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

Optimal. Leaf size=109 \[ \frac {1}{24} \text {RootSum}\left [2 \text {$\#$1}^6-4 \text {$\#$1}^3+1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+2 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-2 \log (x)}{\text {$\#$1}^4-\text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (1-4 x^3\right )}{10 x^5} \]

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Rubi [A]  time = 0.51, antiderivative size = 157, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {28, 6725, 264, 277, 239, 429} \begin {gather*} -\frac {1}{8} \left (3+2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{\sqrt {2}}\right )-\frac {1}{8} \left (3-2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};\frac {x^3}{\sqrt {2}},-x^3\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {\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}}{10 x^5}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + (1 + x^3)^(5/3)/(10*x^5) - ((3 + 2*Sqrt[2])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, -(x
^3/Sqrt[2])])/8 - ((3 - 2*Sqrt[2])*x*AppellF1[1/3, 1, -2/3, 4/3, x^3/Sqrt[2], -x^3])/8 + ArcTan[(1 + (2*x)/(1
+ x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/2

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 6725

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

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx &=\int \frac {\left (-1+x^3\right )^2 \left (1+x^3\right )^{2/3}}{x^6 \left (-2+x^6\right )} \, dx\\ &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (3-2 x^3\right ) \left (1+x^3\right )^{2/3}}{2 \left (-2+x^6\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\right )+\frac {1}{2} \int \frac {\left (3-2 x^3\right ) \left (1+x^3\right )^{2/3}}{-2+x^6} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{2} \int \left (-\frac {\left (3-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (\sqrt {2}-x^3\right )}+\frac {\left (-3-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (\sqrt {2}+x^3\right )}\right ) \, dx+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{8} \left (4-3 \sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\sqrt {2}-x^3} \, dx-\frac {1}{8} \left (4+3 \sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\sqrt {2}+x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}-\frac {1}{8} \left (3+2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{\sqrt {2}}\right )-\frac {1}{8} \left (3-2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};\frac {x^3}{\sqrt {2}},-x^3\right )+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [F]  time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 109, normalized size = 1.00 \begin {gather*} \frac {\left (1-4 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{24} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 4*x^3)*(1 + x^3)^(2/3))/(10*x^5) + RootSum[1 - 4*#1^3 + 2*#1^6 & , (-2*Log[x] + 2*Log[(1 + x^3)^(1/3) -
x*#1] + Log[x]*#1^3 - Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/24

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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^6-2*x^3+1)/x^6/(x^6-2),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} - 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2\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^6-2*x^3+1)/x^6/(x^6-2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 239.11, size = 6590, normalized size = 60.46

method	result	size
risch	\(\text {Expression too large to display}\)	\(6590\)
trager	\(\text {Expression too large to display}\)	\(9174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(x^6-2*x^3+1)/x^6/(x^6-2),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} - 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2\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^6-2*x^3+1)/x^6/(x^6-2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6-2\,x^3+1\right )}{x^6\,\left (x^6-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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