### 3.237 $$\int \frac{1}{\frac{1}{\sqrt [3]{x}}+\frac{1}{\sqrt [4]{x}}} \, dx$$

Optimal. Leaf size=130 $\frac{4 x^{5/4}}{5}-\frac{6 x^{7/6}}{7}+\frac{12 x^{13/12}}{13}+\frac{12 x^{11/12}}{11}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}-\frac{3 x^{2/3}}{2}+\frac{12 x^{7/12}}{7}+\frac{12 x^{5/12}}{5}-x-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right )$

[Out]

12*x^(1/12) - 6*x^(1/6) + 4*x^(1/4) - 3*x^(1/3) + (12*x^(5/12))/5 - 2*Sqrt[x] + (12*x^(7/12))/7 - (3*x^(2/3))/
2 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(11/12))/11 - x + (12*x^(13/12))/13 - (6*x^(7/6))/7 + (4*x^(5/4))/5
- 12*Log[1 + x^(1/12)]

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Rubi [A]  time = 0.0462393, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.231, Rules used = {1593, 266, 43} $\frac{4 x^{5/4}}{5}-\frac{6 x^{7/6}}{7}+\frac{12 x^{13/12}}{13}+\frac{12 x^{11/12}}{11}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}-\frac{3 x^{2/3}}{2}+\frac{12 x^{7/12}}{7}+\frac{12 x^{5/12}}{5}-x-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^(-1/3) + x^(-1/4))^(-1),x]

[Out]

12*x^(1/12) - 6*x^(1/6) + 4*x^(1/4) - 3*x^(1/3) + (12*x^(5/12))/5 - 2*Sqrt[x] + (12*x^(7/12))/7 - (3*x^(2/3))/
2 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(11/12))/11 - x + (12*x^(13/12))/13 - (6*x^(7/6))/7 + (4*x^(5/4))/5
- 12*Log[1 + x^(1/12)]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\frac{1}{\sqrt [3]{x}}+\frac{1}{\sqrt [4]{x}}} \, dx &=\int \frac{\sqrt [3]{x}}{1+\sqrt [12]{x}} \, dx\\ &=12 \operatorname{Subst}\left (\int \frac{x^{15}}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9+x^{10}-x^{11}+x^{12}-x^{13}+x^{14}\right ) \, dx,x,\sqrt [12]{x}\right )\\ &=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 )\\ \end{align*}

Mathematica [A]  time = 0.0366835, size = 130, normalized size = 1. $\frac{4 x^{5/4}}{5}-\frac{6 x^{7/6}}{7}+\frac{12 x^{13/12}}{13}+\frac{12 x^{11/12}}{11}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}-\frac{3 x^{2/3}}{2}+\frac{12 x^{7/12}}{7}+\frac{12 x^{5/12}}{5}-x-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^(-1/3) + x^(-1/4))^(-1),x]

[Out]

12*x^(1/12) - 6*x^(1/6) + 4*x^(1/4) - 3*x^(1/3) + (12*x^(5/12))/5 - 2*Sqrt[x] + (12*x^(7/12))/7 - (3*x^(2/3))/
2 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(11/12))/11 - x + (12*x^(13/12))/13 - (6*x^(7/6))/7 + (4*x^(5/4))/5
- 12*Log[1 + x^(1/12)]

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Maple [A]  time = 0.004, size = 83, normalized size = 0.6 \begin{align*} 12\,{x}^{1/12}-6\,\sqrt [6]{x}+4\,\sqrt [4]{x}-3\,\sqrt [3]{x}+{\frac{12}{5}{x}^{{\frac{5}{12}}}}+{\frac{12}{7}{x}^{{\frac{7}{12}}}}-{\frac{3}{2}{x}^{{\frac{2}{3}}}}+{\frac{4}{3}{x}^{{\frac{3}{4}}}}-{\frac{6}{5}{x}^{{\frac{5}{6}}}}+{\frac{12}{11}{x}^{{\frac{11}{12}}}}-x+{\frac{12}{13}{x}^{{\frac{13}{12}}}}-{\frac{6}{7}{x}^{{\frac{7}{6}}}}+{\frac{4}{5}{x}^{{\frac{5}{4}}}}-12\,\ln \left ( 1+{x}^{1/12} \right ) -2\,\sqrt{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/x^(1/3)+1/x^(1/4)),x)

[Out]

12*x^(1/12)-6*x^(1/6)+4*x^(1/4)-3*x^(1/3)+12/5*x^(5/12)+12/7*x^(7/12)-3/2*x^(2/3)+4/3*x^(3/4)-6/5*x^(5/6)+12/1
1*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))-2*x^(1/2)

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Maxima [A]  time = 0.946955, size = 111, normalized size = 0.85 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/x^(1/3)+1/x^(1/4)),x, algorithm="maxima")

[Out]

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)

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Fricas [A]  time = 2.12897, size = 286, normalized size = 2.2 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/x^(1/3)+1/x^(1/4)),x, algorithm="fricas")

[Out]

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)

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Sympy [A]  time = 2.02183, size = 121, normalized size = 0.93 \begin{align*} \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 )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/x**(1/3)+1/x**(1/4)),x)

[Out]

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*
log(x**(1/12) + 1)

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Giac [A]  time = 1.06202, size = 111, normalized size = 0.85 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/x^(1/3)+1/x^(1/4)),x, algorithm="giac")

[Out]

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)