Integrand size = 17, antiderivative size = 115 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=12 \sqrt [12]{x}+4 \sqrt [4]{x}-3 \sqrt [3]{x}-2 \sqrt {x}+\frac {12 x^{7/12}}{7}+\frac {4 x^{3/4}}{3}-\frac {6 x^{5/6}}{5}+\frac {12 x^{13/12}}{13}+4 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right )-8 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1-\sqrt [12]{x}+\sqrt [6]{x}\right ) \] Output:
12*x^(1/12)+4*x^(1/4)-3*x^(1/3)-2*x^(1/2)+12/7*x^(7/12)+4/3*x^(3/4)-6/5*x^ (5/6)+12/13*x^(13/12)+4*3^(1/2)*arctan(1/3*(1-2*x^(1/12))*3^(1/2))-8*ln(1+ x^(1/12))-2*ln(1-x^(1/12)+x^(1/6))
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=\frac {16380 \sqrt [12]{x}+5460 \sqrt [4]{x}-4095 \sqrt [3]{x}-2730 \sqrt {x}+2340 x^{7/12}+1820 x^{3/4}-1638 x^{5/6}+1260 x^{13/12}}{1365}+4 \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [12]{x}}{\sqrt {3}}\right )-8 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1-\sqrt [12]{x}+\sqrt [6]{x}\right ) \] Input:
Integrate[(1 + x^(1/3))/(1 + x^(1/4)),x]
Output:
(16380*x^(1/12) + 5460*x^(1/4) - 4095*x^(1/3) - 2730*Sqrt[x] + 2340*x^(7/1 2) + 1820*x^(3/4) - 1638*x^(5/6) + 1260*x^(13/12))/1365 + 4*Sqrt[3]*ArcTan [1/Sqrt[3] - (2*x^(1/12))/Sqrt[3]] - 8*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/ 12) + x^(1/6)]
Time = 0.64 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {7267, 2027, 2375, 27, 2426, 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.
\(\displaystyle \int \frac {\sqrt [3]{x}+1}{\sqrt [4]{x}+1} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 12 \int \frac {x^{5/4}+x^{11/12}}{\sqrt [4]{x}+1}d\sqrt [12]{x}\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle 12 \int \frac {\left (\sqrt [3]{x}+1\right ) x^{11/12}}{\sqrt [4]{x}+1}d\sqrt [12]{x}\) |
\(\Big \downarrow \) 2375 |
\(\displaystyle 12 \left (\frac {1}{13} \int \frac {13 \left (1-\sqrt [12]{x}\right ) x^{11/12}}{\sqrt [4]{x}+1}d\sqrt [12]{x}+\frac {x^{13/12}}{13}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 12 \left (\int \frac {\left (1-\sqrt [12]{x}\right ) x^{11/12}}{\sqrt [4]{x}+1}d\sqrt [12]{x}+\frac {x^{13/12}}{13}\right )\) |
\(\Big \downarrow \) 2426 |
\(\displaystyle 12 \left (\int \left (-\frac {\sqrt [6]{x}+1}{\sqrt [4]{x}+1}-x^{3/4}+x^{2/3}+\sqrt {x}-x^{5/12}-\sqrt [4]{x}+\sqrt [6]{x}+1\right )d\sqrt [12]{x}+\frac {x^{13/12}}{13}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 12 \left (\frac {\arctan \left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {x^{13/12}}{13}-\frac {x^{5/6}}{10}+\frac {x^{3/4}}{9}+\frac {x^{7/12}}{7}-\frac {\sqrt {x}}{6}-\frac {\sqrt [3]{x}}{4}+\frac {\sqrt [4]{x}}{3}+\sqrt [12]{x}-\frac {2}{3} \log \left (\sqrt [12]{x}+1\right )-\frac {1}{6} \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )\right )\) |
Input:
Int[(1 + x^(1/3))/(1 + x^(1/4)),x]
Output:
12*(x^(1/12) + x^(1/4)/3 - x^(1/3)/4 - Sqrt[x]/6 + x^(7/12)/7 + x^(3/4)/9 - x^(5/6)/10 + x^(13/12)/13 + ArcTan[(1 - 2*x^(1/12))/Sqrt[3]]/Sqrt[3] - ( 2*Log[1 + x^(1/12)])/3 - Log[1 - x^(1/12) + x^(1/6)]/6)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {12 x^{\frac {13}{12}}}{13}-\frac {6 x^{\frac {5}{6}}}{5}+\frac {4 x^{\frac {3}{4}}}{3}+\frac {12 x^{\frac {7}{12}}}{7}-2 \sqrt {x}-3 x^{\frac {1}{3}}+4 x^{\frac {1}{4}}+12 x^{\frac {1}{12}}-8 \ln \left (1+x^{\frac {1}{12}}\right )-2 \ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{12}}-1\right ) \sqrt {3}}{3}\right )\) | \(81\) |
