3.12.53 \(\int \frac {\sqrt {1-x^6} (1+2 x^6) (1+x^2-x^4-2 x^6-x^8+x^{12})}{(-1+x^6) (-1+2 x^6-3 x^{12}+x^{18})} \, dx\)

Optimal. Leaf size=85 \[ -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {1-x^6}}\right )-\frac {1}{3} \tan ^{-1}\left (\frac {x \sqrt {1-x^6}}{x^6+x^2-1}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x \sqrt {1-x^6}}{x^6-x^2-1}\right )}{\sqrt {3}} \]

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Rubi [F]  time = 2.68, 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 {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[1 - x^6]*(1 + 2*x^6)*(1 + x^2 - x^4 - 2*x^6 - x^8 + x^12))/((-1 + x^6)*(-1 + 2*x^6 - 3*x^12 + x^18))
,x]

[Out]

-((x*(1 - x^2)*Sqrt[(1 + x^2 + x^4)/(1 - (1 + Sqrt[3])*x^2)^2]*EllipticF[ArcCos[(1 - (1 - Sqrt[3])*x^2)/(1 - (
1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(3^(1/4)*Sqrt[-((x^2*(1 - x^2))/(1 - (1 + Sqrt[3])*x^2)^2)]*Sqrt[1 - x^6
])) + Defer[Int][1/(Sqrt[1 - x^6]*(-1 - x^2 + x^6)), x] + (2*Defer[Int][x^2/(Sqrt[1 - x^6]*(-1 - x^2 + x^6)),
x])/3 + 4*Defer[Int][1/(Sqrt[1 - x^6]*(1 - x^2 + x^4 - 2*x^6 + x^8 + x^12)), x] - (10*Defer[Int][x^2/(Sqrt[1 -
x^6]*(1 - x^2 + x^4 - 2*x^6 + x^8 + x^12)), x])/3 + (8*Defer[Int][x^4/(Sqrt[1 - x^6]*(1 - x^2 + x^4 - 2*x^6 +
x^8 + x^12)), x])/3 - 4*Defer[Int][x^6/(Sqrt[1 - x^6]*(1 - x^2 + x^4 - 2*x^6 + x^8 + x^12)), x] + (4*Defer[In
t][x^8/(Sqrt[1 - x^6]*(1 - x^2 + x^4 - 2*x^6 + x^8 + x^12)), x])/3

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx &=-\int \frac {\left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\sqrt {1-x^6} \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx\\ &=-\int \left (\frac {2}{\sqrt {1-x^6}}+\frac {3+x^2-x^4-4 x^6+x^8-2 x^{10}+3 x^{12}-2 x^{14}}{\sqrt {1-x^6} \left (-1+2 x^6-3 x^{12}+x^{18}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt {1-x^6}} \, dx\right )-\int \frac {3+x^2-x^4-4 x^6+x^8-2 x^{10}+3 x^{12}-2 x^{14}}{\sqrt {1-x^6} \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx\\ &=-\frac {x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}-\int \left (\frac {-3-2 x^2}{3 \sqrt {1-x^6} \left (-1-x^2+x^6\right )}-\frac {2 \left (6-5 x^2+4 x^4-6 x^6+2 x^8\right )}{3 \sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=-\frac {x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}-\frac {1}{3} \int \frac {-3-2 x^2}{\sqrt {1-x^6} \left (-1-x^2+x^6\right )} \, dx+\frac {2}{3} \int \frac {6-5 x^2+4 x^4-6 x^6+2 x^8}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )} \, dx\\ &=-\frac {x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}-\frac {1}{3} \int \left (-\frac {3}{\sqrt {1-x^6} \left (-1-x^2+x^6\right )}-\frac {2 x^2}{\sqrt {1-x^6} \left (-1-x^2+x^6\right )}\right ) \, dx+\frac {2}{3} \int \left (\frac {6}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )}-\frac {5 x^2}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )}+\frac {4 x^4}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )}-\frac {6 x^6}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )}+\frac {2 x^8}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=-\frac {x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}+\frac {2}{3} \int \frac {x^2}{\sqrt {1-x^6} \left (-1-x^2+x^6\right )} \, dx+\frac {4}{3} \int \frac {x^8}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )} \, dx+\frac {8}{3} \int \frac {x^4}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )} \, dx-\frac {10}{3} \int \frac {x^2}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )} \, dx+4 \int \frac {1}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )} \, dx-4 \int \frac {x^6}{\sqrt {1-x^6} \left (1-x^2+x^4-2 x^6+x^8+x^{12}\right )} \, dx+\int \frac {1}{\sqrt {1-x^6} \left (-1-x^2+x^6\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[1 - x^6]*(1 + 2*x^6)*(1 + x^2 - x^4 - 2*x^6 - x^8 + x^12))/((-1 + x^6)*(-1 + 2*x^6 - 3*x^12 +
x^18)),x]

[Out]

Integrate[(Sqrt[1 - x^6]*(1 + 2*x^6)*(1 + x^2 - x^4 - 2*x^6 - x^8 + x^12))/((-1 + x^6)*(-1 + 2*x^6 - 3*x^12 +
x^18)), x]

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IntegrateAlgebraic [C]  time = 18.18, size = 105, normalized size = 1.24 \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {1-x^6}}\right )+\frac {1}{3} \left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1-i \sqrt {3}\right ) x}{2 \sqrt {1-x^6}}\right )+\frac {1}{3} \left (1-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1+i \sqrt {3}\right ) x}{2 \sqrt {1-x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[1 - x^6]*(1 + 2*x^6)*(1 + x^2 - x^4 - 2*x^6 - x^8 + x^12))/((-1 + x^6)*(-1 + 2*x^6 -
3*x^12 + x^18)),x]

