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3.17.77 \(\int \frac {(-1+x^4) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx\) [1677]

Optimal. Leaf size=112 \[ \frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+\text {$\#$1}^5}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.66, 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 ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.20, size = 94, normalized size = 0.84 \begin {gather*} \frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(2 + Sqrt[1 + x^2]))/(3*Sqrt[1 + Sqrt[1 + x^2]]) - RootSum[1 + 4*#1^4 + 4*#1^6 + #1^8 & , Log[x/Sqrt[1 +
Sqrt[1 + x^2]] - #1]/(2*#1^3 + #1^5) & ]/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 9.93, size = 6296, normalized size = 56.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*x*sqrt(sqrt(1/4*I - 1/4) + sqrt(-1/4*I - 1/4) - 2*sqrt(-3/16*(2*sqrt(1/4*I - 1/4) + I)^2 - 1/8*(2*sqrt
(1/4*I - 1/4) + I)*(2*sqrt(-1/4*I - 1/4) - I) - 3/16*(2*sqrt(-1/4*I - 1/4) - I)^2 - 1))*log(-1/4*(2*(68*x^4 +
401*x^2 + (277*x^4 + 889*x^2 - 6*(113*x^2 + 59)*sqrt(x^2 + 1) + 354)*(2*sqrt(1/4*I - 1/4) + I) - (277*x^2 + 21
1)*sqrt(x^2 + 1) + 211)*(2*sqrt(-1/4*I - 1/4) - I)^2*sqrt(sqrt(x^2 + 1) + 1) - 2*(37*x^4 - (277*x^4 + 889*x^2
- 6*(113*x^2 + 59)*sqrt(x^2 + 1) + 354)*(2*sqrt(1/4*I - 1/4) + I)^2 + 209*x^2 - 4*(17*x^2 + 31)*sqrt(x^2 + 1)
+ 124)*(2*sqrt(-1/4*I - 1/4) - I)*sqrt(sqrt(x^2 + 1) + 1) + 8*((68*x^4 + 401*x^2 + (277*x^4 + 889*x^2 - 6*(113
*x^2 + 59)*sqrt(x^2 + 1) + 354)*(2*sqrt(1/4*I - 1/4) + I) - (277*x^2 + 211)*sqrt(x^2 + 1) + 211)*(2*sqrt(-1/4*
I - 1/4) - I)*sqrt(sqrt(x^2 + 1) + 1) + (37*x^4 + 209*x^2 + (68*x^4 + 401*x^2 - (277*x^2 + 211)*sqrt(x^2 + 1)
+ 211)*(2*sqrt(1/4*I - 1/4) + I) - 4*(17*x^2 + 31)*sqrt(x^2 + 1) + 124)*sqrt(sqrt(x^2 + 1) + 1))*sqrt(-3/16*(2
*sqrt(1/4*I - 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I - 1/4) + I)*(2*sqrt(-1/4*I - 1/4) - I) - 3/16*(2*sqrt(-1/4*I - 1
/4) - I)^2 - 1) + (12*x^5 + 68*x^3 + 2*(62*x^5 + 143*x^3 - (211*x^3 - 277*x)*sqrt(x^2 + 1) - 339*x)*(2*sqrt(1/
4*I - 1/4) + I)^2 + (124*x^5 + 286*x^3 + (211*x^5 + 154*x^3 - 12*(59*x^3 - 113*x)*sqrt(x^2 + 1) - 1567*x)*(2*s
qrt(1/4*I - 1/4) + I) - 2*(211*x^3 - 277*x)*sqrt(x^2 + 1) - 678*x)*(2*sqrt(-1/4*I - 1/4) - I)^2 - (141*x^5 + 1
74*x^3 - 8*(31*x^3 - 17*x)*sqrt(x^2 + 1) - 277*x)*(2*sqrt(1/4*I - 1/4) + I) - (141*x^5 + 174*x^3 - (211*x^5 +
154*x^3 - 12*(59*x^3 - 113*x)*sqrt(x^2 + 1) - 1567*x)*(2*sqrt(1/4*I - 1/4) + I)^2 - 8*(31*x^3 - 17*x)*sqrt(x^2
 + 1) - 277*x)*(2*sqrt(-1/4*I - 1/4) - I) + 4*(141*x^3 - 37*x)*sqrt(x^2 + 1) + 4*(141*x^5 + 174*x^3 + 2*(62*x^
5 + 143*x^3 - (211*x^3 - 277*x)*sqrt(x^2 + 1) - 339*x)*(2*sqrt(1/4*I - 1/4) + I) + (124*x^5 + 286*x^3 + (211*x
^5 + 154*x^3 - 12*(59*x^3 - 113*x)*sqrt(x^2 + 1) - 1567*x)*(2*sqrt(1/4*I - 1/4) + I) - 2*(211*x^3 - 277*x)*sqr
t(x^2 + 1) - 678*x)*(2*sqrt(-1/4*I - 1/4) - I) - 8*(31*x^3 - 17*x)*sqrt(x^2 + 1) - 277*x)*sqrt(-3/16*(2*sqrt(1
/4*I - 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I - 1/4) + I)*(2*sqrt(-1/4*I - 1/4) - I) - 3/16*(2*sqrt(-1/4*I - 1/4) - I
