3.13.56 \(\int \frac {(-1+x^4) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx\)

Optimal. Leaf size=91 \[ -\sqrt {\frac {1}{6} \left (-1-i \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x}{\left (\sqrt {3}-i\right ) \sqrt {x^4+1}}\right )-\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x}{\left (\sqrt {3}+i\right ) \sqrt {x^4+1}}\right ) \]

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Rubi [F]  time = 0.49, 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+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][Sqrt[1 + x^4]/(-1 - x^2 - 3*x^4 - x^6 - x^8), x] + Defer[Int][(x^4*Sqrt[1 + x^4])/(1 + x^2 + 3*x^4
+ x^6 + x^8), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx &=\int \left (\frac {\sqrt {1+x^4}}{-1-x^2-3 x^4-x^6-x^8}+\frac {x^4 \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8}\right ) \, dx\\ &=\int \frac {\sqrt {1+x^4}}{-1-x^2-3 x^4-x^6-x^8} \, dx+\int \frac {x^4 \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.24, size = 2093, normalized size = 23.00 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((-1)^(1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*x^2]*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1])/Sqrt[1 + x^4]) - ((-1)^(
1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*x^2]*EllipticPi[I/Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0], I*ArcSinh[(-1)
^(1/4)*x], -1])/(Sqrt[1 + x^4]*Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0]*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^
4 & , 1, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0])*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] - Ro
ot[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0])*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] - Root[1 + #1 + 3*#1
^2 + #1^3 + #1^4 & , 4, 0])) + ((-1)^(1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*x^2]*EllipticPi[I/Root[1 + #1 + 3*#1^2 +
#1^3 + #1^4 & , 1, 0], I*ArcSinh[(-1)^(1/4)*x], -1]*Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0]^3)/(Sqrt[1 +
x^4]*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0])*(Root[1 +
#1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0])*(Root[1 + #1 + 3*#1^2 + #1
^3 + #1^4 & , 1, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])) - ((-1)^(1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*
x^2]*EllipticPi[I/Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0], I*ArcSinh[(-1)^(1/4)*x], -1])/(Sqrt[1 + x^4]*R
oot[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0]*(-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] + Root[1 + #1 + 3*#
1^2 + #1^3 + #1^4 & , 2, 0])*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^
4 & , 3, 0])*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])) +
((-1)^(1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*x^2]*EllipticPi[I/Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0], I*ArcSi
nh[(-1)^(1/4)*x], -1]*Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0]^3)/(Sqrt[1 + x^4]*(-Root[1 + #1 + 3*#1^2 +
#1^3 + #1^4 & , 1, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0])*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & ,
2, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0])*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 +
#1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])) - ((-1)^(1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*x^2]*EllipticPi[I/Root[1 + #1
+ 3*#1^2 + #1^3 + #1^4 & , 3, 0], I*ArcSinh[(-1)^(1/4)*x], -1])/(Sqrt[1 + x^4]*Root[1 + #1 + 3*#1^2 + #1^3 + #
1^4 & , 3, 0]*(-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0])*(
-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0])*(Root[1 + #1 + 3
*#1^2 + #1^3 + #1^4 & , 3, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])) + ((-1)^(1/4)*Sqrt[1 - I*x^2]*S
qrt[1 + I*x^2]*EllipticPi[I/Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0], I*ArcSinh[(-1)^(1/4)*x], -1]*Root[1
+ #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0]^3)/(Sqrt[1 + x^4]*(-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] + Root[
1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0])*(-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0] + Root[1 + #1 + 3*#1^2
+ #1^3 + #1^4 & , 3, 0])*(Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0] - Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 &
, 4, 0])) - ((-1)^(1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*x^2]*EllipticPi[I/Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4
, 0], I*ArcSinh[(-1)^(1/4)*x], -1])/(Sqrt[1 + x^4]*Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0]*(-Root[1 + #1
+ 3*#1^2 + #1^3 + #1^4 & , 1, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])*(-Root[1 + #1 + 3*#1^2 + #1^3
+ #1^4 & , 2, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])*(-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3,
0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])) + ((-1)^(1/4)*Sqrt[1 - I*x^2]*Sqrt[1 + I*x^2]*EllipticPi[I
/Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0], I*ArcSinh[(-1)^(1/4)*x], -1]*Root[1 + #1 + 3*#1^2 + #1^3 + #1^4
& , 4, 0]^3)/(Sqrt[1 + x^4]*(-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 1, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1
^4 & , 4, 0])*(-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 2, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0])*(
-Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 3, 0] + Root[1 + #1 + 3*#1^2 + #1^3 + #1^4 & , 4, 0]))

