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

Optimal result

Integrand size = 21, antiderivative size = 129 \[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{2\ 2^{3/4} \sqrt {b}} \]

[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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {406} \[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {2 \sqrt [4]{2} \sqrt {b x^2+2}+2\ 2^{3/4}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {b x^2+2}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{2\ 2^{3/4} \sqrt {b}} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=\frac {\arctan \left (\frac {2^{3/4} b x^2-4 \sqrt [4]{2} \sqrt {2+b x^2}}{4 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/4} \sqrt {b} x \sqrt [4]{2+b x^2}}{\sqrt {2} b x^2+4 \sqrt {2+b x^2}}\right )}{4\ 2^{3/4} \sqrt {b}} \]

[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])

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}+2\right )^{\frac {1}{4}} \left (b \,x^{2}+4\right )}d x\]

[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)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.28 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.93 \[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=-\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} + 2 \, \sqrt {\frac {1}{2}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} + {\left (b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} + 4}\right ) + \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} - 2 \, \sqrt {\frac {1}{2}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} - {\left (b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} + 4}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} + i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} - 2 \, \sqrt {\frac {1}{2}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} + {\left (b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} + 4}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {b x^{2} + 2} b^{2} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} - i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x \left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}} - 2 \, \sqrt {\frac {1}{2}} {\left (b x^{2} + 2\right )}^{\frac {1}{4}} b \sqrt {-\frac {1}{b^{2}}} + {\left (b x^{2} + 2\right )}^{\frac {3}{4}}}{b x^{2} + 4}\right ) \]

[In]

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

[Out]

-1/8*(1/2)^(1/4)*(-1/b^2)^(1/4)*log(((1/2)^(3/4)*sqrt(b*x^2 + 2)*b^2*x*(-1/b^2)^(3/4) - (1/2)^(1/4)*b*x*(-1/b^
2)^(1/4) + 2*sqrt(1/2)*(b*x^2 + 2)^(1/4)*b*sqrt(-1/b^2) + (b*x^2 + 2)^(3/4))/(b*x^2 + 4)) + 1/8*(1/2)^(1/4)*(-
1/b^2)^(1/4)*log(-((1/2)^(3/4)*sqrt(b*x^2 + 2)*b^2*x*(-1/b^2)^(3/4) - (1/2)^(1/4)*b*x*(-1/b^2)^(1/4) - 2*sqrt(
1/2)*(b*x^2 + 2)^(1/4)*b*sqrt(-1/b^2) - (b*x^2 + 2)^(3/4))/(b*x^2 + 4)) + 1/8*I*(1/2)^(1/4)*(-1/b^2)^(1/4)*log
((I*(1/2)^(3/4)*sqrt(b*x^2 + 2)*b^2*x*(-1/b^2)^(3/4) + I*(1/2)^(1/4)*b*x*(-1/b^2)^(1/4) - 2*sqrt(1/2)*(b*x^2 +
2)^(1/4)*b*sqrt(-1/b^2) + (b*x^2 + 2)^(3/4))/(b*x^2 + 4)) - 1/8*I*(1/2)^(1/4)*(-1/b^2)^(1/4)*log((-I*(1/2)^(3
/4)*sqrt(b*x^2 + 2)*b^2*x*(-1/b^2)^(3/4) - I*(1/2)^(1/4)*b*x*(-1/b^2)^(1/4) - 2*sqrt(1/2)*(b*x^2 + 2)^(1/4)*b*
sqrt(-1/b^2) + (b*x^2 + 2)^(3/4))/(b*x^2 + 4))

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=\int \frac {1}{\sqrt [4]{b x^{2} + 2} \left (b x^{2} + 4\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + 4\right )} {\left (b x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + 4\right )} {\left (b x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{2+b x^2} \left (4+b x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+2\right )}^{1/4}\,\left (b\,x^2+4\right )} \,d x \]

[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)