3.30.8 \(\int \frac {(1+x^3)^{2/3} (-1+x^6)}{x^6 (-1-2 x^3+2 x^6)} \, dx\)

Optimal. Leaf size=325 \[ \frac {\log \left (3^{5/6} \sqrt [3]{x^3+1}-3 x\right )}{2\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}}+\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (3^{5/6} \sqrt [3]{x^3+1}+3 x\right )}{2\ 3^{2/3}}-\frac {1}{2} \sqrt [3]{\frac {1}{3} \left (45+26 \sqrt {3}\right )} \tan ^{-1}\left (\frac {3^{2/3} x}{\sqrt [6]{3} x-2 \sqrt [3]{x^3+1}}\right )-\frac {\tan ^{-1}\left (\frac {3^{2/3} x}{2 \sqrt [3]{x^3+1}+\sqrt [6]{3} x}\right )}{2 \sqrt [3]{45+26 \sqrt {3}}}+\frac {\left (x^3+1\right )^{2/3} \left (4 x^3-1\right )}{5 x^5}-\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (3^{5/6} \sqrt [3]{x^3+1} x-3^{2/3} \left (x^3+1\right )^{2/3}-3 x^2\right )}{4\ 3^{2/3}}-\frac {\log \left (3^{5/6} \sqrt [3]{x^3+1} x+3^{2/3} \left (x^3+1\right )^{2/3}+3 x^2\right )}{4\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}} \]

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Rubi [C]  time = 0.56, antiderivative size = 176, normalized size of antiderivative = 0.54, number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6728, 264, 277, 239, 429} \begin {gather*} -\frac {\left (2+\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {3}}\right )}{1-\sqrt {3}}-\frac {\left (2-\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {3}}\right )}{1+\sqrt {3}}+\log \left (\sqrt [3]{x^3+1}-x\right )-\frac {2 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^3+1\right )^{5/3}}{5 x^5}+\frac {\left (x^3+1\right )^{2/3}}{x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-1 - 2*x^3 + 2*x^6)),x]

[Out]

(1 + x^3)^(2/3)/x^2 - (1 + x^3)^(5/3)/(5*x^5) - ((2 + Sqrt[3])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, (2*x^3)/(1
- Sqrt[3])])/(1 - Sqrt[3]) - ((2 - Sqrt[3])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, (2*x^3)/(1 + Sqrt[3])])/(1 + S
qrt[3]) - (2*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + Log[-x + (1 + x^3)^(1/3)]

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^6}-\frac {2 \left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (1+x^3\right )^{2/3} \left (-5+4 x^3\right )}{-1-2 x^3+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\right )+\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {\left (1+x^3\right )^{2/3} \left (-5+4 x^3\right )}{-1-2 x^3+2 x^6} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-2 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (4-2 \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{-2-2 \sqrt {3}+4 x^3}+\frac {\left (4+2 \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{-2+2 \sqrt {3}+4 x^3}\right ) \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (-x+\sqrt [3]{1+x^3}\right )+\left (2 \left (2-\sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-2-2 \sqrt {3}+4 x^3} \, dx+\left (2 \left (2+\sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-2+2 \sqrt {3}+4 x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\left (2+\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {3}}\right )}{1-\sqrt {3}}-\frac {\left (2-\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {3}}\right )}{1+\sqrt {3}}-\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.75, size = 265, normalized size = 0.82 \begin {gather*} \frac {-12 \left (\sqrt {3}-2\right ) \log \left (1-\frac {\sqrt [6]{3} x}{\sqrt [3]{x^3+1}}\right )+12 \left (2+\sqrt {3}\right ) \log \left (\frac {\sqrt [6]{3} x}{\sqrt [3]{x^3+1}}+1\right )-\left (1+\sqrt {3}\right ) \left (3+\sqrt {3}\right ) \left (6 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 x}{\sqrt [3]{3} \sqrt [3]{x^3+1}}\right )+\sqrt {3} \log \left (-\frac {\sqrt [6]{3} x}{\sqrt [3]{x^3+1}}+\frac {\sqrt [3]{3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )\right )+\left (\sqrt {3}-3\right ) \left (\sqrt {3}-1\right ) \left (6 \tan ^{-1}\left (\frac {2 x}{\sqrt [3]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )+\sqrt {3} \log \left (\frac {\sqrt [6]{3} x}{\sqrt [3]{x^3+1}}+\frac {\sqrt [3]{3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )\right )}{24\ 3^{2/3}}+\left (x^3+1\right )^{2/3} \left (\frac {4}{5 x^2}-\frac {1}{5 x^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-1 - 2*x^3 + 2*x^6)),x]

