Integrand size = 23, antiderivative size = 70 \[ \int \frac {1+3 x+3 x^4}{x \sqrt [4]{1+x^4}} \, dx=\left (1+x^4\right )^{3/4}+\frac {3}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \arctan \left (\sqrt [4]{1+x^4}\right )+\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {arctanh}\left (\sqrt [4]{1+x^4}\right ) \]
(x^4+1)^(3/4)+3/2*arctan(x/(x^4+1)^(1/4))+1/2*arctan((x^4+1)^(1/4))+3/2*ar ctanh(x/(x^4+1)^(1/4))-1/2*arctanh((x^4+1)^(1/4))
Time = 5.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {1+3 x+3 x^4}{x \sqrt [4]{1+x^4}} \, dx=\frac {1}{2} \left (2 \left (1+x^4\right )^{3/4}+3 \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\arctan \left (\sqrt [4]{1+x^4}\right )+3 \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\text {arctanh}\left (\sqrt [4]{1+x^4}\right )\right ) \]
(2*(1 + x^4)^(3/4) + 3*ArcTan[x/(1 + x^4)^(1/4)] + ArcTan[(1 + x^4)^(1/4)] + 3*ArcTanh[x/(1 + x^4)^(1/4)] - ArcTanh[(1 + x^4)^(1/4)])/2
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2372, 2009}
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.
\(\displaystyle \int \frac {3 x^4+3 x+1}{x \sqrt [4]{x^4+1}} \, dx\) |
\(\Big \downarrow \) 2372 |
\(\displaystyle \int \left (\frac {3 x^4+1}{x \sqrt [4]{x^4+1}}+\frac {3}{\sqrt [4]{x^4+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \arctan \left (\sqrt [4]{x^4+1}\right )+\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \text {arctanh}\left (\sqrt [4]{x^4+1}\right )+\left (x^4+1\right )^{3/4}\) |
(1 + x^4)^(3/4) + (3*ArcTan[x/(1 + x^4)^(1/4)])/2 + ArcTan[(1 + x^4)^(1/4) ]/2 + (3*ArcTanh[x/(1 + x^4)^(1/4)])/2 - ArcTanh[(1 + x^4)^(1/4)]/2
3.10.18.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 9.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29
method | result | size |
meijerg | \(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi }+\frac {3 x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{4}, 1\right ], \left [2\right ], -x^{4}\right )}{4}+3 x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )\) | \(90\) |
trager | \(\left (x^{4}+1\right )^{\frac {3}{4}}+\frac {\ln \left (-\frac {1-5 x^{2} \left (x^{4}+1\right )^{\frac {1}{4}}+5 \sqrt {x^{4}+1}\, x^{2}-3 x +9 x^{4}-8 x^{7}-4 x^{5}+4 x^{2}-7 x^{3}-\left (x^{4}+1\right )^{\frac {3}{4}}+\sqrt {x^{4}+1}-\left (x^{4}+1\right )^{\frac {1}{4}}+8 x^{8}+4 x^{6}-3 \sqrt {x^{4}+1}\, x +8 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}+4 x^{4} \sqrt {x^{4}+1}-8 x^{6} \left (x^{4}+1\right )^{\frac {1}{4}}-4 x^{4} \left (x^{4}+1\right )^{\frac {1}{4}}+3 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 x \left (x^{4}+1\right )^{\frac {1}{4}}+8 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}-8 x^{4} \left (x^{4}+1\right )^{\frac {3}{4}}+4 x^{3} \left (x^{4}+1\right )^{\frac {3}{4}}+4 \left (x^{4}+1\right )^{\frac {1}{4}} x^{5}-4 \left (x^{4}+1\right )^{\frac {3}{4}} x^{2}+8 \sqrt {x^{4}+1}\, x^{6}-8 \sqrt {x^{4}+1}\, x^{5}-4 \sqrt {x^{4}+1}\, x^{3}+7 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}{x^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-5 x^{2} \left (x^{4}+1\right )^{\frac {1}{4}}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{8}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\left (x^{4}+1\right )^{\frac {3}{4}}-\left (x^{4}+1\right )^{\frac {1}{4}}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{7}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+8 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}-8 x^{6} \left (x^{4}+1\right )^{\frac {1}{4}}-4 x^{4} \left (x^{4}+1\right )^{\frac {1}{4}}-3 \left (x^{4}+1\right )^{\frac {3}{4}} x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{6}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{5}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x +3 x \left (x^{4}+1\right )^{\frac {1}{4}}-8 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+8 x^{4} \left (x^{4}+1\right )^{\frac {3}{4}}-4 x^{3} \left (x^{4}+1\right )^{\frac {3}{4}}+4 \left (x^{4}+1\right )^{\frac {1}{4}} x^{5}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}+7 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}{x^{2}}\right )}{2}\) | \(676\) |
1/8/Pi*2^(1/2)*GAMMA(3/4)*(-1/4*Pi*2^(1/2)/GAMMA(3/4)*x^4*hypergeom([1,1,5 /4],[2,2],-x^4)+(-3*ln(2)-1/2*Pi+4*ln(x))*Pi*2^(1/2)/GAMMA(3/4))+3/4*x^4*h ypergeom([1/4,1],[2],-x^4)+3*x*hypergeom([1/4,1/4],[5/4],-x^4)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (52) = 104\).
