3.4.38 \(\int \frac {x^{2/3}}{1+x^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac {1}{2} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x}}{x^{2/3}+1}\right )-\frac {1}{2} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{x}\right )+\frac {1}{2} \tan ^{-1}\left (2 \sqrt [3]{x}+\sqrt {3}\right )+\tan ^{-1}\left (\sqrt [3]{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.26, antiderivative size = 100, normalized size of antiderivative = 1.37, number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {329, 295, 634, 618, 204, 628, 203} \begin {gather*} \frac {1}{4} \sqrt {3} \log \left (x^{2/3}-\sqrt {3} \sqrt [3]{x}+1\right )-\frac {1}{4} \sqrt {3} \log \left (x^{2/3}+\sqrt {3} \sqrt [3]{x}+1\right )-\frac {1}{2} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{x}\right )+\frac {1}{2} \tan ^{-1}\left (2 \sqrt [3]{x}+\sqrt {3}\right )+\tan ^{-1}\left (\sqrt [3]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(2/3)/(1 + x^2),x]

[Out]

-ArcTan[Sqrt[3] - 2*x^(1/3)]/2 + ArcTan[Sqrt[3] + 2*x^(1/3)]/2 + ArcTan[x^(1/3)] + (Sqrt[3]*Log[1 - Sqrt[3]*x^
(1/3) + x^(2/3)])/4 - (Sqrt[3]*Log[1 + Sqrt[3]*x^(1/3) + x^(2/3)])/4

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 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]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.00, size = 22, normalized size = 0.30 \begin {gather*} \frac {3}{5} x^{5/3} \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)/(1 + x^2),x]

[Out]

(3*x^(5/3)*Hypergeometric2F1[5/6, 1, 11/6, -x^2])/5

________________________________________________________________________________________

IntegrateAlgebraic [C]  time = 0.13, size = 79, normalized size = 1.08 \begin {gather*} \tan ^{-1}\left (\sqrt [3]{x}\right )+\frac {1}{2} \left (1-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [3]{x}\right )+\frac {1}{2} \left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(2/3)/(1 + x^2),x]

[Out]

ArcTan[x^(1/3)] + ((1 - I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x^(1/3))/2])/2 + ((1 + I*Sqrt[3])*ArcTan[((1 + I*Sq
rt[3])*x^(1/3))/2])/2

________________________________________________________________________________________

fricas [B]  time = 0.83, size = 108, normalized size = 1.48 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \log \left (16 \, \sqrt {3} x^{\frac {1}{3}} + 16 \, x^{\frac {2}{3}} + 16\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-16 \, \sqrt {3} x^{\frac {1}{3}} + 16 \, x^{\frac {2}{3}} + 16\right ) - \arctan \left (\sqrt {3} + \frac {1}{2} \, \sqrt {-16 \, \sqrt {3} x^{\frac {1}{3}} + 16 \, x^{\frac {2}{3}} + 16} - 2 \, x^{\frac {1}{3}}\right ) - \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1} - 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2/3)/(x^2+1),x, algorithm="fricas")

[Out]

-1/4*sqrt(3)*log(16*sqrt(3)*x^(1/3) + 16*x^(2/3) + 16) + 1/4*sqrt(3)*log(-16*sqrt(3)*x^(1/3) + 16*x^(2/3) + 16
) - arctan(sqrt(3) + 1/2*sqrt(-16*sqrt(3)*x^(1/3) + 16*x^(2/3) + 16) - 2*x^(1/3)) - arctan(-sqrt(3) + 2*sqrt(s
qrt(3)*x^(1/3) + x^(2/3) + 1) - 2*x^(1/3)) + arctan(x^(1/3))

________________________________________________________________________________________

giac [A]  time = 0.63, size = 68, normalized size = 0.93 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \log \left (\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2/3)/(x^2+1),x, algorithm="giac")

[Out]

-1/4*sqrt(3)*log(sqrt(3)*x^(1/3) + x^(2/3) + 1) + 1/4*sqrt(3)*log(-sqrt(3)*x^(1/3) + x^(2/3) + 1) + 1/2*arctan
(sqrt(3) + 2*x^(1/3)) + 1/2*arctan(-sqrt(3) + 2*x^(1/3)) + arctan(x^(1/3))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 69, normalized size = 0.95 \begin {gather*} \arctan \left (x^{\frac {1}{3}}\right )+\frac {\arctan \left (2 x^{\frac {1}{3}}-\sqrt {3}\right )}{2}+\frac {\arctan \left (2 x^{\frac {1}{3}}+\sqrt {3}\right )}{2}+\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}-\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4}-\frac {\sqrt {3}\, \ln \left (x^{\frac {2}{3}}+\sqrt {3}\, x^{\frac {1}{3}}+1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2/3)/(x^2+1),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.97, size = 68, normalized size = 0.93 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \log \left (\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{4} \, \sqrt {3} \log \left (-\sqrt {3} x^{\frac {1}{3}} + x^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, x^{\frac {1}{3}}\right ) + \arctan \left (x^{\frac {1}{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(2/3)/(x^2+1),x, algorithm="maxima")

[Out]

-1/4*sqrt(3)*log(sqrt(3)*x^(1/3) + x^(2/3) + 1) + 1/4*sqrt(3)*log(-sqrt(3)*x^(1/3) + x^(2/3) + 1) + 1/2*arctan
(sqrt(3) + 2*x^(1/3)) + 1/2*arctan(-sqrt(3) + 2*x^(1/3)) + arctan(x^(1/3))

________________________________________________________________________________________

mupad [B]  time = 4.77, size = 57, normalized size = 0.78 \begin {gather*} \mathrm {atan}\left (x^{1/3}\right )-\mathrm {atan}\left (\frac {486\,x^{1/3}}{-243+\sqrt {3}\,243{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\mathrm {atan}\left (\frac {486\,x^{1/3}}{243+\sqrt {3}\,243{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2/3)/(x^2 + 1),x)

[Out]

atan(x^(1/3)) - atan((486*x^(1/3))/(3^(1/2)*243i - 243))*((3^(1/2)*1i)/2 + 1/2) - atan((486*x^(1/3))/(3^(1/2)*
243i + 243))*((3^(1/2)*1i)/2 - 1/2)

________________________________________________________________________________________

sympy [A]  time = 1.78, size = 94, normalized size = 1.29 \begin {gather*} \frac {\sqrt {3} \log {\left (4 x^{\frac {2}{3}} - 4 \sqrt {3} \sqrt [3]{x} + 4 \right )}}{4} - \frac {\sqrt {3} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt {3} \sqrt [3]{x} + 4 \right )}}{4} + \operatorname {atan}{\left (\sqrt [3]{x} \right )} + \frac {\operatorname {atan}{\left (2 \sqrt [3]{x} - \sqrt {3} \right )}}{2} + \frac {\operatorname {atan}{\left (2 \sqrt [3]{x} + \sqrt {3} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(2/3)/(x**2+1),x)

[Out]

sqrt(3)*log(4*x**(2/3) - 4*sqrt(3)*x**(1/3) + 4)/4 - sqrt(3)*log(4*x**(2/3) + 4*sqrt(3)*x**(1/3) + 4)/4 + atan
(x**(1/3)) + atan(2*x**(1/3) - sqrt(3))/2 + atan(2*x**(1/3) + sqrt(3))/2

________________________________________________________________________________________