3.24.38 \(\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} (1-x^6)} \, dx\)
Optimal. Leaf size=390 \[ \frac {4 \left (x^5-x^3\right )^{3/4}}{3 x^2 \left (x^2-1\right )}+\frac {\sqrt [4]{2} \tan ^{-1}\left (\frac {3^{7/8} \sqrt {2-\sqrt {2}} x \sqrt [4]{x^5-x^3}}{3^{3/4} \sqrt {x^5-x^3}-3 x^2}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tan ^{-1}\left (\frac {3^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{x^5-x^3}}{3^{3/4} \sqrt {x^5-x^3}-3 x^2}\right )+\frac {\sqrt [4]{2} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^5-x^3}}{\sqrt [8]{3} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{x^5-x^3}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {x^5-x^3}}{\sqrt [8]{3} \sqrt {2+\sqrt {2}}}}{x \sqrt [4]{x^5-x^3}}\right ) \]
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Rubi [F] time = 180.00, 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*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
Int[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
$Aborted
Rubi steps
Aborted
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Mathematica [F] time = 1.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
Integrate[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)), x]
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IntegrateAlgebraic [A] time = 4.70, size = 416, normalized size = 1.07 \begin {gather*} \frac {4 \left (x^5-x^3\right )^{3/4}}{3 x^2 \left (x^2-1\right )}+\frac {\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} 3^{7/8} x \sqrt [4]{x^5-x^3}}{\sqrt [8]{17+12 \sqrt {2}} \left (3^{3/4} \sqrt {x^5-x^3}-3 x^2\right )}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} x \sqrt [4]{x^5-x^3}}{3^{3/4} \sqrt {x^5-x^3}-3 x^2}\right )+\frac {\sqrt [4]{2} \tanh ^{-1}\left (\frac {3 \sqrt [8]{\frac {17}{8748}+\frac {\sqrt {2}}{729}} x^2+3^{3/4} \sqrt [8]{\frac {17}{8748}+\frac {\sqrt {2}}{729}} \sqrt {x^5-x^3}}{x \sqrt [4]{x^5-x^3}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{\frac {3}{17+12 \sqrt {2}}} x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^5-x^3}}{\sqrt [4]{2} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}}{x \sqrt [4]{x^5-x^3}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
IntegrateAlgebraic[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
(4*(-x^3 + x^5)^(3/4))/(3*x^2*(-1 + x^2)) + (2^(1/4)*ArcTan[(2^(1/4)*3^(7/8)*x*(-x^3 + x^5)^(1/4))/((17 + 12*S
qrt[2])^(1/8)*(-3*x^2 + 3^(3/4)*Sqrt[-x^3 + x^5]))])/(3*(3*(17 + 12*Sqrt[2]))^(1/8)) + (2^(1/4)*((17 + 12*Sqrt
[2])/3)^(1/8)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x*(-x^3 + x^5)^(1/4))/(-3*x^2 + 3^(3/4)*Sqrt[-x^
3 + x^5])])/3 + (2^(1/4)*ArcTanh[(3*(17/8748 + Sqrt[2]/729)^(1/8)*x^2 + 3^(3/4)*(17/8748 + Sqrt[2]/729)^(1/8)*
Sqrt[-x^3 + x^5])/(x*(-x^3 + x^5)^(1/4))])/(3*(3*(17 + 12*Sqrt[2]))^(1/8)) + (2^(1/4)*((17 + 12*Sqrt[2])/3)^(1
/8)*ArcTanh[(((3/(17 + 12*Sqrt[2]))^(1/8)*x^2)/2^(1/4) + Sqrt[-x^3 + x^5]/(2^(1/4)*(3*(17 + 12*Sqrt[2]))^(1/8)
))/(x*(-x^3 + x^5)^(1/4))])/3
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fricas [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^5-x^3)^(1/4)/(-x^6+1),x, algorithm="fricas")
[Out]
Timed out
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="giac")
[Out]
integrate(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
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maple [C] time = 48.51, size = 2154, normalized size = 5.52 \begin {gather*} \text {Expression too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x)
[Out]
4/3*x/(x^3*(x^2-1))^(1/4)-1/9*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*ln(-108*(16*Root
Of(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^8*x^4-24*RootOf(_Z^2+RootOf(_Z^8
+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^8*x^3-16*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+
RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^8*x^2+54*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^5*RootOf(_Z^2+RootOf(_Z^8+2
187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x-162*(x^5-x^3)^(1/4)*RootOf(_Z^2+Root
Of(_Z^8+2187)^2)*RootOf(_Z^8+2187)^5*x^2-1350*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*
RootOf(_Z^8+2187)^4*x^4-243*RootOf(_Z^8+2187)^4*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)
)*x^3+8100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4+1350*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187
)^2))*RootOf(_Z^8+2187)^4*x^2-24300*(x^5-x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2
187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)*x+72900*(x^5-x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^8+2187)
