3.10.72 \(\int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx\) [972]

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

[Out]

12*x^(1/12)+4*x^(1/4)-3*x^(1/3)+12/7*x^(7/12)+4/3*x^(3/4)-6/5*x^(5/6)+12/13*x^(13/12)-8*ln(1+x^(1/12))-2*ln(1-
x^(1/12)+x^(1/6))+4*arctan(1/3*(1-2*x^(1/12))*3^(1/2))*3^(1/2)-2*x^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 115, normalized size of antiderivative = 1.00, 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 = {1607, 1850, 1901, 1888, 31, 648, 632, 210, 642} \begin {gather*} 4 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right )+\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 ) \end {gather*}

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 31

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

Rule 210

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

Rule 632

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 642

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]

Rule 648

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 1607

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 1850

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 1888

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 1901

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]

Rubi steps

\begin {align*} \int \frac {1+\sqrt [3]{x}}{1+\sqrt [4]{x}} \, dx &=12 \text {Subst}\left (\int \frac {x^{11}+x^{15}}{1+x^3} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \text {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} \text {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} \text {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 \text {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 \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\sqrt [12]{x}\right )-8 \text {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 \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [12]{x}\right )-6 \text {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 \text {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*}

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Mathematica [A]
time = 0.06, size = 115, normalized size = 1.00 \begin {gather*} \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} \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16380*x^(1/12) + 5460*x^(1/4) - 4095*x^(1/3) - 2730*Sqrt[x] + 2340*x^(7/12) + 1820*x^(3/4) - 1638*x^(5/6) + 1
260*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)]

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Maple [A]
time = 0.22, size = 81, normalized size = 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	\(\frac {x^{\frac {1}{4}} \left (4 \sqrt {x}-6 x^{\frac {1}{4}}+12\right )}{3}-4 \ln \left (1+x^{\frac {1}{4}}\right )+\frac {3 x^{\frac {1}{12}} \left (560 x -728 x^{\frac {3}{4}}+1040 \sqrt {x}-1820 x^{\frac {1}{4}}+7280\right )}{1820}-4 x^{\frac {1}{12}} \left (\frac {\ln \left (1+x^{\frac {1}{12}}\right )}{x^{\frac {1}{12}}}-\frac {\ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )}{2 x^{\frac {1}{12}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{12}}}{2-x^{\frac {1}{12}}}\right )}{x^{\frac {1}{12}}}\right )\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^(1/3))/(1+x^(1/4)),x,method=_RETURNVERBOSE)

[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)-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))

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Maxima [A]
time = 0.49, size = 80, normalized size = 0.70 \begin {gather*} -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 {gather*}

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 = 0.35, size = 80, normalized size = 0.70 \begin {gather*} -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 {gather*}

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] Result contains complex when optimal does not.
time = 2.43, size = 221, normalized size = 1.92 \begin {gather*} \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 {gather*}

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 = 3.18, size = 80, normalized size = 0.70 \begin {gather*} -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 {gather*}

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)

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Mupad [B]
time = 0.09, size = 130, normalized size = 1.13 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/3) + 1)/(x^(1/4) + 1),x)

[Out]

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

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