3.24.2 \(\int \frac {(-1+x^2) \sqrt {x^2+\sqrt {1+x^4}}}{(1+x^2) \sqrt {1+x^4}} \, dx\) [2302]

Optimal. Leaf size=176 \[ -\sqrt {2 \left (-1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {2 \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

-(-2+2*2^(1/2))^(1/2)*arctan((2+2*2^(1/2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))+arctanh(2^
(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)-(2+2*2^(1/2))^(1/2)*arctanh((-2+2*2^(1/2))^(1
/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))

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Rubi [C] Result contains complex when optimal does not.
time = 0.67, antiderivative size = 180, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6857, 2157, 212, 2158, 739} \begin {gather*} \frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 + I]*ArcTanh[(1 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/2 - (Sqrt[1 + I]*ArcTanh[(1 + x)/(Sqrt[1 + I]*Sqr
t[1 - I*x^2])])/2 + (Sqrt[1 - I]*ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/2 - (Sqrt[1 - I]*ArcTanh[(1 +
x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/2 + ArcTanh[(Sqrt[2]*x)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[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 2157

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

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]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]
time = 0.68, size = 172, normalized size = 0.98 \begin {gather*} \sqrt {2} \left (-\sqrt {-1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^2+1)/(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^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^2+1)/(x^4+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (138) = 276\).
time = 3.44, size = 368, normalized size = 2.09 \begin {gather*} -\sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {{\left (4 \, x^{2} + 2 \, \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} + 2\right )} \sqrt {-8 \, \sqrt {2} + 12} - 2 \, \sqrt {2} - 4\right )} - {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 3\right )} + 4\right )} \sqrt {-8 \, \sqrt {2} + 12}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} - 2}}{8 \, x}\right ) + \frac {1}{4} \, \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 ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} + {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} + x\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} + 2} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} - {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} + x\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} + 2} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2*sqrt(2) - 2)*arctan(1/8*(4*x^2 + 2*sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1)*((sqrt(2) + 2)*sqrt(-8*sqrt(2) +
12) - 2*sqrt(2) - 4) - (2*x^2 + sqrt(2)*(x^2 + 3) + 4)*sqrt(-8*sqrt(2) + 12))*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(2
*sqrt(2) - 2)/x) + 1/4*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)*sqr
t(x^2 + sqrt(x^4 + 1)) + 1) + 1/4*sqrt(2*sqrt(2) + 2)*log((2*sqrt(2)*x^2 + 4*x^2 + (sqrt(2)*sqrt(x^4 + 1)*x -
sqrt(2)*(x^3 + x) - 2*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(2*sqrt(2) + 2) + 2*sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 +
1)) - 1/4*sqrt(2*sqrt(2) + 2)*log((2*sqrt(2)*x^2 + 4*x^2 - (sqrt(2)*sqrt(x^4 + 1)*x - sqrt(2)*(x^3 + x) - 2*x
)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(2*sqrt(2) + 2) + 2*sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
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^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^2+1)/(x^4+1)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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