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

Optimal. Leaf size=133 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^6+1}-x\right )}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^6+1}+x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\left (x^6+1\right )^{2/3}}{4 x^2}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^6+1} x+2^{2/3} \left (x^6+1\right )^{2/3}+x^2\right )}{12\ 2^{2/3}} \]

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Rubi [C]  time = 0.89, antiderivative size = 241, normalized size of antiderivative = 1.81, number of steps used = 16, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6728, 275, 364, 1438, 429, 465, 510} \begin {gather*} \frac {\left (-\sqrt {15}+i\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {8 x^6}{7-i \sqrt {15}},-x^6\right )}{\sqrt {15}+7 i}+\frac {\left (\sqrt {15}+i\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {8 x^6}{7+i \sqrt {15}},-x^6\right )}{-\sqrt {15}+7 i}+\frac {x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {8 x^6}{7-i \sqrt {15}}\right )}{7-i \sqrt {15}}+\frac {x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {8 x^6}{7+i \sqrt {15}}\right )}{7+i \sqrt {15}}+\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{4 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I - Sqrt[15])*x*AppellF1[1/6, 1, -2/3, 7/6, (-8*x^6)/(7 - I*Sqrt[15]), -x^6])/(7*I + Sqrt[15]) + ((I + Sqrt[
15])*x*AppellF1[1/6, 1, -2/3, 7/6, (-8*x^6)/(7 + I*Sqrt[15]), -x^6])/(7*I - Sqrt[15]) + (x^4*AppellF1[2/3, -2/
3, 1, 5/3, -x^6, (-8*x^6)/(7 - I*Sqrt[15])])/(7 - I*Sqrt[15]) + (x^4*AppellF1[2/3, -2/3, 1, 5/3, -x^6, (-8*x^6
)/(7 + I*Sqrt[15])])/(7 + I*Sqrt[15]) + Hypergeometric2F1[-2/3, -1/3, 2/3, -x^6]/(4*x^2)

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 510

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

Rule 1438

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 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^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx &=\int \left (-\frac {\left (1+x^6\right )^{2/3}}{2 x^3}+\frac {\left (-1+4 x^3\right ) \left (1+x^6\right )^{2/3}}{2 \left (2-x^3+2 x^6\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (1+x^6\right )^{2/3}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+4 x^3\right ) \left (1+x^6\right )^{2/3}}{2-x^3+2 x^6} \, dx\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {\left (1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+\frac {1}{2} \int \left (\frac {4 \left (1+x^6\right )^{2/3}}{-1-i \sqrt {15}+4 x^3}+\frac {4 \left (1+x^6\right )^{2/3}}{-1+i \sqrt {15}+4 x^3}\right ) \, dx\\ &=\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{4 x^2}+2 \int \frac {\left (1+x^6\right )^{2/3}}{-1-i \sqrt {15}+4 x^3} \, dx+2 \int \frac {\left (1+x^6\right )^{2/3}}{-1+i \sqrt {15}+4 x^3} \, dx\\ &=\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{4 x^2}+2 \int \left (\frac {\left (i-\sqrt {15}\right ) \left (1+x^6\right )^{2/3}}{2 \left (7 i+\sqrt {15}+8 i x^6\right )}+\frac {2 x^3 \left (1+x^6\right )^{2/3}}{7-i \sqrt {15}+8 x^6}\right ) \, dx+2 \int \left (\frac {\left (-i-\sqrt {15}\right ) \left (1+x^6\right )^{2/3}}{2 \left (-7 i+\sqrt {15}-8 i x^6\right )}+\frac {2 x^3 \left (1+x^6\right )^{2/3}}{7+i \sqrt {15}+8 x^6}\right ) \, dx\\ &=\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{4 x^2}+4 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{7-i \sqrt {15}+8 x^6} \, dx+4 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{7+i \sqrt {15}+8 x^6} \, dx+\left (-i-\sqrt {15}\right ) \int \frac {\left (1+x^6\right )^{2/3}}{-7 i+\sqrt {15}-8 i x^6} \, dx+\left (i-\sqrt {15}\right ) \int \frac {\left (1+x^6\right )^{2/3}}{7 i+\sqrt {15}+8 i x^6} \, dx\\ &=\frac {\left (i-\sqrt {15}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {8 x^6}{7-i \sqrt {15}},-x^6\right )}{7 i+\sqrt {15}}+\frac {\left (i+\sqrt {15}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {8 x^6}{7+i \sqrt {15}},-x^6\right )}{7 i-\sqrt {15}}+\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{4 x^2}+2 \operatorname {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{7-i \sqrt {15}+8 x^3} \, dx,x,x^2\right )+2 \operatorname {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{7+i \sqrt {15}+8 x^3} \, dx,x,x^2\right )\\ &=\frac {\left (i-\sqrt {15}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {8 x^6}{7-i \sqrt {15}},-x^6\right )}{7 i+\sqrt {15}}+\frac {\left (i+\sqrt {15}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {8 x^6}{7+i \sqrt {15}},-x^6\right )}{7 i-\sqrt {15}}+\frac {x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {8 x^6}{7-i \sqrt {15}}\right )}{7-i \sqrt {15}}+\frac {x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {8 x^6}{7+i \sqrt {15}}\right )}{7+i \sqrt {15}}+\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{4 x^2}\\ \end {align*}

