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

Optimal. Leaf size=21 \[ -\tanh ^{-1}\left (\frac {x}{\sqrt {2 x^4+3 x^2+1}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2112, 206} \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {2 x^4+3 x^2+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[x/Sqrt[1 + 3*x^2 + 2*x^4]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2112

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.33, size = 107, normalized size = 5.10 \begin {gather*} -\frac {i \sqrt {x^2+1} \sqrt {2 x^2+1} \left (F\left (i \sinh ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )-\Pi \left (\frac {1}{2}-\frac {i}{2};i \sinh ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )-\Pi \left (\frac {1}{2}+\frac {i}{2};i \sinh ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )\right )}{\sqrt {4 x^4+6 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 + x^2]*Sqrt[1 + 2*x^2]*(EllipticF[I*ArcSinh[Sqrt[2]*x], 1/2] - EllipticPi[1/2 - I/2, I*ArcSinh[Sq
rt[2]*x], 1/2] - EllipticPi[1/2 + I/2, I*ArcSinh[Sqrt[2]*x], 1/2]))/Sqrt[2 + 6*x^2 + 4*x^4]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.59, size = 21, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[x/Sqrt[1 + 3*x^2 + 2*x^4]]

________________________________________________________________________________________

fricas [B]  time = 0.48, size = 47, normalized size = 2.24 \begin {gather*} \frac {1}{2} \, \log \left (\frac {2 \, x^{4} + 4 \, x^{2} - 2 \, \sqrt {2 \, x^{4} + 3 \, x^{2} + 1} x + 1}{2 \, x^{4} + 2 \, x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log((2*x^4 + 4*x^2 - 2*sqrt(2*x^4 + 3*x^2 + 1)*x + 1)/(2*x^4 + 2*x^2 + 1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^4 - 1)/(sqrt(2*x^4 + 3*x^2 + 1)*(2*x^4 + 2*x^2 + 1)), x)

________________________________________________________________________________________

maple [A]  time = 0.23, size = 22, normalized size = 1.05

method	result	size
elliptic	\(-\arctanh \left (\frac {\sqrt {2 x^{4}+3 x^{2}+1}}{x}\right )\)	\(22\)
trager	\(\frac {\ln \left (-\frac {-2 x^{4}+2 \sqrt {2 x^{4}+3 x^{2}+1}\, x -4 x^{2}-1}{2 x^{4}+2 x^{2}+1}\right )}{2}\)	\(49\)
default	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (i x , \sqrt {2}\right )}{\sqrt {2 x^{4}+3 x^{2}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {2 x^{4}+3 x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {4 i \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (i x , 2 \underline {\hspace {1.25 ex}}\alpha ^{2}+2, i \sqrt {-2}\right )}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )\right )}{4}\)	\(170\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh((2*x^4+3*x^2+1)^(1/2)/x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^4 - 1)/(sqrt(2*x^4 + 3*x^2 + 1)*(2*x^4 + 2*x^2 + 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {2\,x^4-1}{\left (2\,x^4+2\,x^2+1\right )\,\sqrt {2\,x^4+3\,x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________