3.1013 \(\int \frac{1}{x^5 \sqrt [3]{1-x^2} (3+x^2)} \, dx\)

Optimal. Leaf size=172 \[ -\frac{\left (1-x^2\right )^{2/3}}{18 x^2}-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}+\frac{\log \left (x^2+3\right )}{108\ 2^{2/3}}+\frac{1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{18\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{9 \sqrt{3}}-\frac{\log (x)}{27} \]

[Out]

-(1 - x^2)^(2/3)/(12*x^4) - (1 - x^2)^(2/3)/(18*x^2) - ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(18*2^(2/3)*Sqr
t[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) - Log[x]/27 + Log[3 + x^2]/(108*2^(2/3)) + Log[1 -
(1 - x^2)^(1/3)]/18 - Log[2^(2/3) - (1 - x^2)^(1/3)]/(36*2^(2/3))

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Rubi [A]  time = 0.129859, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {446, 103, 151, 156, 55, 618, 204, 31, 617} \[ -\frac{\left (1-x^2\right )^{2/3}}{18 x^2}-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}+\frac{\log \left (x^2+3\right )}{108\ 2^{2/3}}+\frac{1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{18\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{9 \sqrt{3}}-\frac{\log (x)}{27} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

-(1 - x^2)^(2/3)/(12*x^4) - (1 - x^2)^(2/3)/(18*x^2) - ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(18*2^(2/3)*Sqr
t[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) - Log[x]/27 + Log[3 + x^2]/(108*2^(2/3)) + Log[1 -
(1 - x^2)^(1/3)]/18 - Log[2^(2/3) - (1 - x^2)^(1/3)]/(36*2^(2/3))

Rule 446

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

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

Rule 55

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 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 31

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

Rule 617

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]

Rubi steps

\begin{align*} \int \frac{1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x^3 (3+x)} \, dx,x,x^2\right )\\ &=-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{1}{12} \operatorname{Subst}\left (\int \frac{-2-\frac{4 x}{3}}{\sqrt [3]{1-x} x^2 (3+x)} \, dx,x,x^2\right )\\ &=-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{\left (1-x^2\right )^{2/3}}{18 x^2}+\frac{1}{36} \operatorname{Subst}\left (\int \frac{4+\frac{2 x}{3}}{\sqrt [3]{1-x} x (3+x)} \, dx,x,x^2\right )\\ &=-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{\left (1-x^2\right )^{2/3}}{18 x^2}-\frac{1}{54} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )+\frac{1}{27} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right )\\ &=-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{\left (1-x^2\right )^{2/3}}{18 x^2}-\frac{\log (x)}{27}+\frac{\log \left (3+x^2\right )}{108\ 2^{2/3}}-\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}\\ &=-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{\left (1-x^2\right )^{2/3}}{18 x^2}-\frac{\log (x)}{27}+\frac{\log \left (3+x^2\right )}{108\ 2^{2/3}}+\frac{1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{18\ 2^{2/3}}\\ &=-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{\left (1-x^2\right )^{2/3}}{18 x^2}-\frac{\tan ^{-1}\left (\frac{1+\sqrt [3]{2-2 x^2}}{\sqrt{3}}\right )}{18\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1+2 \sqrt [3]{1-x^2}}{\sqrt{3}}\right )}{9 \sqrt{3}}-\frac{\log (x)}{27}+\frac{\log \left (3+x^2\right )}{108\ 2^{2/3}}+\frac{1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0699829, size = 178, normalized size = 1.03 \[ -\frac{12 \left (1-x^2\right )^{2/3} x^2+18 \left (1-x^2\right )^{2/3}+8 x^4 \log (x)-\sqrt [3]{2} x^4 \log \left (x^2+3\right )-12 x^4 \log \left (1-\sqrt [3]{1-x^2}\right )+3 \sqrt [3]{2} x^4 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )+2 \sqrt [3]{2} \sqrt{3} x^4 \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )-8 \sqrt{3} x^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{216 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

-(18*(1 - x^2)^(2/3) + 12*x^2*(1 - x^2)^(2/3) + 2*2^(1/3)*Sqrt[3]*x^4*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]
- 8*Sqrt[3]*x^4*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]] + 8*x^4*Log[x] - 2^(1/3)*x^4*Log[3 + x^2] - 12*x^4*Log
[1 - (1 - x^2)^(1/3)] + 3*2^(1/3)*x^4*Log[2^(2/3) - (1 - x^2)^(1/3)])/(216*x^4)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ({x}^{2}+3 \right ) }{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

int(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="maxima")

[Out]

integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^5), x)

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Fricas [A]  time = 1.54489, size = 664, normalized size = 3.86 \begin{align*} -\frac{4 \cdot 4^{\frac{1}{6}} \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{4} \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 ) + 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x^{4} \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 ) - 2 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x^{4} \log \left (-4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) - 16 \, \sqrt{3} x^{4} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 8 \, x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) - 16 \, x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) + 12 \,{\left (2 \, x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{432 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="fricas")

[Out]

-1/432*(4*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^4*arctan(1/6*4^(1/6)*(2*sqrt(3)*(-1)^(1/3)*(-x^2 + 1)^(1/3) - 4^(1/3)*s
qrt(3))) + 4^(2/3)*(-1)^(1/3)*x^4*log(4^(1/3)*(-1)^(2/3)*(-x^2 + 1)^(1/3) - 4^(2/3)*(-1)^(1/3) + (-x^2 + 1)^(2
/3)) - 2*4^(2/3)*(-1)^(1/3)*x^4*log(-4^(1/3)*(-1)^(2/3) + (-x^2 + 1)^(1/3)) - 16*sqrt(3)*x^4*arctan(2/3*sqrt(3
)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) + 8*x^4*log((-x^2 + 1)^(2/3) + (-x^2 + 1)^(1/3) + 1) - 16*x^4*log((-x^2 + 1)
^(1/3) - 1) + 12*(2*x^2 + 3)*(-x^2 + 1)^(2/3))/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

Integral(1/(x**5*(-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError