3.16.53 \(\int \frac {(1+2 x^6) \sqrt [3]{x+x^3-x^7}}{(-1+x^6)^2} \, dx\)
Optimal. Leaf size=126 \[ -\frac {1}{6} \log \left (\sqrt [3]{-x^7+x^3+x}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^7+x^3+x}+x}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{-x^7+x^3+x} x}{2 \left (x^6-1\right )}+\frac {1}{12} \log \left (x^2+\sqrt [3]{-x^7+x^3+x} x+\left (-x^7+x^3+x\right )^{2/3}\right ) \]
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Rubi [F] time = 8.91, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]
[Out]
(2*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(-1 + I*Sqrt[3] - 2*x)^2, x], x, x^(2/3
)])/(9*x^(1/3)*(1 + x^2 - x^6)^(1/3)) - (((2*I)/9)*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^
9)^(1/3)/(-1 + I*Sqrt[3] - 2*x), x], x, x^(2/3)])/(Sqrt[3]*x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((x + x^3 - x^7)^(
1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(-1 + x)^2, x], x, x^(2/3)])/(18*x^(1/3)*(1 + x^2 - x^6)^(1
/3)) - ((x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(-1 + x), x], x, x^(2/3)])/(18*x^(
1/3)*(1 + x^2 - x^6)^(1/3)) + ((3 - I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(
1/3)/(1 - I*Sqrt[3] + 2*x), x], x, x^(2/3)])/(54*x^(1/3)*(1 + x^2 - x^6)^(1/3)) + (2*(x + x^3 - x^7)^(1/3)*Def
er[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(1 + I*Sqrt[3] + 2*x)^2, x], x, x^(2/3)])/(9*x^(1/3)*(1 + x^2 - x^6
)^(1/3)) - (((2*I)/9)*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(1 + I*Sqrt[3] + 2*x
), x], x, x^(2/3)])/(Sqrt[3]*x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((3 + I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Sub
st][Defer[Int][(1 + x^3 - x^9)^(1/3)/(1 + I*Sqrt[3] + 2*x), x], x, x^(2/3)])/(54*x^(1/3)*(1 + x^2 - x^6)^(1/3)
) - (2*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 + x^3 - x^9)^(1/3))/(-1 + I*Sqrt[3] - 2*x^3)^2, x],
x, x^(2/3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((1 - I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int]
[(x*(1 + x^3 - x^9)^(1/3))/(-1 + I*Sqrt[3] - 2*x^3)^2, x], x, x^(2/3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3)) - (2*(
x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 + x^3 - x^9)^(1/3))/(1 + I*Sqrt[3] + 2*x^3)^2, x], x, x^(2/
3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((1 + I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 +
x^3 - x^9)^(1/3))/(1 + I*Sqrt[3] + 2*x^3)^2, x], x, x^(2/3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3))
Rubi steps
\begin {align*} \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx &=\frac {\sqrt [3]{x+x^3-x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2-x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6-x^{18}} \left (1+2 x^{18}\right )}{\left (-1+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9} \left (1+2 x^9\right )}{\left (-1+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1+x^3-x^9}}{27 (-1+x)^2}-\frac {\sqrt [3]{1+x^3-x^9}}{27 (-1+x)}-\frac {\sqrt [3]{1+x^3-x^9}}{9 \left (1+x+x^2\right )^2}+\frac {(1+x) \sqrt [3]{1+x^3-x^9}}{27 \left (1+x+x^2\right )}+\frac {x \left (1+x^3\right ) \sqrt [3]{1+x^3-x^9}}{\left (1+x^3+x^6\right )^2}-\frac {x \sqrt [3]{1+x^3-x^9}}{3 \left (1+x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}
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Mathematica [F] time = 2.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]
[Out]
Integrate[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2, x]
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IntegrateAlgebraic [A] time = 0.22, size = 126, normalized size = 1.00 \begin {gather*} -\frac {1}{6} \log \left (\sqrt [3]{-x^7+x^3+x}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^7+x^3+x}+x}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{-x^7+x^3+x} x}{2 \left (x^6-1\right )}+\frac {1}{12} \log \left (x^2+\sqrt [3]{-x^7+x^3+x} x+\left (-x^7+x^3+x\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]
[Out]
-1/2*(x*(x + x^3 - x^7)^(1/3))/(-1 + x^6) - ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3 - x^7)^(1/3))]/(2*Sqrt[3]) - Lo
g[-x + (x + x^3 - x^7)^(1/3)]/6 + Log[x^2 + x*(x + x^3 - x^7)^(1/3) + (x + x^3 - x^7)^(2/3)]/12
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fricas [A] time = 1.08, size = 151, normalized size = 1.20 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x^{6} - 1\right )} \arctan \left (-\frac {4 \, \sqrt {3} {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{6} - x^{2} - 1\right )} - 2 \, \sqrt {3} {\left (-x^{7} + x^{3} + x\right )}^{\frac {2}{3}}}{x^{6} - 9 \, x^{2} - 1}\right ) - {\left (x^{6} - 1\right )} \log \left (\frac {x^{6} - 3 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} - 1}\right ) - 6 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x}{12 \, {\left (x^{6} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="fricas")
[Out]
1/12*(2*sqrt(3)*(x^6 - 1)*arctan(-(4*sqrt(3)*(-x^7 + x^3 + x)^(1/3)*x - sqrt(3)*(x^6 - x^2 - 1) - 2*sqrt(3)*(-
x^7 + x^3 + x)^(2/3))/(x^6 - 9*x^2 - 1)) - (x^6 - 1)*log((x^6 - 3*(-x^7 + x^3 + x)^(1/3)*x + 3*(-x^7 + x^3 + x
)^(2/3) - 1)/(x^6 - 1)) - 6*(-x^7 + x^3 + x)^(1/3)*x)/(x^6 - 1)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + 1\right )}}{{\left (x^{6} - 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="giac")
[Out]
integrate((-x^7 + x^3 + x)^(1/3)*(2*x^6 + 1)/(x^6 - 1)^2, x)
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maple [C] time = 19.72, size = 1137, normalized size = 9.02
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x)
[Out]
-1/2*x/(x^6-1)*(-x*(x^6-x^2-1))^(1/3)+(1/6*ln(-(10282164833688*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)*RootOf(
_Z^2+59*_Z+3481)*x^2-10282164833688*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)*RootOf(_Z^2+59*_Z+3481)*x^6+162990
246567115*x^12-325980493134230*x^6+314126657020258*x^2-314126657020258*x^8+151136410453143*x^4-11413752794033*
RootOf(_Z^2+59*_Z+3481)+13821682980*RootOf(_Z^2+59*_Z+3481)^2*x^12+13821682980*RootOf(_Z^2+59*_Z+3481)^2+24975
485320739*RootOf(_Z^2+59*_Z+3481)*x^8+22827505588066*RootOf(_Z^2+59*_Z+3481)*x^6-13561732526706*RootOf(_Z^2+59
*_Z+3481)*x^4-24975485320739*RootOf(_Z^2+59*_Z+3481)*x^2-11413752794033*RootOf(_Z^2+59*_Z+3481)*x^12-231513189
915*RootOf(_Z^2+59*_Z+3481)^2*x^8-27643365960*RootOf(_Z^2+59*_Z+3481)^2*x^6+217691506935*RootOf(_Z^2+59*_Z+348
1)^2*x^4+231513189915*RootOf(_Z^2+59*_Z+3481)^2*x^2-951278629528797*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)+15
57926354716389*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(2/3)+951278629528797*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)
*x^6+16123366602183*RootOf(_Z^2+59*_Z+3481)*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(2/3)-951278629528797*(x^14-2*x^
10-2*x^8+x^6+2*x^4+x^2)^(1/3)*x^2+10282164833688*RootOf(_Z^2+59*_Z+3481)*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/
3)+162990246567115)/(x^6-x^2-1)/(-1+x)/(1+x)/(x^2+x+1)/(x^2-x+1))+1/354*RootOf(_Z^2+59*_Z+3481)*ln((-264055314
35871*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)*RootOf(_Z^2+59*_Z+3481)*x^2+26405531435871*(x^14-2*x^10-2*x^8+x^
6+2*x^4+x^2)^(1/3)*RootOf(_Z^2+59*_Z+3481)*x^6+709670857187355*x^12-1419341714374710*x^6+2080899293108685*x^2-
2080899293108685*x^8+1371228435921330*x^4-2147979732673*RootOf(_Z^2+59*_Z+3481)-3405296212*RootOf(_Z^2+59*_Z+3
481)^2*x^12-3405296212*RootOf(_Z^2+59*_Z+3481)^2-4969813596014*RootOf(_Z^2+59*_Z+3481)*x^8+4295959465346*RootO
f(_Z^2+59*_Z+3481)*x^6+7117793328687*RootOf(_Z^2+59*_Z+3481)*x^4+4969813596014*RootOf(_Z^2+59*_Z+3481)*x^2-214
7979732673*RootOf(_Z^2+59*_Z+3481)*x^12+57038711551*RootOf(_Z^2+59*_Z+3481)^2*x^8+6810592424*RootOf(_Z^2+59*_Z
+3481)^2*x^6-53633415339*RootOf(_Z^2+59*_Z+3481)^2*x^4-57038711551*RootOf(_Z^2+59*_Z+3481)^2*x^2-9512786295287
97*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)-606647725187592*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(2/3)+95127862952
8797*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)*x^6+16123366602183*RootOf(_Z^2+59*_Z+3481)*(x^14-2*x^10-2*x^8+x^6
+2*x^4+x^2)^(2/3)-951278629528797*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)*x^2-26405531435871*RootOf(_Z^2+59*_Z
+3481)*(x^14-2*x^10-2*x^8+x^6+2*x^4+x^2)^(1/3)+709670857187355)/(x^6-x^2-1)/(-1+x)/(1+x)/(x^2+x+1)/(x^2-x+1)))
*(-x*(x^6-x^2-1))^(1/3)*(x^2*(x^6-x^2-1)^2)^(1/3)/x/(x^6-x^2-1)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + 1\right )}}{{\left (x^{6} - 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="maxima")
[Out]
integrate((-x^7 + x^3 + x)^(1/3)*(2*x^6 + 1)/(x^6 - 1)^2, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^6+1\right )\,{\left (-x^7+x^3+x\right )}^{1/3}}{{\left (x^6-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((2*x^6 + 1)*(x + x^3 - x^7)^(1/3))/(x^6 - 1)^2,x)
[Out]
int(((2*x^6 + 1)*(x + x^3 - x^7)^(1/3))/(x^6 - 1)^2, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{- x \left (x^{6} - x^{2} - 1\right )} \left (2 x^{6} + 1\right )}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x**6+1)*(-x**7+x**3+x)**(1/3)/(x**6-1)**2,x)
[Out]
Integral((-x*(x**6 - x**2 - 1))**(1/3)*(2*x**6 + 1)/((x - 1)**2*(x + 1)**2*(x**2 - x + 1)**2*(x**2 + x + 1)**2
), x)
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