Integrand size = 21, antiderivative size = 124 \[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2+3 x^2}}{2+\sqrt {2} \sqrt {2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}} \] Output:
1/12*arctan(2^(1/4)*3^(1/2)*x*(3*x^2+2)^(1/4)/(2+2^(1/2)*(3*x^2+2)^(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))*3^ (1/2)/x/(3*x^2+2)^(1/4))*2^(1/4)*3^(1/2)
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=\frac {\arctan \left (\frac {3 \sqrt {2} x^2-4 \sqrt {2+3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )+\text {arctanh}\left (\frac {2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2+3 x^2}}{3 \sqrt {2} x^2+4 \sqrt {2+3 x^2}}\right )}{4\ 2^{3/4} \sqrt {3}} \] Input:
Integrate[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]
Output:
(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])
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {308}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt [4]{3 x^2+2} \left (3 x^2+4\right )} \, dx\) |
\(\Big \downarrow \) 308 |
\(\displaystyle -\frac {\arctan \left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}}\) |
Input:
Int[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]
Output:
-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]) - ArcTanh[(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])
Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{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*Sqrt[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.51 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.50
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right ) \ln \left (\frac {\left (3 x^{2}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{3}+6 \left (3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2} x +18 \sqrt {3 x^{2}+2}\, x}{3 x^{2}+4}\right )}{24}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}\right ) \ln \left (-\frac {\left (3 x^{2}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}-6 \left (3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2}\right )+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+72\right )^{2} x -18 \sqrt {3 x^{2}+2}\, x}{3 x^{2}+4}\right )}{24}\) | \(186\) |
Input:
int(1/(3*x^2+2)^(1/4)/(3*x^2+4),x,method=_RETURNVERBOSE)
Output:
1/24*RootOf(_Z^4+72)*ln(((3*x^2+2)^(1/4)*RootOf(_Z^4+72)^3+6*(3*x^2+2)^(3/ 4)*RootOf(_Z^4+72)+3*RootOf(_Z^4+72)^2*x+18*(3*x^2+2)^(1/2)*x)/(3*x^2+4))+ 1/24*RootOf(_Z^2+RootOf(_Z^4+72)^2)*ln(-((3*x^2+2)^(1/4)*RootOf(_Z^2+RootO f(_Z^4+72)^2)*RootOf(_Z^4+72)^2-6*(3*x^2+2)^(3/4)*RootOf(_Z^2+RootOf(_Z^4+ 72)^2)+3*RootOf(_Z^4+72)^2*x-18*(3*x^2+2)^(1/2)*x)/(3*x^2+4))
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (88) = 176\).
Time = 1.77 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=-\frac {1}{288} \cdot 72^{\frac {3}{4}} \arctan \left (\frac {72^{\frac {3}{4}} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} + 18 \, x^{4} + 4 \, \sqrt {2} {\left (3 \, x^{2} + 4\right )} \sqrt {3 \, x^{2} + 2} + 8 \cdot 72^{\frac {1}{4}} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x + 24 \, x^{2}}{2 \, {\left (9 \, x^{4} - 24 \, x^{2} - 16\right )}}\right ) - \frac {1}{288} \cdot 72^{\frac {3}{4}} \arctan \left (\frac {72^{\frac {3}{4}} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} - 18 \, x^{4} - 4 \, \sqrt {2} {\left (3 \, x^{2} + 4\right )} \sqrt {3 \, x^{2} + 2} + 8 \cdot 72^{\frac {1}{4}} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x - 24 \, x^{2}}{2 \, {\left (9 \, x^{4} - 24 \, x^{2} - 16\right )}}\right ) + \frac {1}{576} \cdot 72^{\frac {3}{4}} \log \left (\frac {2 \, {\left (3 \, \sqrt {2} x^{2} + 2 \cdot 72^{\frac {1}{4}} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}\right )}}{3 \, x^{2} + 4}\right ) - \frac {1}{576} \cdot 72^{\frac {3}{4}} \log \left (\frac {2 \, {\left (3 \, \sqrt {2} x^{2} - 2 \cdot 72^{\frac {1}{4}} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}\right )}}{3 \, x^{2} + 4}\right ) \] Input:
integrate(1/(3*x^2+2)^(1/4)/(3*x^2+4),x, algorithm="fricas")
Output:
-1/288*72^(3/4)*arctan(1/2*(72^(3/4)*(3*x^2 + 2)^(1/4)*x^3 + 18*x^4 + 4*sq rt(2)*(3*x^2 + 4)*sqrt(3*x^2 + 2) + 8*72^(1/4)*(3*x^2 + 2)^(3/4)*x + 24*x^ 2)/(9*x^4 - 24*x^2 - 16)) - 1/288*72^(3/4)*arctan(1/2*(72^(3/4)*(3*x^2 + 2 )^(1/4)*x^3 - 18*x^4 - 4*sqrt(2)*(3*x^2 + 4)*sqrt(3*x^2 + 2) + 8*72^(1/4)* (3*x^2 + 2)^(3/4)*x - 24*x^2)/(9*x^4 - 24*x^2 - 16)) + 1/576*72^(3/4)*log( 2*(3*sqrt(2)*x^2 + 2*72^(1/4)*(3*x^2 + 2)^(1/4)*x + 4*sqrt(3*x^2 + 2))/(3* x^2 + 4)) - 1/576*72^(3/4)*log(2*(3*sqrt(2)*x^2 - 2*72^(1/4)*(3*x^2 + 2)^( 1/4)*x + 4*sqrt(3*x^2 + 2))/(3*x^2 + 4))
\[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=\int \frac {1}{\sqrt [4]{3 x^{2} + 2} \cdot \left (3 x^{2} + 4\right )}\, dx \] Input:
integrate(1/(3*x**2+2)**(1/4)/(3*x**2+4),x)
Output:
Integral(1/((3*x**2 + 2)**(1/4)*(3*x**2 + 4)), x)
\[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 4\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(3*x^2+2)^(1/4)/(3*x^2+4),x, algorithm="maxima")
Output:
integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 4\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(3*x^2+2)^(1/4)/(3*x^2+4),x, algorithm="giac")
Output:
integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=\int \frac {1}{{\left (3\,x^2+2\right )}^{1/4}\,\left (3\,x^2+4\right )} \,d x \] Input:
int(1/((3*x^2 + 2)^(1/4)*(3*x^2 + 4)),x)
Output:
int(1/((3*x^2 + 2)^(1/4)*(3*x^2 + 4)), x)
\[ \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx=\int \frac {1}{3 \left (3 x^{2}+2\right )^{\frac {1}{4}} x^{2}+4 \left (3 x^{2}+2\right )^{\frac {1}{4}}}d x \] Input:
int(1/(3*x^2+2)^(1/4)/(3*x^2+4),x)
Output:
int(1/(3*(3*x**2 + 2)**(1/4)*x**2 + 4*(3*x**2 + 2)**(1/4)),x)