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IntegrateAlgebraic [A]  time = 0.60, size = 65, normalized size = 0.71 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {1+x^4}}{1-x^2+x^4}\right )}{2 \sqrt {3}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x \sqrt {1+x^4}}{1+x^2+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 95, normalized size = 1.04 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{4} - x^{2} + 1\right )} \sqrt {x^{4} + 1}}{3 \, {\left (x^{5} + x\right )}}\right ) + \frac {1}{4} \, \log \left (\frac {x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} - 2 \, {\left (x^{5} + x^{3} + x\right )} \sqrt {x^{4} + 1} + 1}{x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(x^4 - x^2 + 1)*sqrt(x^4 + 1)/(x^5 + x)) + 1/4*log((x^8 + 3*x^6 + 3*x^4 + 3*x^2
- 2*(x^5 + x^3 + x)*sqrt(x^4 + 1) + 1)/(x^8 + x^6 + 3*x^4 + x^2 + 1))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x, algorithm="giac")

[Out]

Exception raised: AttributeError >> type

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maple [A]  time = 1.40, size = 126, normalized size = 1.38

method	result	size
default	\(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}+\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}-\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}\right ) \sqrt {2}}{2}\)	\(126\)
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}+\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}-\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}\right ) \sqrt {2}}{2}\)	\(126\)
trager	\(\RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \ln \left (\frac {x^{4}+12 \sqrt {x^{4}+1}\, \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +6 \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+4 \sqrt {x^{4}+1}\, x +x^{2}+1}{-x^{4}+6 \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+x^{2}-1}\right )-\frac {\ln \left (-\frac {-x^{4}+12 \sqrt {x^{4}+1}\, \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +6 \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x +2 x^{2}-1}{x^{4}+6 \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 x^{2}+1}\right )}{2}-\ln \left (-\frac {-x^{4}+12 \sqrt {x^{4}+1}\, \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +6 \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x +2 x^{2}-1}{x^{4}+6 \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 x^{2}+1}\right ) \RootOf \left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1/4*2^(1/2)*ln(1/x^2*(x^4+1)+(x^4+1)^(1/2)/x+1)+1/6*6^(1/2)*arctan(1/6*(2*2^(1/2)/x*(x^4+1)^(1/2)+2^(1/2
))*6^(1/2))+1/4*2^(1/2)*ln(1/x^2*(x^4+1)-(x^4+1)^(1/2)/x+1)+1/6*6^(1/2)*arctan(1/6*(2*2^(1/2)/x*(x^4+1)^(1/2)-
2^(1/2))*6^(1/2)))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + 1)*(x^4 - 1)/(x^8 + x^6 + 3*x^4 + 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^4-1\right )\,\sqrt {x^4+1}}{x^8+x^6+3\,x^4+x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - 1)*(x^4 + 1)^(1/2))/(x^2 + 3*x^4 + x^6 + x^8 + 1),x)

[Out]

int(((x^4 - 1)*(x^4 + 1)^(1/2))/(x^2 + 3*x^4 + x^6 + x^8 + 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)*(x**4+1)**(1/2)/(x**8+x**6+3*x**4+x**2+1),x)

[Out]

Timed out

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