Integrand size = 13, antiderivative size = 130 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=12 \sqrt [12]{x}-6 \sqrt [6]{x}+4 \sqrt [4]{x}-3 \sqrt [3]{x}+\frac {12 x^{5/12}}{5}-2 \sqrt {x}+\frac {12 x^{7/12}}{7}-\frac {3 x^{2/3}}{2}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{11/12}}{11}-x+\frac {12 x^{13/12}}{13}-\frac {6 x^{7/6}}{7}+\frac {4 x^{5/4}}{5}-12 \log \left (1+\sqrt [12]{x}\right ) \] Output:
12*x^(1/12)-6*x^(1/6)+4*x^(1/4)-3*x^(1/3)+12/5*x^(5/12)-2*x^(1/2)+12/7*x^( 7/12)-3/2*x^(2/3)+4/3*x^(3/4)-6/5*x^(5/6)+12/11*x^(11/12)-x+12/13*x^(13/12 )-6/7*x^(7/6)+4/5*x^(5/4)-12*ln(1+x^(1/12))
Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=\frac {360360 \sqrt [12]{x}-180180 \sqrt [6]{x}+120120 \sqrt [4]{x}-90090 \sqrt [3]{x}+72072 x^{5/12}-60060 \sqrt {x}+51480 x^{7/12}-45045 x^{2/3}+40040 x^{3/4}-36036 x^{5/6}+32760 x^{11/12}-30030 x+27720 x^{13/12}-25740 x^{7/6}+24024 x^{5/4}}{30030}-12 \log \left (1+\sqrt [12]{x}\right ) \] Input:
Integrate[(x^(-1/3) + x^(-1/4))^(-1),x]
Output:
(360360*x^(1/12) - 180180*x^(1/6) + 120120*x^(1/4) - 90090*x^(1/3) + 72072 *x^(5/12) - 60060*Sqrt[x] + 51480*x^(7/12) - 45045*x^(2/3) + 40040*x^(3/4) - 36036*x^(5/6) + 32760*x^(11/12) - 30030*x + 27720*x^(13/12) - 25740*x^( 7/6) + 24024*x^(5/4))/30030 - 12*Log[1 + x^(1/12)]
Time = 0.39 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2027, 798, 49, 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.
\(\displaystyle \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\frac {1}{\sqrt [3]{x}}} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {\sqrt [3]{x}}{\sqrt [12]{x}+1}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 12 \int \frac {x^{5/4}}{\sqrt [12]{x}+1}d\sqrt [12]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 12 \int \left (x^{7/6}-x^{13/12}+x-x^{11/12}+x^{5/6}-x^{3/4}+x^{2/3}-x^{7/12}+\sqrt {x}-x^{5/12}+\sqrt [3]{x}-\sqrt [4]{x}+\sqrt [6]{x}-\sqrt [12]{x}+\frac {1}{-\sqrt [12]{x}-1}+1\right )d\sqrt [12]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 12 \left (\frac {x^{5/4}}{15}-\frac {x^{7/6}}{14}+\frac {x^{13/12}}{13}+\frac {x^{11/12}}{11}-\frac {x^{5/6}}{10}+\frac {x^{3/4}}{9}-\frac {x^{2/3}}{8}+\frac {x^{7/12}}{7}+\frac {x^{5/12}}{5}-\frac {x}{12}-\frac {\sqrt {x}}{6}-\frac {\sqrt [3]{x}}{4}+\frac {\sqrt [4]{x}}{3}-\frac {\sqrt [6]{x}}{2}+\sqrt [12]{x}-\log \left (\sqrt [12]{x}+1\right )\right )\) |
Input:
Int[(x^(-1/3) + x^(-1/4))^(-1),x]
Output:
12*(x^(1/12) - x^(1/6)/2 + x^(1/4)/3 - x^(1/3)/4 + x^(5/12)/5 - Sqrt[x]/6 + x^(7/12)/7 - x^(2/3)/8 + x^(3/4)/9 - x^(5/6)/10 + x^(11/12)/11 - x/12 + x^(13/12)/13 - x^(7/6)/14 + x^(5/4)/15 - Log[1 + x^(1/12)])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(12 x^{\frac {1}{12}}-6 x^{\frac {1}{6}}+4 x^{\frac {1}{4}}-3 x^{\frac {1}{3}}+\frac {12 x^{\frac {5}{12}}}{5}-2 \sqrt {x}+\frac {12 x^{\frac {7}{12}}}{7}-\frac {3 x^{\frac {2}{3}}}{2}+\frac {4 x^{\frac {3}{4}}}{3}-\frac {6 x^{\frac {5}{6}}}{5}+\frac {12 x^{\frac {11}{12}}}{11}-x +\frac {12 x^{\frac {13}{12}}}{13}-\frac {6 x^{\frac {7}{6}}}{7}+\frac {4 x^{\frac {5}{4}}}{5}-12 \ln \left (1+x^{\frac {1}{12}}\right )\) | \(83\) |
default | \(12 x^{\frac {1}{12}}-6 x^{\frac {1}{6}}+4 x^{\frac {1}{4}}-3 x^{\frac {1}{3}}+\frac {12 x^{\frac {5}{12}}}{5}-2 \sqrt {x}+\frac {12 x^{\frac {7}{12}}}{7}-\frac {3 x^{\frac {2}{3}}}{2}+\frac {4 x^{\frac {3}{4}}}{3}-\frac {6 x^{\frac {5}{6}}}{5}+\frac {12 x^{\frac {11}{12}}}{11}-x +\frac {12 x^{\frac {13}{12}}}{13}-\frac {6 x^{\frac {7}{6}}}{7}+\frac {4 x^{\frac {5}{4}}}{5}-12 \ln \left (1+x^{\frac {1}{12}}\right )\) | \(83\) |
meijerg | \(\frac {x^{\frac {1}{12}} \left (48048 x^{\frac {7}{6}}-51480 x^{\frac {13}{12}}+55440 x -60060 x^{\frac {11}{12}}+65520 x^{\frac {5}{6}}-72072 x^{\frac {3}{4}}+80080 x^{\frac {2}{3}}-90090 x^{\frac {7}{12}}+102960 \sqrt {x}-120120 x^{\frac {5}{12}}+144144 x^{\frac {1}{3}}-180180 x^{\frac {1}{4}}+240240 x^{\frac {1}{6}}-360360 x^{\frac {1}{12}}+720720\right )}{60060}-12 \ln \left (1+x^{\frac {1}{12}}\right )\) | \(85\) |
Input:
int(1/(1/x^(1/3)+1/x^(1/4)),x,method=_RETURNVERBOSE)
Output:
12*x^(1/12)-6*x^(1/6)+4*x^(1/4)-3*x^(1/3)+12/5*x^(5/12)-2*x^(1/2)+12/7*x^( 7/12)-3/2*x^(2/3)+4/3*x^(3/4)-6/5*x^(5/6)+12/11*x^(11/12)-x+12/13*x^(13/12 )-6/7*x^(7/6)+4/5*x^(5/4)-12*ln(1+x^(1/12))
Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=\frac {4}{5} \, {\left (x + 5\right )} x^{\frac {1}{4}} - \frac {6}{7} \, {\left (x + 7\right )} x^{\frac {1}{6}} + \frac {12}{13} \, {\left (x + 13\right )} x^{\frac {1}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \] Input:
integrate(1/(1/x^(1/3)+1/x^(1/4)),x, algorithm="fricas")
Output:
4/5*(x + 5)*x^(1/4) - 6/7*(x + 7)*x^(1/6) + 12/13*(x + 13)*x^(1/12) - x + 12/11*x^(11/12) - 6/5*x^(5/6) + 4/3*x^(3/4) - 3/2*x^(2/3) + 12/7*x^(7/12) - 2*sqrt(x) + 12/5*x^(5/12) - 3*x^(1/3) - 12*log(x^(1/12) + 1)
Time = 1.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=\frac {12 x^{\frac {13}{12}}}{13} + \frac {12 x^{\frac {11}{12}}}{11} + \frac {12 x^{\frac {7}{12}}}{7} + \frac {12 x^{\frac {5}{12}}}{5} + 12 \sqrt [12]{x} - \frac {6 x^{\frac {7}{6}}}{7} - \frac {6 x^{\frac {5}{6}}}{5} - 6 \sqrt [6]{x} + \frac {4 x^{\frac {5}{4}}}{5} + \frac {4 x^{\frac {3}{4}}}{3} + 4 \sqrt [4]{x} - \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} - 2 \sqrt {x} - x - 12 \log {\left (\sqrt [12]{x} + 1 \right )} \] Input:
integrate(1/(1/x**(1/3)+1/x**(1/4)),x)
Output:
12*x**(13/12)/13 + 12*x**(11/12)/11 + 12*x**(7/12)/7 + 12*x**(5/12)/5 + 12 *x**(1/12) - 6*x**(7/6)/7 - 6*x**(5/6)/5 - 6*x**(1/6) + 4*x**(5/4)/5 + 4*x **(3/4)/3 + 4*x**(1/4) - 3*x**(2/3)/2 - 3*x**(1/3) - 2*sqrt(x) - x - 12*lo g(x**(1/12) + 1)
