3.4.4 \(\int \frac {1}{\sqrt [4]{2+3 x^2} (4+3 x^2)} \, dx\) [304]

Optimal. Leaf size=129 \[ -\frac {\tan ^{-1}\left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \]

[Out]

-1/12*arctan(1/6*(2*2^(3/4)+2*2^(1/4)*(3*x^2+2)^(1/2))/x/(3*x^2+2)^(1/4)*3^(1/2))*2^(1/4)*3^(1/2)-1/12*arctanh
(1/6*(2*2^(3/4)-2*2^(1/4)*(3*x^2+2)^(1/2))/x/(3*x^2+2)^(1/4)*3^(1/2))*2^(1/4)*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {406} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 406

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b^2/a, 4]}, Simp[(-b/(2*a
*d*q))*ArcTan[(b + q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x] - Simp[(b/(2*a*d*q))*ArcTanh[(b - q^2*S
qrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a
]

Rubi steps

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

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Mathematica [A]
time = 0.26, size = 119, normalized size = 0.92 \begin {gather*} \frac {\tan ^{-1}\left (\frac {3 \sqrt {2} x^2-4 \sqrt {2+3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )+\tanh ^{-1}\left (\frac {2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2+3 x^2}}{3 \sqrt {2} x^2+4 \sqrt {2+3 x^2}}\right )}{4\ 2^{3/4} \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.55, size = 186, normalized size = 1.44

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right ) \ln \left (\frac {6 \left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right )-\left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right )+18 \sqrt {3 x^{2}+2}\, x -3 \RootOf \left (\textit {\_Z}^{4}+72\right )^{2} x}{3 x^{2}+4}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{4}+72\right ) \ln \left (\frac {6 \left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )+\left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )^{3}+18 \sqrt {3 x^{2}+2}\, x +3 \RootOf \left (\textit {\_Z}^{4}+72\right )^{2} x}{3 x^{2}+4}\right )}{24}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*RootOf(_Z^2+RootOf(_Z^4+72)^2)*ln((6*(3*x^2+2)^(3/4)*RootOf(_Z^2+RootOf(_Z^4+72)^2)-(3*x^2+2)^(1/4)*RootO
f(_Z^4+72)^2*RootOf(_Z^2+RootOf(_Z^4+72)^2)+18*(3*x^2+2)^(1/2)*x-3*RootOf(_Z^4+72)^2*x)/(3*x^2+4))+1/24*RootOf
(_Z^4+72)*ln((6*(3*x^2+2)^(3/4)*RootOf(_Z^4+72)+(3*x^2+2)^(1/4)*RootOf(_Z^4+72)^3+18*(3*x^2+2)^(1/2)*x+3*RootO
f(_Z^4+72)^2*x)/(3*x^2+4))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (88) = 176\).
time = 2.55, size = 553, normalized size = 4.29 \begin {gather*} \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} + 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x + 12 \, \sqrt {2} {\left (3 \, x^{2} + 4\right )} \sqrt {3 \, x^{2} + 2} + 72 \, x^{2} - {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} \sqrt {3 \, x^{2} + 2} + 72 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} + 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} + 48 \, \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}}{3 \, x^{2} + 4}}}{6 \, {\left (9 \, x^{4} - 24 \, x^{2} - 16\right )}}\right ) - \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} - 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x - 12 \, \sqrt {2} {\left (3 \, x^{2} + 4\right )} \sqrt {3 \, x^{2} + 2} - 72 \, x^{2} - {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} \sqrt {3 \, x^{2} + 2} - 72 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} + 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} - 48 \, \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}}{3 \, x^{2} + 4}}}{6 \, {\left (9 \, x^{4} - 24 \, x^{2} - 16\right )}}\right ) + \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {36 \, {\left (3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}\right )}}{3 \, x^{2} + 4}\right ) - \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {36 \, {\left (3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}\right )}}{3 \, x^{2} + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*18^(3/4)*sqrt(2)*arctan(-1/6*(6*18^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x^3 + 54*x^4 + 24*18^(1/4)*sqrt(2)*(3*
x^2 + 2)^(3/4)*x + 12*sqrt(2)*(3*x^2 + 4)*sqrt(3*x^2 + 2) + 72*x^2 - (18^(3/4)*sqrt(2)*(3*x^3 - 4*x)*sqrt(3*x^
2 + 2) + 72*(3*x^2 + 2)^(1/4)*x^2 + 6*18^(1/4)*sqrt(2)*(3*x^3 + 4*x) + 48*sqrt(2)*(3*x^2 + 2)^(3/4))*sqrt((3*s
qrt(2)*x^2 + 2*18^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 4*sqrt(3*x^2 + 2))/(3*x^2 + 4)))/(9*x^4 - 24*x^2 - 16))
- 1/72*18^(3/4)*sqrt(2)*arctan(1/6*(6*18^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x^3 - 54*x^4 + 24*18^(1/4)*sqrt(2)*(3
*x^2 + 2)^(3/4)*x - 12*sqrt(2)*(3*x^2 + 4)*sqrt(3*x^2 + 2) - 72*x^2 - (18^(3/4)*sqrt(2)*(3*x^3 - 4*x)*sqrt(3*x
^2 + 2) - 72*(3*x^2 + 2)^(1/4)*x^2 + 6*18^(1/4)*sqrt(2)*(3*x^3 + 4*x) - 48*sqrt(2)*(3*x^2 + 2)^(3/4))*sqrt((3*
sqrt(2)*x^2 - 2*18^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 4*sqrt(3*x^2 + 2))/(3*x^2 + 4)))/(9*x^4 - 24*x^2 - 16))
+ 1/288*18^(3/4)*sqrt(2)*log(36*(3*sqrt(2)*x^2 + 2*18^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 4*sqrt(3*x^2 + 2))/
(3*x^2 + 4)) - 1/288*18^(3/4)*sqrt(2)*log(36*(3*sqrt(2)*x^2 - 2*18^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 4*sqrt(
3*x^2 + 2))/(3*x^2 + 4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (3\,x^2+2\right )}^{1/4}\,\left (3\,x^2+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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