3.1.13 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx\) [13]

Optimal. Leaf size=81 \[ -\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right ) \]

[Out]

-1/2*arctanh((1+I*x)/(1-I)^(1/2)/(1-I*x^2)^(1/2))*(1-I)^(1/2)-1/2*arctanh((1-I*x)/(1+I)^(1/2)/(1+I*x^2)^(1/2))
*(1+I)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2158, 739, 212} \begin {gather*} -\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 2158

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_S
ymbol] :> Dist[(1 - I)/2, Int[(c + d*x)^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Dist[(1 + I)/2, Int[(c + d*x)^m/Sq
rt[Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx &=\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(1+x) \sqrt {1-i x^2}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(1+x) \sqrt {1+i x^2}} \, dx\\ &=\left (-\frac {1}{2}-\frac {i}{2}\right ) \text {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-i x}{\sqrt {1+i x^2}}\right )+\left (-\frac {1}{2}+\frac {i}{2}\right ) \text {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+i x}{\sqrt {1-i x^2}}\right )\\ &=-\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(81)=162\).
time = 0.86, size = 205, normalized size = 2.53 \begin {gather*} \frac {\sqrt {-1+\sqrt {2}} \left (\tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )-\tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} 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]

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

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (1+x \right ) \sqrt {x^{4}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (49) = 98\).
time = 3.05, size = 369, normalized size = 4.56 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {{\left (2 \, x^{2} - \sqrt {2} {\left (x^{3} - x^{2} + x + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} - 2} + {\left (x^{2} + \sqrt {2} \sqrt {x^{4} + 1} + 1\right )} \sqrt {2 \, \sqrt {2} + 2} \sqrt {2 \, \sqrt {2} - 2}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\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 (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\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 (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2*sqrt(2) - 2)*arctan(1/2*((2*x^2 - sqrt(2)*(x^3 - x^2 + x + 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2)
- 2*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(2*sqrt(2) - 2) + (x^2 + sqrt(2)*sqrt(x^4 + 1) + 1)*sqrt(2*sqrt(2) + 2)*
sqrt(2*sqrt(2) - 2))/(x^2 - 2*x + 1)) - 1/8*sqrt(2*sqrt(2) + 2)*log(-((2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + s
qrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) + (x^2 - sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1)
*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2 + 2*x + 1)) + 1/8*sqrt(2*sqrt(2) + 2)*log(-((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)) - (x^2 - sqrt(2)*(x^2
+ 1) + sqrt(x^4 + 1)*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2 + 2*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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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}}{\sqrt {x^4+1}\,\left (x+1\right )} \,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^4 + 1)^(1/2)*(x + 1)),x)

[Out]

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

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