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]
________________________________________________________________________________________
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]
[Out]
Rule 1593
Rule 266
Rule 43
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]