3.4.6 \(\int \frac {1}{\sqrt [4]{2+b x^2} (4+b x^2)} \, dx\) [306]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 {b x^2+2}+2\ 2^{3/4}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {b x^2+2}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.22, size = 119, normalized size = 0.92 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2^{3/4} b x^2-4 \sqrt [4]{2} \sqrt {2+b x^2}}{4 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )+\tanh ^{-1}\left (\frac {2\ 2^{3/4} \sqrt {b} x \sqrt [4]{2+b x^2}}{\sqrt {2} b x^2+4 \sqrt {2+b x^2}}\right )}{4\ 2^{3/4} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{2}+2\right )^{\frac {1}{4}} \left (b \,x^{2}+4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((b*x^2 + 4)*(b*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. 755 vs. \(2 (86) = 172\).
time = 7.25, size = 755, normalized size = 5.85 \begin {gather*} \frac {1}{4} \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} \frac {1}{b^{2}}^{\frac {1}{4}} \arctan \left (-\frac {2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{2}}^{\frac {1}{4}} x^{3} + b^{2} x^{4} + 8 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {3}{4}} b^{2} \frac {1}{b^{2}}^{\frac {3}{4}} x + 4 \, b x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (b^{2} x^{2} + 4 \, b\right )} \sqrt {b x^{2} + 2} \sqrt {\frac {1}{b^{2}}} - 2 \, \sqrt {\frac {1}{2}} {\left (4 \, {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b x^{2} + 2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b^{3} x^{3} + 4 \, b^{2} x\right )} \frac {1}{b^{2}}^{\frac {3}{4}} + 16 \, \sqrt {\frac {1}{2}} {\left (b x^{2} + 2\right )}^{\frac {3}{4}} b \sqrt {\frac {1}{b^{2}}} + \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (b^{2} x^{3} - 4 \, b x\right )} \sqrt {b x^{2} + 2} \frac {1}{b^{2}}^{\frac {1}{4}}\right )} \sqrt {\frac {2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{2}}^{\frac {3}{4}} x + \sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{b^{2}}} x^{2} + 2 \, \sqrt {b x^{2} + 2}}{b x^{2} + 4}}}{b^{2} x^{4} - 8 \, b x^{2} - 16}\right ) - \frac {1}{4} \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} \frac {1}{b^{2}}^{\frac {1}{4}} \arctan \left (\frac {2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{2}}^{\frac {1}{4}} x^{3} - b^{2} x^{4} + 8 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {3}{4}} b^{2} \frac {1}{b^{2}}^{\frac {3}{4}} x - 4 \, b x^{2} - 4 \, \sqrt {\frac {1}{2}} {\left (b^{2} x^{2} + 4 \, b\right )} \sqrt {b x^{2} + 2} \sqrt {\frac {1}{b^{2}}} + 2 \, \sqrt {\frac {1}{2}} {\left (4 \, {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b x^{2} - 2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b^{3} x^{3} + 4 \, b^{2} x\right )} \frac {1}{b^{2}}^{\frac {3}{4}} + 16 \, \sqrt {\frac {1}{2}} {\left (b x^{2} + 2\right )}^{\frac {3}{4}} b \sqrt {\frac {1}{b^{2}}} - \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (b^{2} x^{3} - 4 \, b x\right )} \sqrt {b x^{2} + 2} \frac {1}{b^{2}}^{\frac {1}{4}}\right )} \sqrt {-\frac {2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{2}}^{\frac {3}{4}} x - \sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{b^{2}}} x^{2} - 2 \, \sqrt {b x^{2} + 2}}{b x^{2} + 4}}}{b^{2} x^{4} - 8 \, b x^{2} - 16}\right ) + \frac {1}{16} \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} \frac {1}{b^{2}}^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{2}}^{\frac {3}{4}} x + \sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{b^{2}}} x^{2} + 2 \, \sqrt {b x^{2} + 2}}{2 \, {\left (b x^{2} + 4\right )}}\right ) - \frac {1}{16} \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {1}{4}} \frac {1}{b^{2}}^{\frac {1}{4}} \log \left (-\frac {2 \, \sqrt {2} \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{2}}^{\frac {3}{4}} x - \sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{b^{2}}} x^{2} - 2 \, \sqrt {b x^{2} + 2}}{2 \, {\left (b x^{2} + 4\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*(1/2)^(1/4)*(b^(-2))^(1/4)*arctan(-(2*sqrt(2)*(1/2)^(1/4)*(b*x^2 + 2)^(1/4)*b^2*(b^(-2))^(1/4)*x^3
+ b^2*x^4 + 8*sqrt(2)*(1/2)^(3/4)*(b*x^2 + 2)^(3/4)*b^2*(b^(-2))^(3/4)*x + 4*b*x^2 + 4*sqrt(1/2)*(b^2*x^2 + 4
*b)*sqrt(b*x^2 + 2)*sqrt(b^(-2)) - 2*sqrt(1/2)*(4*(b*x^2 + 2)^(1/4)*b*x^2 + 2*sqrt(2)*(1/2)^(3/4)*(b^3*x^3 + 4
*b^2*x)*(b^(-2))^(3/4) + 16*sqrt(1/2)*(b*x^2 + 2)^(3/4)*b*sqrt(b^(-2)) + sqrt(2)*(1/2)^(1/4)*(b^2*x^3 - 4*b*x)
*sqrt(b*x^2 + 2)*(b^(-2))^(1/4))*sqrt((2*sqrt(2)*(1/2)^(3/4)*(b*x^2 + 2)^(1/4)*b^2*(b^(-2))^(3/4)*x + sqrt(1/2
)*b^2*sqrt(b^(-2))*x^2 + 2*sqrt(b*x^2 + 2))/(b*x^2 + 4)))/(b^2*x^4 - 8*b*x^2 - 16)) - 1/4*sqrt(2)*(1/2)^(1/4)*
(b^(-2))^(1/4)*arctan((2*sqrt(2)*(1/2)^(1/4)*(b*x^2 + 2)^(1/4)*b^2*(b^(-2))^(1/4)*x^3 - b^2*x^4 + 8*sqrt(2)*(1
/2)^(3/4)*(b*x^2 + 2)^(3/4)*b^2*(b^(-2))^(3/4)*x - 4*b*x^2 - 4*sqrt(1/2)*(b^2*x^2 + 4*b)*sqrt(b*x^2 + 2)*sqrt(
b^(-2)) + 2*sqrt(1/2)*(4*(b*x^2 + 2)^(1/4)*b*x^2 - 2*sqrt(2)*(1/2)^(3/4)*(b^3*x^3 + 4*b^2*x)*(b^(-2))^(3/4) +
16*sqrt(1/2)*(b*x^2 + 2)^(3/4)*b*sqrt(b^(-2)) - sqrt(2)*(1/2)^(1/4)*(b^2*x^3 - 4*b*x)*sqrt(b*x^2 + 2)*(b^(-2))
^(1/4))*sqrt(-(2*sqrt(2)*(1/2)^(3/4)*(b*x^2 + 2)^(1/4)*b^2*(b^(-2))^(3/4)*x - sqrt(1/2)*b^2*sqrt(b^(-2))*x^2 -
2*sqrt(b*x^2 + 2))/(b*x^2 + 4)))/(b^2*x^4 - 8*b*x^2 - 16)) + 1/16*sqrt(2)*(1/2)^(1/4)*(b^(-2))^(1/4)*log(1/2*
(2*sqrt(2)*(1/2)^(3/4)*(b*x^2 + 2)^(1/4)*b^2*(b^(-2))^(3/4)*x + sqrt(1/2)*b^2*sqrt(b^(-2))*x^2 + 2*sqrt(b*x^2
+ 2))/(b*x^2 + 4)) - 1/16*sqrt(2)*(1/2)^(1/4)*(b^(-2))^(1/4)*log(-1/2*(2*sqrt(2)*(1/2)^(3/4)*(b*x^2 + 2)^(1/4)
*b^2*(b^(-2))^(3/4)*x - sqrt(1/2)*b^2*sqrt(b^(-2))*x^2 - 2*sqrt(b*x^2 + 2))/(b*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]{b x^{2} + 2} \left (b x^{2} + 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((b*x^2 + 4)*(b*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 (b\,x^2+2\right )}^{1/4}\,\left (b\,x^2+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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