^3 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 8.45, size = 600, normalized size = 0.99 \begin {gather*} \frac {x \left (-5+9 x^2-52 x^4+72 x^6-96 x^8+72 x^{10}-48 x^{12}\right ) \sqrt {x^2+\sqrt {1+x^4}}+\left (2 x^2+x^4+21 x^6-12 x^8+36 x^{10}-16 x^{12}+16 x^{14}\right ) \sqrt {x^2+\sqrt {1+x^4}}+\sqrt {1+x^4} \left (x \left (-22 x^2+36 x^4-72 x^6+72 x^8-48 x^{10}\right ) \sqrt {x^2+\sqrt {1+x^4}}+\left (x^2+9 x^4-4 x^6+28 x^8-16 x^{10}+16 x^{12}\right ) \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 x \left (-1+x^2\right )^2 \left (x^2+\sqrt {1+x^4}\right )^5}-\frac {\tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )}{\sqrt {-1+\sqrt {2}}}-\frac {5 \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-3 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {1}{2} \sqrt {\frac {73}{2}+\frac {53}{\sqrt {2}}} \tan ^{-1}\left (\frac {\sqrt {-2+2 \sqrt {2}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )}{\sqrt {1+\sqrt {2}}}-\frac {5 \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2+2 \sqrt {2}}}+\frac {1}{2} \sqrt {-\frac {73}{2}+\frac {53}{\sqrt {2}}} \tanh ^{-1}\left (\frac {\sqrt {2+2 \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[1 + x^4]/((1 + x)^3*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

(x*(-5 + 9*x^2 - 52*x^4 + 72*x^6 - 96*x^8 + 72*x^10 - 48*x^12)*Sqrt[x^2 + Sqrt[1 + x^4]] + (2*x^2 + x^4 + 21*x
^6 - 12*x^8 + 36*x^10 - 16*x^12 + 16*x^14)*Sqrt[x^2 + Sqrt[1 + x^4]] + Sqrt[1 + x^4]*(x*(-22*x^2 + 36*x^4 - 72
*x^6 + 72*x^8 - 48*x^10)*Sqrt[x^2 + Sqrt[1 + x^4]] + (x^2 + 9*x^4 - 4*x^6 + 28*x^8 - 16*x^10 + 16*x^12)*Sqrt[x
^2 + Sqrt[1 + x^4]]))/(2*x*(-1 + x^2)^2*(x^2 + Sqrt[1 + x^4])^5) - ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1
+ x^4]]]/Sqrt[-1 + Sqrt[2]] - (5*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]])/(2*Sqrt[-2 + 2*Sqrt[2]])
- 3*Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] + (Sqrt[73/2 + 53/Sqrt[2]
]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/2 + ArcTanh[Sqrt[-1 +
Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[1 + Sqrt[2]] - (5*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]
]])/(2*Sqrt[2 + 2*Sqrt[2]]) + (Sqrt[-73/2 + 53/Sqrt[2]]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4
]])/(1 + x^2 + Sqrt[1 + x^4])])/2

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fricas [A]  time = 5.88, size = 562, normalized size = 0.93 \begin {gather*} -\frac {4 \, \sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \sqrt {53 \, \sqrt {2} + 73} \arctan \left (\frac {\sqrt {2} {\left (14 \, x^{2} - 9 \, \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (7 \, \sqrt {2} - 9\right )} + 14\right )} \sqrt {53 \, \sqrt {2} + 73} \sqrt {\sqrt {2} + 1} + 2 \, {\left (9 \, x^{3} + 5 \, x^{2} - \sqrt {2} {\left (7 \, x^{3} + 2 \, x^{2} - 2 \, x + 7\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (7 \, x + 2\right )} - 9 \, x - 5\right )} - 5 \, x + 9\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {53 \, \sqrt {2} + 73}}{34 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - \sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \sqrt {53 \, \sqrt {2} - 73} \log \left (-\frac {17 \, {\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (4 \, x^{2} + 5 \, \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} + 5\right )} + 4\right )} \sqrt {53 \, \sqrt {2} - 73}}{x^{2} + 2 \, x + 1}\right ) + \sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \sqrt {53 \, \sqrt {2} - 73} \log \left (-\frac {17 \, {\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (4 \, x^{2} + 5 \, \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} + 5\right )} + 4\right )} \sqrt {53 \, \sqrt {2} - 73}}{x^{2} + 2 \, x + 1}\right ) - 24 \, \sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) - 8 \, {\left (2 \, x^{4} + 8 \, x^{3} + 5 \, x^{2} - \sqrt {x^{4} + 1} {\left (2 \, x^{2} + 8 \, x + 5\right )} + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 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+1)^(1/2)/(1+x)^3/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/16*(4*sqrt(2)*(x^2 + 2*x + 1)*sqrt(53*sqrt(2) + 73)*arctan(1/34*(sqrt(2)*(14*x^2 - 9*sqrt(2)*(x^2 + 1) + 2*
sqrt(x^4 + 1)*(7*sqrt(2) - 9) + 14)*sqrt(53*sqrt(2) + 73)*sqrt(sqrt(2) + 1) + 2*(9*x^3 + 5*x^2 - sqrt(2)*(7*x^
3 + 2*x^2 - 2*x + 7) + sqrt(x^4 + 1)*(sqrt(2)*(7*x + 2) - 9*x - 5) - 5*x + 9)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(5
3*sqrt(2) + 73))/(x^2 - 2*x + 1)) - sqrt(2)*(x^2 + 2*x + 1)*sqrt(53*sqrt(2) - 73)*log(-(17*(2*x^3 - sqrt(2)*(x
^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) + (4*x^2 + 5*sqrt(2)*
(x^2 + 1) + 2*sqrt(x^4 + 1)*(2*sqrt(2) + 5) + 4)*sqrt(53*sqrt(2) - 73))/(x^2 + 2*x + 1)) + sqrt(2)*(x^2 + 2*x
+ 1)*sqrt(53*sqrt(2) - 73)*log(-(17*(2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*
x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (4*x^2 + 5*sqrt(2)*(x^2 + 1) + 2*sqrt(x^4 + 1)*(2*sqrt(2) + 5) + 4)*sqrt(5
3*sqrt(2) - 73))/(x^2 + 2*x + 1)) - 24*sqrt(2)*(x^2 + 2*x + 1)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 + 1
))*sqrt(x^2 + sqrt(x^4 + 1))/x) - 8*(2*x^4 + 8*x^3 + 5*x^2 - sqrt(x^4 + 1)*(2*x^2 + 8*x + 5) + x)*sqrt(x^2 + s
qrt(x^4 + 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} + 1}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x + 1\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/2)/(1+x)^3/(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 + 1)^3), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{4}+1}}{\left (1+x \right )^{3} \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^4+1)^(1/2)/(1+x)^3/(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 \frac {\sqrt {x^{4} + 1}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x + 1\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/2)/(1+x)^3/(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 + 1)^3), x)

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

[Out]

int((x^4 + 1)^(1/2)/(((x^4 + 1)^(1/2) + x^2)^(1/2)*(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} + 1}}{\left (x + 1\right )^{3} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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