∫ 1+x6 ___________________________4√−x3 +x5 (1−x6) dx

3.30.87 $\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} (1-x^6)} \, dx$

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 = 4.51, size = 416, normalized size = 1.07 \begin {gather*} \frac {4 \left (-x^3+x^5\right )^{3/4}}{3 x^2 \left (-1+x^2\right )}+\frac {\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} 3^{7/8} x \sqrt [4]{-x^3+x^5}}{\sqrt [8]{17+12 \sqrt {2}} \left (-3 x^2+3^{3/4} \sqrt {-x^3+x^5}\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^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\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^3+x^5}}{x \sqrt [4]{-x^3+x^5}}\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^3+x^5}}{\sqrt [4]{2} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}}{x \sqrt [4]{-x^3+x^5}}\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 = 124.08, size = 2178, normalized size = 5.58


method	result	size

risch	$\text {Expression too large to display}$	$2178$
trager	$\text {Expression too large to display}$	$2187$

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,method=_RETURNVERBOSE)

[Out]

4/3*x/(x^3*(x^2-1))^(1/4)+1/19683*RootOf(_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+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^11*x^3-26*RootOf(_Z^2+RootOf(_Z^8+218

7)^2)*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^11*x^2+243*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^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+RootOf(_Z^8+2187)^2)*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+218

7)^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^2-524

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^4-104976*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+21

87)^2))*RootOf(_Z^8+2187)^3*x^3+656100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4+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^2+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*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+135*x+108))+1/9*RootOf(_Z^2-RootOf

(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*ln(-108*(-16*RootOf(_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+5

4*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^5*RootOf(_Z^2+RootOf(_Z^8+2187)^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+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^5*x^2+1350*R

ootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^4*x^4+243*RootOf(_Z^2-RootOf(

_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^4*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+2187)*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-Roo

tOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^4-56862*RootOf(_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*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+135*x+

108))-1/9*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*ln(-108*(-16*x^4*RootOf(_Z^8+2187)^8*RootOf(_Z^2+RootOf(_Z^8+2187)^

2)+24*x^3*RootOf(_Z^8+2187)^8*RootOf(_Z^2+RootOf(_Z^8+2187)^2)+16*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)^6*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x-162*(x^5-x^3)^(1/4)*Roo

tOf(_Z^8+2187)^6*x^2-1350*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4*x^4-243*x^3*RootOf(_Z^8+2187)^4

*RootOf(_Z^2+RootOf(_Z^8+2187)^2)+8100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4+1350*RootOf(_Z^2+RootOf(_Z^8+2187)^

2)*RootOf(_Z^8+2187)^4*x^2+24300*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^2*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x-72900*

(x^5-x^3)^(1/4)*RootOf(_Z^8+2187)^2*x^2-28431*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x^4-56862*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)^2)*x^2)/x^2/(x*RootOf(_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))-1/9*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)*Root

Of(_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*RootOf(_Z^8+218

7)^5*x^3+8100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4+1350*RootOf(_Z^8+2187)^5*x^2-24300*(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^3-39366*(x^5-x^3)^(3/4)+28431*RootOf(_Z^8+2187)*x^2)/x^2/(x*RootOf(_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))

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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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3.30.87 \(\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} (1-x^6)} \, dx\)