3.17.14 \(\int \frac {(-1+x^3)^{2/3} (2+x^3)}{x^6 (4+x^6)} \, dx\)

Optimal. Leaf size=110 \[ \frac {1}{96} \text {RootSum}\left [4 \text {$\#$1}^6-8 \text {$\#$1}^3+5\& ,\frac {-6 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+6 \text {$\#$1}^3 \log (x)+5 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-5 \log (x)}{\text {$\#$1}^4-\text {$\#$1}}\& \right ]+\frac {\left (x^3-1\right )^{2/3} \left (-x^3-4\right )}{40 x^5} \]

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Rubi [C]  time = 0.41, antiderivative size = 185, normalized size of antiderivative = 1.68, number of steps used = 11, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6725, 264, 277, 239, 430, 429} \begin {gather*} -\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {i x^3}{2}\right )}{\left (1-x^3\right )^{2/3}}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};-\frac {i x^3}{2},x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {1}{8} \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 )}{4 \sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{8 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)),x]

[Out]

-1/8*(-1 + x^3)^(2/3)/x^2 + (-1 + x^3)^(5/3)/(10*x^5) - ((1/16 - I/16)*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3,
1, 4/3, x^3, (I/2)*x^3])/(1 - x^3)^(2/3) - ((1/16 + I/16)*x*(-1 + x^3)^(2/3)*AppellF1[1/3, 1, -2/3, 4/3, (-1/2
*I)*x^3, x^3])/(1 - x^3)^(2/3) + ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(4*Sqrt[3]) - Log[-x + (-1 + x^3
)^(1/3)]/8

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 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 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 (2+x^3\right )}{x^6 \left (4+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}}{4 x^3}+\frac {\left (-2-x^3\right ) \left (-1+x^3\right )^{2/3}}{4 \left (4+x^6\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{4} \int \frac {\left (-2-x^3\right ) \left (-1+x^3\right )^{2/3}}{4+x^6} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{4} \int \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-1+x^3\right )^{2/3}}{2 i-x^3}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-1+x^3\right )^{2/3}}{2 i+x^3}\right ) \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 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 )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\left (-\frac {1}{8}-\frac {i}{8}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2 i+x^3} \, dx+\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2 i-x^3} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 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 )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+-\frac {\left (\left (\frac {1}{8}+\frac {i}{8}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2 i+x^3} \, dx}{\left (1-x^3\right )^{2/3}}+\frac {\left (\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2 i-x^3} \, dx}{\left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {i x^3}{2}\right )}{\left (1-x^3\right )^{2/3}}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};-\frac {i x^3}{2},x^3\right )}{\left (1-x^3\right )^{2/3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \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 (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)),x]

[Out]

Integrate[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)), x]

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IntegrateAlgebraic [A]  time = 0.00, size = 110, normalized size = 1.00 \begin {gather*} \frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{96} \text {RootSum}\left [5-8 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)),x]

[Out]

((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[5 - 8*#1^3 + 4*#1^6 & , (-5*Log[x] + 5*Log[(-1 + x^3)^(1/3) -
x*#1] + 6*Log[x]*#1^3 - 6*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/96

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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^3+2)/x^6/(x^6+4),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^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4\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^3+2)/x^6/(x^6+4),x, algorithm="giac")

[Out]

integrate((x^3 + 2)*(x^3 - 1)^(2/3)/((x^6 + 4)*x^6), x)

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maple [B]  time = 269.17, size = 6835, normalized size = 62.14

method	result	size
risch	\(\text {Expression too large to display}\)	\(6835\)
trager	\(\text {Expression too large to display}\)	\(10408\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^3+2)/x^6/(x^6+4),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^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4\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^3+2)/x^6/(x^6+4),x, algorithm="maxima")

[Out]

integrate((x^3 + 2)*(x^3 - 1)^(2/3)/((x^6 + 4)*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^3+2\right )}{x^6\,\left (x^6+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(x^3 + 2))/(x^6*(x^6 + 4)),x)

[Out]

int(((x^3 - 1)^(2/3)*(x^3 + 2))/(x^6*(x^6 + 4)), 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^{3} + 2\right )}{x^{6} \left (x^{6} + 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(2/3)*(x**3+2)/x**6/(x**6+4),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**3 + 2)/(x**6*(x**6 + 4)), x)

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