3.7.50 \(\int \frac {-1+7 x^8}{(1+x^8) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx\)

Optimal. Leaf size=51 \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {3} x^8-\sqrt {3 x^{16}-x^9+6 x^8+x^2-x+3}+\sqrt {3}}\right ) \]

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Rubi [F]  time = 2.06, 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 {-1+7 x^8}{\left (1+x^8\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + 7*x^8)/((1 + x^8)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]),x]

[Out]

7*Defer[Int][1/Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16], x] - (-1)^(1/8)*Defer[Int][1/(((-1)^(1/8) - x)*Sqrt[3
- x + x^2 + 6*x^8 - x^9 + 3*x^16]), x] - (-1)^(3/8)*Defer[Int][1/(((-1)^(3/8) - x)*Sqrt[3 - x + x^2 + 6*x^8 -
x^9 + 3*x^16]), x] + (-1)^(5/8)*Defer[Int][1/((-(-1)^(5/8) - x)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]), x]
+ (-1)^(7/8)*Defer[Int][1/((-(-1)^(7/8) - x)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]), x] - (-1)^(1/8)*Defer
[Int][1/(((-1)^(1/8) + x)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]), x] - (-1)^(3/8)*Defer[Int][1/(((-1)^(3/8)
+ x)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]), x] + (-1)^(5/8)*Defer[Int][1/((-(-1)^(5/8) + x)*Sqrt[3 - x +
x^2 + 6*x^8 - x^9 + 3*x^16]), x] + (-1)^(7/8)*Defer[Int][1/((-(-1)^(7/8) + x)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 +
3*x^16]), x]

Rubi steps

\begin {align*} \int \frac {-1+7 x^8}{\left (1+x^8\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx &=\int \left (\frac {7}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}-\frac {8}{\left (1+x^8\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx\\ &=7 \int \frac {1}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-8 \int \frac {1}{\left (1+x^8\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx\\ &=7 \int \frac {1}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-8 \int \left (\frac {i}{2 \left (i-x^4\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}+\frac {i}{2 \left (i+x^4\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx\\ &=-\left (4 i \int \frac {1}{\left (i-x^4\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx\right )-4 i \int \frac {1}{\left (i+x^4\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx+7 \int \frac {1}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx\\ &=-\left (4 i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx\right )-4 i \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx+7 \int \frac {1}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx\\ &=7 \int \frac {1}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-\left (2 \sqrt [4]{-1}\right ) \int \frac {1}{\left (\sqrt [4]{-1}-x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-\left (2 \sqrt [4]{-1}\right ) \int \frac {1}{\left (\sqrt [4]{-1}+x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx+\left (2 (-1)^{3/4}\right ) \int \frac {1}{\left (-(-1)^{3/4}-x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx+\left (2 (-1)^{3/4}\right ) \int \frac {1}{\left (-(-1)^{3/4}+x^2\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx\\ &=7 \int \frac {1}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {(-1)^{7/8}}{2 \left (\sqrt [8]{-1}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}-\frac {(-1)^{7/8}}{2 \left (\sqrt [8]{-1}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx-\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {(-1)^{3/8}}{2 \left (-(-1)^{5/8}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}-\frac {(-1)^{3/8}}{2 \left (-(-1)^{5/8}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx+\left (2 (-1)^{3/4}\right ) \int \left (\frac {(-1)^{5/8}}{2 \left ((-1)^{3/8}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}+\frac {(-1)^{5/8}}{2 \left ((-1)^{3/8}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx+\left (2 (-1)^{3/4}\right ) \int \left (\frac {\sqrt [8]{-1}}{2 \left (-(-1)^{7/8}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}+\frac {\sqrt [8]{-1}}{2 \left (-(-1)^{7/8}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \, dx\\ &=7 \int \frac {1}{\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-\sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-\sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-(-1)^{3/8} \int \frac {1}{\left ((-1)^{3/8}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx-(-1)^{3/8} \int \frac {1}{\left ((-1)^{3/8}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx+(-1)^{5/8} \int \frac {1}{\left (-(-1)^{5/8}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx+(-1)^{5/8} \int \frac {1}{\left (-(-1)^{5/8}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx+(-1)^{7/8} \int \frac {1}{\left (-(-1)^{7/8}-x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx+(-1)^{7/8} \int \frac {1}{\left (-(-1)^{7/8}+x\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+7 x^8}{\left (1+x^8\right ) \sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-1 + 7*x^8)/((1 + x^8)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]),x]