)^2 - 1) + 136*x)*sqrt(sqrt(1/4*I - 1/4) + sqrt(-1/4*I - 1/4) - 2*sqrt(-3/16*(2*sqrt(1/4*I - 1/4) + I)^2 - 1/8
*(2*sqrt(1/4*I - 1/4) + I)*(2*sqrt(-1/4*I - 1/4) - I) - 3/16*(2*sqrt(-1/4*I - 1/4) - I)^2 - 1)) - 2*(316*x^4 -
 (68*x^4 + 401*x^2 - (277*x^2 + 211)*sqrt(x^2 + 1) + 211)*(2*sqrt(1/4*I - 1/4) + I)^2 + 62*x^2 + (37*x^4 + 209
*x^2 - 4*(17*x^2 + 31)*sqrt(x^2 + 1) + 124)*(2*sqrt(1/4*I - 1/4) + I) - 2*(37*x^2 + 141)*sqrt(x^2 + 1) + 282)*
sqrt(sqrt(x^2 + 1) + 1))/(x^5 + x)) - 3*x*sqrt(sqrt(1/4*I - 1/4) + sqrt(-1/4*I - 1/4) - 2*sqrt(-3/16*(2*sqrt(1
/4*I - 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I - 1/4) + I)*(2*sqrt(-1/4*I - 1/4) - I) - 3/16*(2*sqrt(-1/4*I - 1/4) - I
)^2 - 1))*log(-1/4*(2*(68*x^4 + 401*x^2 + (277*x^4 + 889*x^2 - 6*(113*x^2 + 59)*sqrt(x^2 + 1) + 354)*(2*sqrt(1
/4*I - 1/4) + I) - (277*x^2 + 211)*sqrt(x^2 + 1) + 211)*(2*sqrt(-1/4*I - 1/4) - I)^2*sqrt(sqrt(x^2 + 1) + 1) -
 2*(37*x^4 - (277*x^4 + 889*x^2 - 6*(113*x^2 + 59)*sqrt(x^2 + 1) + 354)*(2*sqrt(1/4*I - 1/4) + I)^2 + 209*x^2
- 4*(17*x^2 + 31)*sqrt(x^2 + 1) + 124)*(2*sqrt(-1/4*I - 1/4) - I)*sqrt(sqrt(x^2 + 1) + 1) + 8*((68*x^4 + 401*x
^2 + (277*x^4 + 889*x^2 - 6*(113*x^2 + 59)*sqrt(x^2 + 1) + 354)*(2*sqrt(1/4*I - 1/4) + I) - (277*x^2 + 211)*sq
rt(x^2 + 1) + 211)*(2*sqrt(-1/4*I - 1/4) - I)*sqrt(sqrt(x^2 + 1) + 1) + (37*x^4 + 209*x^2 + (68*x^4 + 401*x^2
- (277*x^2 + 211)*sqrt(x^2 + 1) + 211)*(2*sqrt(1/4*I - 1/4) + I) - 4*(17*x^2 + 31)*sqrt(x^2 + 1) + 124)*sqrt(s
qrt(x^2 + 1) + 1))*sqrt(-3/16*(2*sqrt(1/4*I - 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I - 1/4) + I)*(2*sqrt(-1/4*I - 1/4
) - I) - 3/16*(2*sqrt(-1/4*I - 1/4) - I)^2 - 1) - (12*x^5 + 68*x^3 + 2*(62*x^5 + 143*x^3 - (211*x^3 - 277*x)*s
qrt(x^2 + 1) - 339*x)*(2*sqrt(1/4*I - 1/4) + I)^2 + (124*x^5 + 286*x^3 + (211*x^5 + 154*x^3 - 12*(59*x^3 - 113
*x)*sqrt(x^2 + 1) - 1567*x)*(2*sqrt(1/4*I - 1/4) + I) - 2*(211*x^3 - 277*x)*sqrt(x^2 + 1) - 678*x)*(2*sqrt(-1/
4*I - 1/4) - I)^2 - (141*x^5 + 174*x^3 - 8*(31*x^3 - 17*x)*sqrt(x^2 + 1) - 277*x)*(2*sqrt(1/4*I - 1/4) + I) -
(141*x^5 + 174*x^3 - (211*x^5 + 154*x^3 - 12*(59*x^3 - 113*x)*sqrt(x^2 + 1) - 1567*x)*(2*sqrt(1/4*I - 1/4) + I
)^2 - 8*(31*x^3 - 17*x)*sqrt(x^2 + 1) - 277*x)*(2*sqrt(-1/4*I - 1/4) - I) + 4*(141*x^3 - 37*x)*sqrt(x^2 + 1) +
 4*(141*x^5 + 174*x^3 + 2*(62*x^5 + 143*x^3 - (211*x^3 - 277*x)*sqrt(x^2 + 1) - 339*x)*(2*sqrt(1/4*I - 1/4) +
I) + (124*x^5 + 286*x^3 + (211*x^5 + 154*x^3 - 12*(59*x^3 - 113*x)*sqrt(x^2 + 1) - 1567*x)*(2*sqrt(1/4*I - 1/4
) + I) - 2*(211*x^3 - 277*x)*sqrt(x^2 + 1) - 678*x)*(2*sqrt(-1/4*I - 1/4) - I) - 8*(31*x^3 - 17*x)*sqrt(x^2 +
1) - 277*x)*sqrt(-3/16*(2*sqrt(1/4*I - 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I - 1/4) + I)*(2*sqrt(-1/4*I - 1/4) - I)
- 3/16*(2*sqrt(-1/4*I - 1/4) - I)^2 - 1) + 136*x)*sqrt(sqrt(1/4*I - 1/4) + sqrt(-1/4*I - 1/4) - 2*sqrt(-3/16*(
2*sqrt(1/4*I - 1/4) + I)^2 - 1/8*(2*sqrt(1/4*I ...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((x^4 - 1)*sqrt(sqrt(x^2 + 1) + 1)/(x^4 + 1), 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 )\,\sqrt {\sqrt {x^2+1}+1}}{x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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