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

Optimal. Leaf size=189 \[ -\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{\sqrt {2}}+\frac {\left (6 x^8+12 x^4+4\right ) x^2+\sqrt {x^4+1} \left (6 x^6+9 x^2\right ) x^2}{2 x \left (2 x^8+3 x^4+1\right ) \sqrt {\sqrt {x^4+1}+x^2}+2 x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \left (2 x^6+2 x^2\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 2.75, 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 {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-4*ArcTanh[x/Sqrt[x^2 + Sqrt[1 + x^4]]] + Defer[Int][1/Sqrt[x^2 + Sqrt[1 + x^4]], x] + (I/4)*Defer[Int][1/(((-
1)^(1/4) - x)^2*Sqrt[x^2 + Sqrt[1 + x^4]]), x] + (3*(-1)^(1/4)*Defer[Int][1/(((-1)^(1/4) - x)*Sqrt[x^2 + Sqrt[
1 + x^4]]), x])/4 - (I/4)*Defer[Int][1/((-(-1)^(3/4) - x)^2*Sqrt[x^2 + Sqrt[1 + x^4]]), x] - (3*(-1)^(3/4)*Def
er[Int][1/((-(-1)^(3/4) - x)*Sqrt[x^2 + Sqrt[1 + x^4]]), x])/4 + (I/4)*Defer[Int][1/(((-1)^(1/4) + x)^2*Sqrt[x
^2 + Sqrt[1 + x^4]]), x] + (3*(-1)^(1/4)*Defer[Int][1/(((-1)^(1/4) + x)*Sqrt[x^2 + Sqrt[1 + x^4]]), x])/4 - (I
/4)*Defer[Int][1/((-(-1)^(3/4) + x)^2*Sqrt[x^2 + Sqrt[1 + x^4]]), x] - (3*(-1)^(3/4)*Defer[Int][1/((-(-1)^(3/4
) + x)*Sqrt[x^2 + Sqrt[1 + x^4]]), x])/4

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx &=\int \left (\frac {1}{\sqrt {x^2+\sqrt {1+x^4}}}+\frac {4}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {4}{\left (1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=4 \int \frac {1}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-4 \int \frac {1}{\left (1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=4 \int \left (-\frac {1}{4 \left (i-x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{4 \left (i+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{2 \left (-1-x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx-4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \int \frac {1}{\left (-1-x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{\left (i-x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{\left (i+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \int \left (-\frac {i}{2 \left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{2 \left (i+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \left (\frac {i}{4 \left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {i}{4 \left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {i}{2 \left (-i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx-\int \left (-\frac {i}{4 \left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{4 \left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{2 \left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} i \int \frac {1}{\left (-i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} i \int \frac {1}{\left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+i \int \frac {1}{\left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+i \int \frac {1}{\left (i+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+i \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.70, size = 202, normalized size = 1.07 \begin {gather*} \frac {x^2 \sqrt {1+x^4} \left (9 x^2+6 x^6\right )+x^2 \left (4+12 x^4+6 x^8\right )}{2 x \sqrt {1+x^4} \left (2 x^2+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+2 x \left (1+3 x^4+2 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[1 + x^4]*(9*x^2 + 6*x^6) + x^2*(4 + 12*x^4 + 6*x^8))/(2*x*Sqrt[1 + x^4]*(2*x^2 + 2*x^6)*Sqrt[x^2 + S
qrt[1 + x^4]] + 2*x*(1 + 3*x^4 + 2*x^8)*Sqrt[x^2 + Sqrt[1 + x^4]]) - (3*ArcTanh[x/Sqrt[x^2 + Sqrt[1 + x^4]]])/
2 + ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2]

________________________________________________________________________________________

fricas [A]  time = 1.37, size = 180, normalized size = 0.95 \begin {gather*} \frac {\sqrt {2} {\left (x^{4} + 1\right )} \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 ) + 3 \, {\left (x^{4} + 1\right )} \log \left (-\frac {9 \, x^{4} + 8 \, \sqrt {x^{4} + 1} x^{2} - 4 \, {\left (2 \, x^{3} + \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1}{x^{4} + 1}\right ) - 4 \, {\left (x^{7} + 3 \, x^{3} - {\left (x^{5} + 4 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(sqrt(2)*(x^4 + 1)*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) + 3*(x^4 + 1)*log(-(9*x^4 + 8*sqrt(x^4 + 1)*x^2 - 4*(2*x^3 + sqrt(x^4 + 1)*x)*sqrt(x^2 + s
qrt(x^4 + 1)) + 1)/(x^4 + 1)) - 4*(x^7 + 3*x^3 - (x^5 + 4*x)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1)))/(x^4 +
1)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 1\right )}^{2}}{{\left (x^{4} + 1\right )}^{2} \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)^2/(x^4+1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right )^{2}}{\left (x^{4}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 1\right )}^{2}}{{\left (x^{4} + 1\right )}^{2} \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)^2/(x^4+1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________