3.11.11 \(\int \frac {1}{x \sqrt [3]{1-x^2} (3+x^2)} \, dx\) [1011]

Optimal. Leaf size=136 \[ -\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\log (x)}{6}+\frac {\log \left (3+x^2\right )}{12\ 2^{2/3}}+\frac {1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}} \]

[Out]

-1/6*ln(x)+1/24*ln(x^2+3)*2^(1/3)+1/4*ln(1-(-x^2+1)^(1/3))-1/8*ln(2^(2/3)-(-x^2+1)^(1/3))*2^(1/3)-1/12*arctan(
1/3*(1+(-2*x^2+2)^(1/3))*3^(1/2))*3^(1/2)*2^(1/3)+1/6*arctan(1/3*(1+2*(-x^2+1)^(1/3))*3^(1/2))*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {457, 88, 57, 632, 210, 31, 631} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\log \left (x^2+3\right )}{12\ 2^{2/3}}+\frac {1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac {\log (x)}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

-1/2*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(2*Sq
rt[3]) - Log[x]/6 + Log[3 + x^2]/(12*2^(2/3)) + Log[1 - (1 - x^2)^(1/3)]/4 - Log[2^(2/3) - (1 - x^2)^(1/3)]/(4
*2^(2/3))

Rule 31

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 88

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.17, size = 163, normalized size = 1.20 \begin {gather*} \frac {1}{24} \left (-2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-2 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )+\sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )+4 \log \left (-1+\sqrt [3]{1-x^2}\right )-2 \log \left (1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

(-2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 4*Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]
] - 2*2^(1/3)*Log[-2 + (2 - 2*x^2)^(1/3)] + 2^(1/3)*Log[4 + 2*(2 - 2*x^2)^(1/3) + (2 - 2*x^2)^(2/3)] + 4*Log[-
1 + (1 - x^2)^(1/3)] - 2*Log[1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3)])/24

________________________________________________________________________________________

Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.68, size = 177, normalized size = 1.30 \begin {gather*} -\frac {1}{12} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (2 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{48} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{12} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/6*4^(1/6)*(2*sqrt(3)*(-1)^(1/3)*(-x^2 + 1)^(1/3) - 4^(1/3)*sqrt(3)))
- 1/48*4^(2/3)*(-1)^(1/3)*log(4^(1/3)*(-1)^(2/3)*(-x^2 + 1)^(1/3) - 4^(2/3)*(-1)^(1/3) + (-x^2 + 1)^(2/3)) +
1/24*4^(2/3)*(-1)^(1/3)*log(-4^(1/3)*(-1)^(2/3) + (-x^2 + 1)^(1/3)) + 1/6*sqrt(3)*arctan(2/3*sqrt(3)*(-x^2 + 1
)^(1/3) + 1/3*sqrt(3)) - 1/12*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) + 1/6*log((-x^2 + 1)^(1/3) - 1)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*(-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)

________________________________________________________________________________________

Giac [A]
time = 0.57, size = 149, normalized size = 1.10 \begin {gather*} -\frac {1}{24} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{48} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{12} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3))) + 1/48*4^(2/3)*log(4^(2/3) +
4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) - 1/24*4^(2/3)*log(4^(1/3) - (-x^2 + 1)^(1/3)) + 1/6*sqrt(3)*arc
tan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) - 1/12*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) + 1/6*log(-(-x^2
+ 1)^(1/3) + 1)

________________________________________________________________________________________

Mupad [B]
time = 0.55, size = 256, normalized size = 1.88 \begin {gather*} \frac {\ln \left (\frac {405}{8}-\frac {405\,{\left (1-x^2\right )}^{1/3}}{8}\right )}{6}+\ln \left ({\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^3\,\left (393660\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2-\frac {37179\,{\left (1-x^2\right )}^{1/3}}{4}\right )-\frac {243\,{\left (1-x^2\right )}^{1/3}}{32}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (-{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^3\,\left (393660\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2-\frac {37179\,{\left (1-x^2\right )}^{1/3}}{4}\right )-\frac {243\,{\left (1-x^2\right )}^{1/3}}{32}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {2^{1/3}\,\ln \left (\frac {405\,{\left (1-x^2\right )}^{1/3}}{128}-\frac {405\,2^{2/3}}{128}\right )}{12}+\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (\frac {405\,{\left (1-x^2\right )}^{1/3}}{128}-\frac {405\,{\left (-1\right )}^{2/3}\,2^{2/3}}{128}\right )}{12}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (-\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^3\,\left (\frac {37179\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {10935\,{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )}{6912}-\frac {243\,{\left (1-x^2\right )}^{1/3}}{32}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(405/8 - (405*(1 - x^2)^(1/3))/8)/6 + log(((3^(1/2)*1i)/12 - 1/12)^3*(393660*((3^(1/2)*1i)/12 - 1/12)^2 - (
37179*(1 - x^2)^(1/3))/4) - (243*(1 - x^2)^(1/3))/32)*((3^(1/2)*1i)/12 - 1/12) - log(- ((3^(1/2)*1i)/12 + 1/12
)^3*(393660*((3^(1/2)*1i)/12 + 1/12)^2 - (37179*(1 - x^2)^(1/3))/4) - (243*(1 - x^2)^(1/3))/32)*((3^(1/2)*1i)/
12 + 1/12) - (2^(1/3)*log((405*(1 - x^2)^(1/3))/128 - (405*2^(2/3))/128))/12 + ((-1)^(1/3)*2^(1/3)*log((405*(1
- x^2)^(1/3))/128 - (405*(-1)^(2/3)*2^(2/3))/128))/12 - ((-1)^(1/3)*2^(1/3)*log(- ((3^(1/2)*1i + 1)^3*((37179
*(1 - x^2)^(1/3))/4 - (10935*(-1)^(2/3)*2^(2/3)*(3^(1/2)*1i + 1)^2)/16))/6912 - (243*(1 - x^2)^(1/3))/32)*(3^(
1/2)*1i + 1))/24

________________________________________________________________________________________