[Out]

(-1/5*1/x^5 + 4/(5*x^2))*(1 + x^3)^(2/3) + (-12*(-2 + Sqrt[3])*Log[1 - (3^(1/6)*x)/(1 + x^3)^(1/3)] - (1 + Sqr
t[3])*(3 + Sqrt[3])*(6*ArcTan[1/Sqrt[3] - (2*x)/(3^(1/3)*(1 + x^3)^(1/3))] + Sqrt[3]*Log[1 + (3^(1/3)*x^2)/(1
+ x^3)^(2/3) - (3^(1/6)*x)/(1 + x^3)^(1/3)]) + 12*(2 + Sqrt[3])*Log[1 + (3^(1/6)*x)/(1 + x^3)^(1/3)] + (-3 + S
qrt[3])*(-1 + Sqrt[3])*(6*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/3)*(1 + x^3)^(1/3))] + Sqrt[3]*Log[1 + (3^(1/3)*x^2)/
(1 + x^3)^(2/3) + (3^(1/6)*x)/(1 + x^3)^(1/3)]))/(24*3^(2/3))

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IntegrateAlgebraic [A]  time = 2.05, size = 325, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (-1+4 x^3\right )}{5 x^5}-\frac {1}{2} \sqrt [3]{\frac {1}{3} \left (45+26 \sqrt {3}\right )} \tan ^{-1}\left (\frac {3^{2/3} x}{\sqrt [6]{3} x-2 \sqrt [3]{1+x^3}}\right )-\frac {\tan ^{-1}\left (\frac {3^{2/3} x}{\sqrt [6]{3} x+2 \sqrt [3]{1+x^3}}\right )}{2 \sqrt [3]{45+26 \sqrt {3}}}+\frac {\log \left (-3 x+3^{5/6} \sqrt [3]{1+x^3}\right )}{2\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}}+\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (3 x+3^{5/6} \sqrt [3]{1+x^3}\right )}{2\ 3^{2/3}}-\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (-3 x^2+3^{5/6} x \sqrt [3]{1+x^3}-3^{2/3} \left (1+x^3\right )^{2/3}\right )}{4\ 3^{2/3}}-\frac {\log \left (3 x^2+3^{5/6} x \sqrt [3]{1+x^3}+3^{2/3} \left (1+x^3\right )^{2/3}\right )}{4\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-1 - 2*x^3 + 2*x^6)),x]

[Out]

((1 + x^3)^(2/3)*(-1 + 4*x^3))/(5*x^5) - (((45 + 26*Sqrt[3])/3)^(1/3)*ArcTan[(3^(2/3)*x)/(3^(1/6)*x - 2*(1 + x
^3)^(1/3))])/2 - ArcTan[(3^(2/3)*x)/(3^(1/6)*x + 2*(1 + x^3)^(1/3))]/(2*(45 + 26*Sqrt[3])^(1/3)) + Log[-3*x +
3^(5/6)*(1 + x^3)^(1/3)]/(2*3^(2/3)*(26 + 15*Sqrt[3])^(1/3)) + ((26 + 15*Sqrt[3])^(1/3)*Log[3*x + 3^(5/6)*(1 +
x^3)^(1/3)])/(2*3^(2/3)) - ((26 + 15*Sqrt[3])^(1/3)*Log[-3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) - 3^(2/3)*(1 + x^3
)^(2/3)])/(4*3^(2/3)) - Log[3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) + 3^(2/3)*(1 + x^3)^(2/3)]/(4*3^(2/3)*(26 + 15*S
qrt[3])^(1/3))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x, algorithm="giac")

[Out]

integrate((x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^6 - 2*x^3 - 1)*x^6), x)

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maple [C]  time = 178.67, size = 8453, normalized size = 26.01

method	result	size
risch	\(\text {Expression too large to display}\)	\(8453\)
trager	\(\text {Expression too large to display}\)	\(11228\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x, algorithm="maxima")

[Out]

integrate((x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^6 - 2*x^3 - 1)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6-1\right )}{x^6\,\left (-2\,x^6+2\,x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^3 + 1)^(2/3)*(x^6 - 1))/(x^6*(2*x^3 - 2*x^6 + 1)),x)

[Out]

-int(((x^3 + 1)^(2/3)*(x^6 - 1))/(x^6*(2*x^3 - 2*x^6 + 1)), x)

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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**3+1)**(2/3)*(x**6-1)/x**6/(2*x**6-2*x**3-1),x)

[Out]

Timed out

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