3.14.49 \(\int \frac {(-3+x^4) \sqrt [3]{1+x^4}}{x^9} \, dx\)

Optimal. Leaf size=97 \[ \frac {1}{6} \log \left (\sqrt [3]{x^4+1}-1\right )-\frac {1}{12} \log \left (\left (x^4+1\right )^{2/3}+\sqrt [3]{x^4+1}+1\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^4+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\sqrt [3]{x^4+1} \left (3-x^4\right )}{8 x^8} \]

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Rubi [A]  time = 0.06, antiderivative size = 86, normalized size of antiderivative = 0.89, 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, 47, 57, 618, 204, 31} \begin {gather*} -\frac {\sqrt [3]{x^4+1}}{2 x^4}+\frac {1}{4} \log \left (1-\sqrt [3]{x^4+1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {3 \left (x^4+1\right )^{4/3}}{8 x^8}-\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-3 + x^4)*(1 + x^4)^(1/3))/x^9,x]

[Out]

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

Rule 31

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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 (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(-3+x) \sqrt [3]{1+x}}{x^3} \, dx,x,x^4\right )\\ &=\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}-\frac {\log (x)}{3}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^4}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^4}\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}-\frac {\log (x)}{3}+\frac {1}{4} \log \left (1-\sqrt [3]{1+x^4}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^4}\right )\\ &=-\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{4} \log \left (1-\sqrt [3]{1+x^4}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 35, normalized size = 0.36 \begin {gather*} \frac {3 \left (x^4+1\right )^{4/3} \left (x^8 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};x^4+1\right )+1\right )}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-3 + x^4)*(1 + x^4)^(1/3))/x^9,x]

[Out]

(3*(1 + x^4)^(4/3)*(1 + x^8*Hypergeometric2F1[4/3, 2, 7/3, 1 + x^4]))/(8*x^8)

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IntegrateAlgebraic [A]  time = 0.10, size = 97, normalized size = 1.00 \begin {gather*} \frac {\left (3-x^4\right ) \sqrt [3]{1+x^4}}{8 x^8}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (-1+\sqrt [3]{1+x^4}\right )-\frac {1}{12} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(1/3))/x^9,x]

[Out]

((3 - x^4)*(1 + x^4)^(1/3))/(8*x^8) - ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[-1 + (
1 + x^4)^(1/3)]/6 - Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/12

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fricas [A]  time = 0.46, size = 84, normalized size = 0.87 \begin {gather*} -\frac {4 \, \sqrt {3} x^{8} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 2 \, x^{8} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, x^{8} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^4+1)^(1/3)/x^9,x, algorithm="fricas")

[Out]

-1/24*(4*sqrt(3)*x^8*arctan(2/3*sqrt(3)*(x^4 + 1)^(1/3) + 1/3*sqrt(3)) + 2*x^8*log((x^4 + 1)^(2/3) + (x^4 + 1)
^(1/3) + 1) - 4*x^8*log((x^4 + 1)^(1/3) - 1) + 3*(x^4 + 1)^(1/3)*(x^4 - 3))/x^8

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giac [A]  time = 0.24, size = 76, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} - 4 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{8 \, x^{8}} - \frac {1}{12} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^4+1)^(1/3)/x^9,x, algorithm="giac")

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 + 1)^(1/3) + 1)) - 1/8*((x^4 + 1)^(4/3) - 4*(x^4 + 1)^(1/3))/x^8 - 1/1
2*log((x^4 + 1)^(2/3) + (x^4 + 1)^(1/3) + 1) + 1/6*log((x^4 + 1)^(1/3) - 1)

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maple [C]  time = 3.91, size = 69, normalized size = 0.71

method	result	size
risch	\(-\frac {x^{8}-2 x^{4}-3}{8 x^{8} \left (x^{4}+1\right )^{\frac {2}{3}}}+\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{6 \Gamma \left (\frac {2}{3}\right )}\)	\(69\)
meijerg	\(\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}-\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}\)	\(115\)
trager	\(-\frac {\left (x^{4}-3\right ) \left (x^{4}+1\right )^{\frac {1}{3}}}{8 x^{8}}+\frac {\ln \left (-\frac {333 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+393 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+60 x^{4}-351 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-333 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-144 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+48 \left (x^{4}+1\right )^{\frac {2}{3}}+384 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-165 \left (x^{4}+1\right )^{\frac {1}{3}}+80}{x^{4}}\right )}{6}+\frac {\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-162 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+40 x^{4}-351 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-153 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+495 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}-165 \left (x^{4}+1\right )^{\frac {2}{3}}-195 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+48 \left (x^{4}+1\right )^{\frac {1}{3}}+100}{x^{4}}\right )}{2}\)	\(299\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3)*(x^4+1)^(1/3)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/8*(x^8-2*x^4-3)/x^8/(x^4+1)^(2/3)+1/6/GAMMA(2/3)*(-2/3*GAMMA(2/3)*x^4*hypergeom([1,1,5/3],[2,2],-x^4)+(1/6*
Pi*3^(1/2)-3/2*ln(3)+4*ln(x))*GAMMA(2/3))

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maxima [A]  time = 0.41, size = 103, normalized size = 1.06 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{8 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} - \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} - \frac {1}{12} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^4+1)^(1/3)/x^9,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.27, size = 121, normalized size = 1.25 \begin {gather*} \frac {\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{16}-\frac {1}{16}\right )}{6}-\frac {\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {{\left (x^4+1\right )}^{4/3}}{8}}{2\,x^4-{\left (x^4+1\right )}^2+1}-\frac {{\left (x^4+1\right )}^{1/3}}{4\,x^4}+\ln \left (\frac {3\,{\left (x^4+1\right )}^{1/3}}{4}+\frac {3}{8}-\frac {\sqrt {3}\,3{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (\frac {3\,{\left (x^4+1\right )}^{1/3}}{4}+\frac {3}{8}+\frac {\sqrt {3}\,3{}\mathrm {i}}{8}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 1)^(1/3)*(x^4 - 3))/x^9,x)

[Out]

log((x^4 + 1)^(1/3)/16 - 1/16)/6 - ((x^4 + 1)^(1/3)/4 + (x^4 + 1)^(4/3)/8)/(2*x^4 - (x^4 + 1)^2 + 1) - (x^4 +
1)^(1/3)/(4*x^4) + log((3*(x^4 + 1)^(1/3))/4 - (3^(1/2)*3i)/8 + 3/8)*((3^(1/2)*1i)/12 - 1/12) - log((3^(1/2)*3
i)/8 + (3*(x^4 + 1)^(1/3))/4 + 3/8)*((3^(1/2)*1i)/12 + 1/12)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-3)*(x**4+1)**(1/3)/x**9,x)

[Out]

Timed out

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