default | \(\frac {12 x^{\frac {13}{12}}}{13}-\frac {6 x^{\frac {5}{6}}}{5}+\frac {4 x^{\frac {3}{4}}}{3}+\frac {12 x^{\frac {7}{12}}}{7}-2 \sqrt {x}-3 x^{\frac {1}{3}}+4 x^{\frac {1}{4}}+12 x^{\frac {1}{12}}-8 \ln \left (1+x^{\frac {1}{12}}\right )-2 \ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{12}}-1\right ) \sqrt {3}}{3}\right )\) | \(81\) |
meijerg | \(\frac {x^{\frac {1}{4}} \left (4 \sqrt {x}-6 x^{\frac {1}{4}}+12\right )}{3}-4 \ln \left (1+x^{\frac {1}{4}}\right )+\frac {3 x^{\frac {1}{12}} \left (560 x -728 x^{\frac {3}{4}}+1040 \sqrt {x}-1820 x^{\frac {1}{4}}+7280\right )}{1820}-4 x^{\frac {1}{12}} \left (\frac {\ln \left (1+x^{\frac {1}{12}}\right )}{x^{\frac {1}{12}}}-\frac {\ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )}{2 x^{\frac {1}{12}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{12}}}{2-x^{\frac {1}{12}}}\right )}{x^{\frac {1}{12}}}\right )\) | \(109\) |
Input:
int((1+x^(1/3))/(1+x^(1/4)),x,method=_RETURNVERBOSE)
Output:
12/13*x^(13/12)-6/5*x^(5/6)+4/3*x^(3/4)+12/7*x^(7/12)-2*x^(1/2)-3*x^(1/3)+ 4*x^(1/4)+12*x^(1/12)-8*ln(1+x^(1/12))-2*ln(1-x^(1/12)+x^(1/6))-4*3^(1/2)* arctan(1/3*(2*x^(1/12)-1)*3^(1/2))
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=-4 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{12}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {12}{13} \, {\left (x + 13\right )} x^{\frac {1}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac {1}{12}} + 1\right ) \] Input:
integrate((1+x^(1/3))/(1+x^(1/4)),x, algorithm="fricas")
Output:
-4*sqrt(3)*arctan(2/3*sqrt(3)*x^(1/12) - 1/3*sqrt(3)) + 12/13*(x + 13)*x^( 1/12) - 6/5*x^(5/6) + 4/3*x^(3/4) + 12/7*x^(7/12) - 2*sqrt(x) - 3*x^(1/3) + 4*x^(1/4) - 2*log(x^(1/6) - x^(1/12) + 1) - 8*log(x^(1/12) + 1)
Result contains complex when optimal does not.
Time = 3.18 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.92 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=\frac {64 x^{\frac {13}{12}} \Gamma \left (\frac {16}{3}\right )}{13 \Gamma \left (\frac {19}{3}\right )} + \frac {64 x^{\frac {7}{12}} \Gamma \left (\frac {16}{3}\right )}{7 \Gamma \left (\frac {19}{3}\right )} + \frac {64 \sqrt [12]{x} \Gamma \left (\frac {16}{3}\right )}{\Gamma \left (\frac {19}{3}\right )} - \frac {32 x^{\frac {5}{6}} \Gamma \left (\frac {16}{3}\right )}{5 \Gamma \left (\frac {19}{3}\right )} + \frac {4 x^{\frac {3}{4}}}{3} + 4 \sqrt [4]{x} - \frac {16 \sqrt [3]{x} \Gamma \left (\frac {16}{3}\right )}{\Gamma \left (\frac {19}{3}\right )} - 2 \sqrt {x} - 4 \log {\left (\sqrt [4]{x} + 1 \right )} + \frac {64 e^{- \frac {i \pi }{3}} \log {\left (- \sqrt [12]{x} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {16}{3}\right )}{3 \Gamma \left (\frac {19}{3}\right )} - \frac {64 \log {\left (- \sqrt [12]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac {16}{3}\right )}{3 \Gamma \left (\frac {19}{3}\right )} + \frac {64 e^{\frac {i \pi }{3}} \log {\left (- \sqrt [12]{x} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {16}{3}\right )}{3 \Gamma \left (\frac {19}{3}\right )} \] Input:
integrate((1+x**(1/3))/(1+x**(1/4)),x)
Output:
64*x**(13/12)*gamma(16/3)/(13*gamma(19/3)) + 64*x**(7/12)*gamma(16/3)/(7*g amma(19/3)) + 64*x**(1/12)*gamma(16/3)/gamma(19/3) - 32*x**(5/6)*gamma(16/ 3)/(5*gamma(19/3)) + 4*x**(3/4)/3 + 4*x**(1/4) - 16*x**(1/3)*gamma(16/3)/g amma(19/3) - 2*sqrt(x) - 4*log(x**(1/4) + 1) + 64*exp(-I*pi/3)*log(-x**(1/ 12)*exp_polar(I*pi/3) + 1)*gamma(16/3)/(3*gamma(19/3)) - 64*log(-x**(1/12) *exp_polar(I*pi) + 1)*gamma(16/3)/(3*gamma(19/3)) + 64*exp(I*pi/3)*log(-x* *(1/12)*exp_polar(5*I*pi/3) + 1)*gamma(16/3)/(3*gamma(19/3))
Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=-4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {12}{13} \, x^{\frac {13}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} + 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac {1}{12}} + 1\right ) \] Input:
integrate((1+x^(1/3))/(1+x^(1/4)),x, algorithm="maxima")
Output:
-4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 12/13*x^(13/12) - 6/5*x^ (5/6) + 4/3*x^(3/4) + 12/7*x^(7/12) - 2*sqrt(x) - 3*x^(1/3) + 4*x^(1/4) + 12*x^(1/12) - 2*log(x^(1/6) - x^(1/12) + 1) - 8*log(x^(1/12) + 1)
Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=-4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {12}{13} \, x^{\frac {13}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} + 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) - 8 \, \log \left (x^{\frac {1}{12}} + 1\right ) \] Input:
integrate((1+x^(1/3))/(1+x^(1/4)),x, algorithm="giac")
Output:
-4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 12/13*x^(13/12) - 6/5*x^ (5/6) + 4/3*x^(3/4) + 12/7*x^(7/12) - 2*sqrt(x) - 3*x^(1/3) + 4*x^(1/4) + 12*x^(1/12) - 2*log(x^(1/6) - x^(1/12) + 1) - 8*log(x^(1/12) + 1)
Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.13 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=4\,x^{1/4}+\ln \left (\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (54-36\,x^{1/12}+\sqrt {3}\,18{}\mathrm {i}\right )-144\,x^{1/12}+144\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )-\ln \left (\left (2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (36\,x^{1/12}-54+\sqrt {3}\,18{}\mathrm {i}\right )-144\,x^{1/12}+144\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )-2\,\sqrt {x}-3\,x^{1/3}-8\,\ln \left (144\,x^{1/12}+144\right )+\frac {4\,x^{3/4}}{3}-\frac {6\,x^{5/6}}{5}+12\,x^{1/12}+\frac {12\,x^{7/12}}{7}+\frac {12\,x^{13/12}}{13} \] Input:
int((x^(1/3) + 1)/(x^(1/4) + 1),x)
Output:
log((3^(1/2)*2i - 2)*(3^(1/2)*18i - 36*x^(1/12) + 54) - 144*x^(1/12) + 144 )*(3^(1/2)*2i - 2) - 8*log(144*x^(1/12) + 144) - log((3^(1/2)*2i + 2)*(3^( 1/2)*18i + 36*x^(1/12) - 54) - 144*x^(1/12) + 144)*(3^(1/2)*2i + 2) - 2*x^ (1/2) - 3*x^(1/3) + 4*x^(1/4) + (4*x^(3/4))/3 - (6*x^(5/6))/5 + 12*x^(1/12 ) + (12*x^(7/12))/7 + (12*x^(13/12))/13
Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx=-4 \sqrt {3}\, \mathit {atan} \left (\frac {2 x^{\frac {1}{12}}-1}{\sqrt {3}}\right )+\frac {12 x^{\frac {7}{12}}}{7}+\frac {12 x^{\frac {13}{12}}}{13}+12 x^{\frac {1}{12}}-\frac {6 x^{\frac {5}{6}}}{5}+\frac {4 x^{\frac {3}{4}}}{3}+4 x^{\frac {1}{4}}-3 x^{\frac {1}{3}}-2 \sqrt {x}-4 \,\mathrm {log}\left (x^{\frac {1}{12}}+1\right )-4 \,\mathrm {log}\left (x^{\frac {1}{4}}+1\right )+2 \,\mathrm {log}\left (-x^{\frac {1}{12}}+x^{\frac {1}{6}}+1\right ) \] Input:
int((1+x^(1/3))/(1+x^(1/4)),x)
Output:
( - 5460*sqrt(3)*atan((2*x**(1/12) - 1)/sqrt(3)) + 2340*x**(7/12) + 1260*x **(1/12)*x + 16380*x**(1/12) - 1638*x**(5/6) + 1820*x**(3/4) + 5460*x**(1/ 4) - 4095*x**(1/3) - 2730*sqrt(x) - 5460*log(x**(1/12) + 1) - 5460*log(x** (1/4) + 1) + 2730*log( - x**(1/12) + x**(1/6) + 1))/1365