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

Optimal. Leaf size=170 \[ -\frac {3}{2} \tan ^{-1}\left (x \sqrt {\sqrt {x^4+1}+x^2}\right )+\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}\right )}{2 \sqrt {2}}+\frac {x \left (2 x^8+8 x^4+4\right ) \sqrt {\sqrt {x^4+1}+x^2}+x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \left (2 x^6+7 x^2\right )}{2 \left (2 x^8+3 x^4+1\right )+2 \sqrt {x^4+1} \left (2 x^6+2 x^2\right )} \]

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Rubi [F]  time = 3.15, 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 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.79, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.44, size = 170, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {1+x^4} \left (7 x^2+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (4+8 x^4+2 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+x^4} \left (2 x^2+2 x^6\right )+2 \left (1+3 x^4+2 x^8\right )}-\frac {3}{2} \tan ^{-1}\left (x \sqrt {x^2+\sqrt {1+x^4}}\right )+\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 + x^4]*(7*x^2 + 2*x^6)*Sqrt[x^2 + Sqrt[1 + x^4]] + x*(4 + 8*x^4 + 2*x^8)*Sqrt[x^2 + Sqrt[1 + x^4]])/
(2*Sqrt[1 + x^4]*(2*x^2 + 2*x^6) + 2*(1 + 3*x^4 + 2*x^8)) - (3*ArcTan[x*Sqrt[x^2 + Sqrt[1 + x^4]]])/2 + ArcTan
[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]]/(2*Sqrt[2])

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fricas [A]  time = 1.68, size = 164, normalized size = 0.96 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (x^{4} + 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 ) - 3 \, {\left (x^{4} + 1\right )} \arctan \left (-\frac {4 \, {\left (3 \, x^{9} - 12 \, x^{5} - {\left (3 \, x^{7} - 5 \, x^{3}\right )} \sqrt {x^{4} + 1} + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{17 \, x^{8} - 46 \, x^{4} + 1}\right ) - 4 \, {\left (2 \, x^{5} - \sqrt {x^{4} + 1} x^{3} + 4 \, x\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^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^2,x, algorithm="fricas")

[Out]

-1/8*(2*sqrt(2)*(x^4 + 1)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))/x) - 3*(
x^4 + 1)*arctan(-4*(3*x^9 - 12*x^5 - (3*x^7 - 5*x^3)*sqrt(x^4 + 1) + x)*sqrt(x^2 + sqrt(x^4 + 1))/(17*x^8 - 46
*x^4 + 1)) - 4*(2*x^5 - sqrt(x^4 + 1)*x^3 + 4*x)*sqrt(x^2 + sqrt(x^4 + 1)))/(x^4 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{4}+1\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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