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Mathematica [F]  time = 1.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.10, size = 133, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^6\right )^{2/3}}{4 x^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^6}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^6}\right )}{6\ 2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^6}+2^{2/3} \left (1+x^6\right )^{2/3}\right )}{12\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 149.19, size = 349, normalized size = 2.62 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{13} + x^{10} + 3 \, x^{7} + x^{4} + 2 \, x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (8 \, x^{18} + 60 \, x^{15} + 48 \, x^{12} + 119 \, x^{9} + 48 \, x^{6} + 60 \, x^{3} + 8\right )} + 12 \, \sqrt {3} {\left (4 \, x^{14} + 14 \, x^{11} + 9 \, x^{8} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{18} - 12 \, x^{15} - 24 \, x^{12} - 25 \, x^{9} - 24 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{6} - x^{3} + 2\right )} - 12 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{6} - x^{3} + 2}\right ) + 4^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{7} + x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (4 \, x^{12} + 14 \, x^{9} + 9 \, x^{6} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{8} + x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{4 \, x^{12} - 4 \, x^{9} + 9 \, x^{6} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{144 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(4*4^(1/6)*sqrt(3)*x^2*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^13 + x^10 + 3*x^7 + x^4 + 2*x)*(x^6
+ 1)^(2/3) + 4^(1/3)*sqrt(3)*(8*x^18 + 60*x^15 + 48*x^12 + 119*x^9 + 48*x^6 + 60*x^3 + 8) + 12*sqrt(3)*(4*x^14
+ 14*x^11 + 9*x^8 + 14*x^5 + 4*x^2)*(x^6 + 1)^(1/3))/(8*x^18 - 12*x^15 - 24*x^12 - 25*x^9 - 24*x^6 - 12*x^3 +
8)) - 2*4^(2/3)*x^2*log(-(6*4^(1/3)*(x^6 + 1)^(1/3)*x^2 + 4^(2/3)*(2*x^6 - x^3 + 2) - 12*(x^6 + 1)^(2/3)*x)/(
2*x^6 - x^3 + 2)) + 4^(2/3)*x^2*log((6*4^(2/3)*(x^7 + x^4 + x)*(x^6 + 1)^(2/3) + 4^(1/3)*(4*x^12 + 14*x^9 + 9*
x^6 + 14*x^3 + 4) + 6*(4*x^8 + x^5 + 4*x^2)*(x^6 + 1)^(1/3))/(4*x^12 - 4*x^9 + 9*x^6 - 4*x^3 + 4)) - 36*(x^6 +
1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 5.09, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{6}-1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{x^{3} \left (2 x^{6}-x^{3}+2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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