[Out]

Integrate[(-1 + 7*x^8)/((1 + x^8)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]), x]

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IntegrateAlgebraic [A]  time = 0.54, size = 51, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{\sqrt {3}+\sqrt {3} x^8-\sqrt {3-x+x^2+6 x^8-x^9+3 x^{16}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 7*x^8)/((1 + x^8)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]),x]

[Out]

2*ArcTanh[x/(Sqrt[3] + Sqrt[3]*x^8 - Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16])]

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fricas [A]  time = 1.36, size = 45, normalized size = 0.88 \begin {gather*} \log \left (-\frac {x^{8} - 2 \, x + 2 \, \sqrt {3 \, x^{16} - x^{9} + 6 \, x^{8} + x^{2} - x + 3} + 1}{x^{8} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^8-1)/(x^8+1)/(3*x^16-x^9+6*x^8+x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

log(-(x^8 - 2*x + 2*sqrt(3*x^16 - x^9 + 6*x^8 + x^2 - x + 3) + 1)/(x^8 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {7 \, x^{8} - 1}{\sqrt {3 \, x^{16} - x^{9} + 6 \, x^{8} + x^{2} - x + 3} {\left (x^{8} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^8-1)/(x^8+1)/(3*x^16-x^9+6*x^8+x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

integrate((7*x^8 - 1)/(sqrt(3*x^16 - x^9 + 6*x^8 + x^2 - x + 3)*(x^8 + 1)), x)

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maple [A]  time = 0.23, size = 46, normalized size = 0.90

method	result	size
trager	\(\ln \left (-\frac {x^{8}+2 \sqrt {3 x^{16}-x^{9}+6 x^{8}+x^{2}-x +3}-2 x +1}{x^{8}+1}\right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x^8-1)/(x^8+1)/(3*x^16-x^9+6*x^8+x^2-x+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

ln(-(x^8+2*(3*x^16-x^9+6*x^8+x^2-x+3)^(1/2)-2*x+1)/(x^8+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {7 \, x^{8} - 1}{\sqrt {3 \, x^{16} - x^{9} + 6 \, x^{8} + x^{2} - x + 3} {\left (x^{8} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^8-1)/(x^8+1)/(3*x^16-x^9+6*x^8+x^2-x+3)^(1/2),x, algorithm="maxima")

[Out]

integrate((7*x^8 - 1)/(sqrt(3*x^16 - x^9 + 6*x^8 + x^2 - x + 3)*(x^8 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {7\,x^8-1}{\left (x^8+1\right )\,\sqrt {3\,x^{16}-x^9+6\,x^8+x^2-x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x^8 - 1)/((x^8 + 1)*(x^2 - x + 6*x^8 - x^9 + 3*x^16 + 3)^(1/2)),x)

[Out]

int((7*x^8 - 1)/((x^8 + 1)*(x^2 - x + 6*x^8 - x^9 + 3*x^16 + 3)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {7 x^{8} - 1}{\left (x^{8} + 1\right ) \sqrt {3 x^{16} - x^{9} + 6 x^{8} + x^{2} - x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x**8-1)/(x**8+1)/(3*x**16-x**9+6*x**8+x**2-x+3)**(1/2),x)

[Out]

Integral((7*x**8 - 1)/((x**8 + 1)*sqrt(3*x**16 - x**9 + 6*x**8 + x**2 - x + 3)), x)

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