^2)*RootOf(_Z^8+2187)*x^2+28431*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^4+56862*Root
Of(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^3+39366*(x^5-x^3)^(3/4)-28431*RootOf(_Z^2+RootOf
(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^2)/x^2/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+81*x)/(x*R
ootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+135*x+108))-1/9*RootOf(_Z^8+2187)*ln(108*(16*RootOf(_Z^8+2187)^9*x^4-
24*RootOf(_Z^8+2187)^9*x^3-16*RootOf(_Z^8+2187)^9*x^2+54*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^7*x-162*(x^5-x^3)^(
1/4)*RootOf(_Z^8+2187)^6*x^2+1350*RootOf(_Z^8+2187)^5*x^4+243*x^3*RootOf(_Z^8+2187)^5-1350*RootOf(_Z^8+2187)^5
*x^2-8100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4+24300*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^3*x-72900*(x^5-x^3)^(1/4
)*RootOf(_Z^8+2187)^2*x^2+28431*RootOf(_Z^8+2187)*x^4+56862*RootOf(_Z^8+2187)*x^3-28431*RootOf(_Z^8+2187)*x^2+
39366*(x^5-x^3)^(3/4))/x^2/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4-135*x-108)/(x*RootOf(_Z^8+2187)^4-2*Ro
otOf(_Z^8+2187)^4-81*x))-1/9*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*ln(108*(16*x^4*RootOf(_Z^8+2187)^8*RootOf(_Z^2+R
ootOf(_Z^8+2187)^2)-24*x^3*RootOf(_Z^8+2187)^8*RootOf(_Z^2+RootOf(_Z^8+2187)^2)-16*x^2*RootOf(_Z^8+2187)^8*Roo
tOf(_Z^2+RootOf(_Z^8+2187)^2)-54*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^6*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x+1350*R
ootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4*x^4+162*(x^5-x^3)^(1/4)*RootOf(_Z^8+2187)^6*x^2+243*RootOf
(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4*x^3-1350*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4*x
^2-8100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4-24300*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^2*RootOf(_Z^2+RootOf(_Z^8+
2187)^2)*x+28431*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x^4+72900*(x^5-x^3)^(1/4)*RootOf(_Z^8+2187)^2*x^2+56862*Root
Of(_Z^2+RootOf(_Z^8+2187)^2)*x^3-28431*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x^2+39366*(x^5-x^3)^(3/4))/x^2/(x*Root
Of(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4-135*x-108)/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4-81*x))+1/19683*R
ootOf(_Z^8+2187)^7*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187
)^2))*ln(-108*(26*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)
^2))*RootOf(_Z^8+2187)^11*x^4-39*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+Ro
otOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^11*x^3-26*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^11*RootOf(_
Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^2+243*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+R
ootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^7*x^4-4050*RootOf(_Z^2+RootOf(_Z^8+2187)^
2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^7*x^3-243*RootOf(_Z^2+Roo
tOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^7*x^2+4374
*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^6*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x+13122*(
x^5-x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^5*x^2-52488*RootOf(_Z^2+RootOf(_Z^8+2187)^2)
*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^3*x^4-104976*RootOf(_Z^2+Ro
otOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^3*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^3+524
88*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^
8+2187)^3*x^2+656100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4-1968300*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^2*RootOf(_Z
^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x-5904900*(x^5-x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^8+2187)
^2)*RootOf(_Z^8+2187)*x^2+3188646*(x^5-x^3)^(3/4))/x^2/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+81*x)/(x*R
ootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+135*x+108))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="maxima")
[Out]
-integrate((x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x^6+1}{\left (x^6-1\right )\,{\left (x^5-x^3\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)),x)
[Out]
int(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{6}}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx - \int \frac {1}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**6+1)/(x**5-x**3)**(1/4)/(-x**6+1),x)
[Out]
-Integral(x**6/(x**6*(x**5 - x**3)**(1/4) - (x**5 - x**3)**(1/4)), x) - Integral(1/(x**6*(x**5 - x**3)**(1/4)
- (x**5 - x**3)**(1/4)), x)
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