Time = 6.69 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.01 \[ \int \frac {1+3 x+3 x^4}{x \sqrt [4]{1+x^4}} \, dx={\left (x^{4} + 1\right )}^{\frac {3}{4}} + \frac {3}{4} \, \arctan \left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x\right ) - \frac {1}{4} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} + 1\right )}^{\frac {3}{4}} + {\left (x^{4} + 1\right )}^{\frac {1}{4}}\right )}}{x^{4}}\right ) + \frac {3}{4} \, \log \left (2 \, x^{4} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1\right ) + \frac {1}{4} \, \log \left (-\frac {x^{4} - 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {x^{4} + 1} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 2}{x^{4}}\right ) \]
(x^4 + 1)^(3/4) + 3/4*arctan(2*(x^4 + 1)^(1/4)*x^3 + 2*(x^4 + 1)^(3/4)*x) - 1/4*arctan(2*((x^4 + 1)^(3/4) + (x^4 + 1)^(1/4))/x^4) + 3/4*log(2*x^4 + 2*(x^4 + 1)^(1/4)*x^3 + 2*sqrt(x^4 + 1)*x^2 + 2*(x^4 + 1)^(3/4)*x + 1) + 1 /4*log(-(x^4 - 2*(x^4 + 1)^(3/4) + 2*sqrt(x^4 + 1) - 2*(x^4 + 1)^(1/4) + 2 )/x^4)
Result contains complex when optimal does not.
Time = 1.64 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {1+3 x+3 x^4}{x \sqrt [4]{1+x^4}} \, dx=\frac {3 x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \left (x^{4} + 1\right )^{\frac {3}{4}} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \]
3*x*gamma(1/4)*hyper((1/4, 1/4), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/ 4)) + (x**4 + 1)**(3/4) - gamma(1/4)*hyper((1/4, 1/4), (5/4,), exp_polar(I *pi)/x**4)/(4*x*gamma(5/4))
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.26 \[ \int \frac {1+3 x+3 x^4}{x \sqrt [4]{1+x^4}} \, dx={\left (x^{4} + 1\right )}^{\frac {3}{4}} + \frac {1}{2} \, \arctan \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{2} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{4} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{4}} - 1\right ) + \frac {3}{4} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {3}{4} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \]
(x^4 + 1)^(3/4) + 1/2*arctan((x^4 + 1)^(1/4)) - 3/2*arctan((x^4 + 1)^(1/4) /x) - 1/4*log((x^4 + 1)^(1/4) + 1) + 1/4*log((x^4 + 1)^(1/4) - 1) + 3/4*lo g((x^4 + 1)^(1/4)/x + 1) - 3/4*log((x^4 + 1)^(1/4)/x - 1)
\[ \int \frac {1+3 x+3 x^4}{x \sqrt [4]{1+x^4}} \, dx=\int { \frac {3 \, x^{4} + 3 \, x + 1}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} x} \,d x } \]
Time = 5.59 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59 \[ \int \frac {1+3 x+3 x^4}{x \sqrt [4]{1+x^4}} \, dx=\frac {\mathrm {atan}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}-\frac {\mathrm {atanh}\left ({\left (x^4+1\right )}^{1/4}\right )}{2}+3\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -x^4\right )+{\left (x^4+1\right )}^{3/4} \]