3.972 \(\int \frac{1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx\)

Optimal. Leaf size=115 \[ \frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}-8 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

[Out]

12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*Sqrt[x] + (12*x^(7/12))/7 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(13/
12))/13 + 4*Sqrt[3]*ArcTan[(1 - 2*x^(1/12))/Sqrt[3]] - 8*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]

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Rubi [A]  time = 0.149687, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {1593, 1836, 1887, 1874, 31, 634, 618, 204, 628} \[ \frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}-8 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^(1/3))/(1 + x^(1/4)),x]

[Out]

12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*Sqrt[x] + (12*x^(7/12))/7 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(13/
12))/13 + 4*Sqrt[3]*ArcTan[(1 - 2*x^(1/12))/Sqrt[3]] - 8*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]

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 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + 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]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

Rule 1887

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]

Rule 1874

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2], q = (a/b)^(1/3)}, Dist[(q*(A - B*q + C*q^2))/(3*a), Int[1/(q + x), x], x] + Dist[q/(3*a), Int[(q*(2*A + B
*q - C*q^2) - (A - B*q - 2*C*q^2)*x)/(q^2 - q*x + x^2), x], x] /; NeQ[a*B^3 - b*A^3, 0] && NeQ[A - B*q + C*q^2
, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2] && GtQ[a/b, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.0806612, size = 123, normalized size = 1.07 \[ \frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}+12 \sqrt [12]{x}+4 \left (\sqrt [3]{-1}-1\right ) \log \left (\sqrt [3]{-1}-\sqrt [12]{x}\right )-4 \left (1+(-1)^{2/3}\right ) \log \left (-\sqrt [12]{x}-(-1)^{2/3}\right )-8 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*Sqrt[x] + (12*x^(7/12))/7 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(13/
12))/13 + 4*(-1 + (-1)^(1/3))*Log[(-1)^(1/3) - x^(1/12)] - 4*(1 + (-1)^(2/3))*Log[-(-1)^(2/3) - x^(1/12)] - 8*
Log[1 + x^(1/12)]

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Maple [A]  time = 0.006, size = 81, normalized size = 0.7 \begin{align*}{\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\,\sqrt [3]{x}+4\,\sqrt [4]{x}+12\,{x}^{1/12}-2\,\ln \left ( 1-{x}^{1/12}+\sqrt [6]{x} \right ) -4\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,{x}^{1/12}-1 \right ) \sqrt{3} \right ) -8\,\ln \left ( 1+{x}^{1/12} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)-2*ln(1-x^(1/12
)+x^(1/6))-4*3^(1/2)*arctan(1/3*(2*x^(1/12)-1)*3^(1/2))-8*ln(1+x^(1/12))

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Maxima [A]  time = 1.67745, size = 108, normalized size = 0.94 \begin{align*} -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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Fricas [A]  time = 1.46299, size = 290, normalized size = 2.52 \begin{align*} -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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Sympy [C]  time = 4.03954, size = 221, normalized size = 1.92 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

64*x**(13/12)*gamma(16/3)/(13*gamma(19/3)) + 64*x**(7/12)*gamma(16/3)/(7*gamma(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)
/gamma(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(1
6/3)/(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))

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Giac [A]  time = 1.1758, size = 108, normalized size = 0.94 \begin{align*} -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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)