[Out]

-1/3*ArcTan[x/Sqrt[1 - x^6]] + ((1 + I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/(2*Sqrt[1 - x^6])])/3 + ((1 - I*Sqr
t[3])*ArcTan[((1 + I*Sqrt[3])*x)/(2*Sqrt[1 - x^6])])/3

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fricas [B]  time = 1.19, size = 440, normalized size = 5.18 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \log \left (\frac {16 \, {\left (x^{12} - 5 \, x^{8} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {3} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 5 \, x^{2} + 1\right )}}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (\frac {16 \, {\left (x^{12} - 5 \, x^{8} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {3} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 5 \, x^{2} + 1\right )}}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {-x^{6} + 1} x}{x^{6} + x^{2} - 1}\right ) + \frac {1}{3} \, \arctan \left (-\frac {\sqrt {-x^{6} + 1} x + {\left (x^{6} - \sqrt {3} \sqrt {-x^{6} + 1} x - x^{2} - 1\right )} \sqrt {\frac {x^{12} - 5 \, x^{8} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {3} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 5 \, x^{2} + 1}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}}}{x^{6} + x^{2} - 1}\right ) - \frac {1}{3} \, \arctan \left (\frac {\sqrt {-x^{6} + 1} x + {\left (x^{6} + \sqrt {3} \sqrt {-x^{6} + 1} x - x^{2} - 1\right )} \sqrt {\frac {x^{12} - 5 \, x^{8} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {3} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 5 \, x^{2} + 1}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}}}{x^{6} + x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^18-3*x^12+2*x^6-1),x, algorithm="fric
as")

[Out]

-1/12*sqrt(3)*log(16*(x^12 - 5*x^8 - 2*x^6 + x^4 + 2*sqrt(3)*(x^7 - x^3 - x)*sqrt(-x^6 + 1) + 5*x^2 + 1)/(x^12
+ x^8 - 2*x^6 + x^4 - x^2 + 1)) + 1/12*sqrt(3)*log(16*(x^12 - 5*x^8 - 2*x^6 + x^4 - 2*sqrt(3)*(x^7 - x^3 - x)
*sqrt(-x^6 + 1) + 5*x^2 + 1)/(x^12 + x^8 - 2*x^6 + x^4 - x^2 + 1)) + 1/6*arctan(2*sqrt(-x^6 + 1)*x/(x^6 + x^2
- 1)) + 1/3*arctan(-(sqrt(-x^6 + 1)*x + (x^6 - sqrt(3)*sqrt(-x^6 + 1)*x - x^2 - 1)*sqrt((x^12 - 5*x^8 - 2*x^6
+ x^4 + 2*sqrt(3)*(x^7 - x^3 - x)*sqrt(-x^6 + 1) + 5*x^2 + 1)/(x^12 + x^8 - 2*x^6 + x^4 - x^2 + 1)))/(x^6 + x^
2 - 1)) - 1/3*arctan((sqrt(-x^6 + 1)*x + (x^6 + sqrt(3)*sqrt(-x^6 + 1)*x - x^2 - 1)*sqrt((x^12 - 5*x^8 - 2*x^6
+ x^4 - 2*sqrt(3)*(x^7 - x^3 - x)*sqrt(-x^6 + 1) + 5*x^2 + 1)/(x^12 + x^8 - 2*x^6 + x^4 - x^2 + 1)))/(x^6 + x
^2 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{12} - x^{8} - 2 \, x^{6} - x^{4} + x^{2} + 1\right )} {\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{18} - 3 \, x^{12} + 2 \, x^{6} - 1\right )} {\left (x^{6} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^18-3*x^12+2*x^6-1),x, algorithm="giac
")

[Out]

integrate((x^12 - x^8 - 2*x^6 - x^4 + x^2 + 1)*(2*x^6 + 1)*sqrt(-x^6 + 1)/((x^18 - 3*x^12 + 2*x^6 - 1)*(x^6 -
1)), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-x^{6}+1}\, \left (2 x^{6}+1\right ) \left (x^{12}-x^{8}-2 x^{6}-x^{4}+x^{2}+1\right )}{\left (x^{6}-1\right ) \left (x^{18}-3 x^{12}+2 x^{6}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^18-3*x^12+2*x^6-1),x)

[Out]

int((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^18-3*x^12+2*x^6-1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{12} - x^{8} - 2 \, x^{6} - x^{4} + x^{2} + 1\right )} {\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{18} - 3 \, x^{12} + 2 \, x^{6} - 1\right )} {\left (x^{6} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^18-3*x^12+2*x^6-1),x, algorithm="maxi
ma")

[Out]

integrate((x^12 - x^8 - 2*x^6 - x^4 + x^2 + 1)*(2*x^6 + 1)*sqrt(-x^6 + 1)/((x^18 - 3*x^12 + 2*x^6 - 1)*(x^6 -
1)), 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^{12}-x^8-2\,x^6-x^4+x^2+1\right )}{\sqrt {1-x^6}\,\left (x^{18}-3\,x^{12}+2\,x^6-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x^6 + 1)*(x^2 - x^4 - 2*x^6 - x^8 + x^12 + 1))/((1 - x^6)^(1/2)*(2*x^6 - 3*x^12 + x^18 - 1)),x)

[Out]

int(-((2*x^6 + 1)*(x^2 - x^4 - 2*x^6 - x^8 + x^12 + 1))/((1 - x^6)^(1/2)*(2*x^6 - 3*x^12 + x^18 - 1)), 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)**(1/2)*(2*x**6+1)*(x**12-x**8-2*x**6-x**4+x**2+1)/(x**6-1)/(x**18-3*x**12+2*x**6-1),x)

[Out]

Timed out

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