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=\frac {4}{5} \, x^{\frac {5}{4}} - \frac {6}{7} \, x^{\frac {7}{6}} + \frac {12}{13} \, x^{\frac {13}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 6 \, x^{\frac {1}{6}} + 12 \, x^{\frac {1}{12}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \] Input:
integrate(1/(1/x^(1/3)+1/x^(1/4)),x, algorithm="maxima")
Output:
4/5*x^(5/4) - 6/7*x^(7/6) + 12/13*x^(13/12) - x + 12/11*x^(11/12) - 6/5*x^ (5/6) + 4/3*x^(3/4) - 3/2*x^(2/3) + 12/7*x^(7/12) - 2*sqrt(x) + 12/5*x^(5/ 12) - 3*x^(1/3) + 4*x^(1/4) - 6*x^(1/6) + 12*x^(1/12) - 12*log(x^(1/12) + 1)
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=\frac {4}{5} \, x^{\frac {5}{4}} - \frac {6}{7} \, x^{\frac {7}{6}} + \frac {12}{13} \, x^{\frac {13}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 6 \, x^{\frac {1}{6}} + 12 \, x^{\frac {1}{12}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \] Input:
integrate(1/(1/x^(1/3)+1/x^(1/4)),x, algorithm="giac")
Output:
4/5*x^(5/4) - 6/7*x^(7/6) + 12/13*x^(13/12) - x + 12/11*x^(11/12) - 6/5*x^ (5/6) + 4/3*x^(3/4) - 3/2*x^(2/3) + 12/7*x^(7/12) - 2*sqrt(x) + 12/5*x^(5/ 12) - 3*x^(1/3) + 4*x^(1/4) - 6*x^(1/6) + 12*x^(1/12) - 12*log(x^(1/12) + 1)
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=4\,x^{1/4}-12\,\ln \left (x^{1/12}+1\right )-2\,\sqrt {x}-3\,x^{1/3}-x-\frac {3\,x^{2/3}}{2}-6\,x^{1/6}+\frac {4\,x^{3/4}}{3}+\frac {4\,x^{5/4}}{5}-\frac {6\,x^{5/6}}{5}+12\,x^{1/12}-\frac {6\,x^{7/6}}{7}+\frac {12\,x^{5/12}}{5}+\frac {12\,x^{7/12}}{7}+\frac {12\,x^{11/12}}{11}+\frac {12\,x^{13/12}}{13} \] Input:
int(1/(1/x^(1/3) + 1/x^(1/4)),x)
Output:
4*x^(1/4) - 12*log(x^(1/12) + 1) - 2*x^(1/2) - 3*x^(1/3) - x - (3*x^(2/3)) /2 - 6*x^(1/6) + (4*x^(3/4))/3 + (4*x^(5/4))/5 - (6*x^(5/6))/5 + 12*x^(1/1 2) - (6*x^(7/6))/7 + (12*x^(5/12))/5 + (12*x^(7/12))/7 + (12*x^(11/12))/11 + (12*x^(13/12))/13
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx=\frac {12 x^{\frac {11}{12}}}{11}+\frac {12 x^{\frac {7}{12}}}{7}+\frac {12 x^{\frac {5}{12}}}{5}+\frac {12 x^{\frac {13}{12}}}{13}+12 x^{\frac {1}{12}}-\frac {6 x^{\frac {5}{6}}}{5}-\frac {6 x^{\frac {7}{6}}}{7}-6 x^{\frac {1}{6}}+\frac {4 x^{\frac {3}{4}}}{3}+\frac {4 x^{\frac {5}{4}}}{5}+4 x^{\frac {1}{4}}-\frac {3 x^{\frac {2}{3}}}{2}-3 x^{\frac {1}{3}}-2 \sqrt {x}-12 \,\mathrm {log}\left (x^{\frac {1}{12}}+1\right )-x \] Input:
int(1/(1/x^(1/3)+1/x^(1/4)),x)
Output:
(32760*x**(11/12) + 51480*x**(7/12) + 72072*x**(5/12) + 27720*x**(1/12)*x + 360360*x**(1/12) - 36036*x**(5/6) - 25740*x**(1/6)*x - 180180*x**(1/6) + 40040*x**(3/4) + 24024*x**(1/4)*x + 120120*x**(1/4) - 45045*x**(2/3) - 90 090*x**(1/3) - 60060*sqrt(x) - 360360*log(x**(1/12) + 1) - 30030*x)/30030