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

Optimal. Leaf size=97 \[ \frac {\sqrt [3]{x^3+1} \left (3 x^3+1\right )}{3 x^3}+\frac {2}{9} \log \left (\sqrt [3]{x^3+1}-1\right )-\frac {1}{9} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 83, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {446, 78, 50, 57, 618, 204, 31} \begin {gather*} \frac {\left (x^3+1\right )^{4/3}}{3 x^3}+\frac {2}{3} \sqrt [3]{x^3+1}+\frac {1}{3} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)*(1 + x^3)^(1/3))/x^4,x]

[Out]

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

Rule 31

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), 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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 98, normalized size = 1.01 \begin {gather*} \frac {\sqrt [3]{x^3+1}}{3 x^3}+\sqrt [3]{x^3+1}+\frac {2}{9} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {1}{9} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^3)^(1/3) + (1 + x^3)^(1/3)/(3*x^3) - (2*ArcTan[(1 + 2*(1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (2*Log[1
- (1 + x^3)^(1/3)])/9 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/9

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IntegrateAlgebraic [A]  time = 0.08, size = 97, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{1+x^3} \left (1+3 x^3\right )}{3 x^3}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{9} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(1/3))/x^4,x]

[Out]

((1 + x^3)^(1/3)*(1 + 3*x^3))/(3*x^3) - (2*ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (2*L
og[-1 + (1 + x^3)^(1/3)])/9 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/9

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fricas [A]  time = 0.53, size = 85, normalized size = 0.88 \begin {gather*} -\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (3 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(2*sqrt(3)*x^3*arctan(2/3*sqrt(3)*(x^3 + 1)^(1/3) + 1/3*sqrt(3)) + x^3*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1
/3) + 1) - 2*x^3*log((x^3 + 1)^(1/3) - 1) - 3*(3*x^3 + 1)*(x^3 + 1)^(1/3))/x^3

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giac [A]  time = 0.18, size = 74, normalized size = 0.76 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + (x^3 + 1)^(1/3) + 1/3*(x^3 + 1)^(1/3)/x^3 - 1/9*log
((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1) + 2/9*log(abs((x^3 + 1)^(1/3) - 1))

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maple [C]  time = 2.19, size = 71, normalized size = 0.73

method	result	size
risch	\(\frac {3 x^{6}+4 x^{3}+1}{3 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}}+\frac {-\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{27}+\frac {2 \left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{9}}{\Gamma \left (\frac {2}{3}\right )}\)	\(71\)
meijerg	\(\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {-\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\)	\(103\)
trager	\(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (3 x^{3}+1\right )}{3 x^{3}}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+11 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{9}-\frac {2 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{9}-\frac {2 \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-19 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{9}\)	\(368\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*x^6+4*x^3+1)/x^3/(x^3+1)^(2/3)+2/9/GAMMA(2/3)*(-2/3*GAMMA(2/3)*x^3*hypergeom([1,1,5/3],[2,2],-x^3)+(1/6
*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*GAMMA(2/3))

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maxima [A]  time = 0.43, size = 73, normalized size = 0.75 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + (x^3 + 1)^(1/3) + 1/3*(x^3 + 1)^(1/3)/x^3 - 1/9*log
((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1) + 2/9*log((x^3 + 1)^(1/3) - 1)

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mupad [B]  time = 1.24, size = 152, normalized size = 1.57 \begin {gather*} \frac {\ln \left ({\left (x^3+1\right )}^{1/3}-1\right )}{3}-\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+{\left (x^3+1\right )}^{1/3}-\ln \left ({\left (x^3+1\right )}^{1/3}+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\ln \left ({\left (x^3+1\right )}^{1/3}+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\frac {{\left (x^3+1\right )}^{1/3}}{3\,x^3}+\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((x^3 + 1)^(1/3) - 1)/3 - log((x^3 + 1)^(1/3)/9 - 1/9)/9 + (x^3 + 1)^(1/3) - log((x^3 + 1)^(1/3) - (3^(1/2)
*1i)/2 + 1/2)*((3^(1/2)*1i)/18 - 1/18) + log((3^(1/2)*1i)/2 + (x^3 + 1)^(1/3) + 1/2)*((3^(1/2)*1i)/18 + 1/18)
+ (x^3 + 1)^(1/3)/(3*x^3) + log(3*(x^3 + 1)^(1/3) - (3^(1/2)*3i)/2 + 3/2)*((3^(1/2)*1i)/6 - 1/6) - log((3^(1/2
)*3i)/2 + 3*(x^3 + 1)^(1/3) + 3/2)*((3^(1/2)*1i)/6 + 1/6)

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sympy [C]  time = 87.87, size = 65, normalized size = 0.67 \begin {gather*} - \frac {x \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {2}{3}\right )} + \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*gamma(-1/3)*hyper((-1/3, -1/3), (2/3,), exp_polar(I*pi)/x**3)/(3*gamma(2/3)) + gamma(2/3)*hyper((-1/3, 2/3)
, (5/3,), exp_polar(I*pi)/x**3)/(3*x**2*gamma(5/3))

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