3.284 \(\int \frac{3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx\)

Optimal. Leaf size=71 \[ \frac{1}{2} \left (\left (1+\sqrt{2}\right ) \log \left (-x^7+\sqrt{2} x^2+\sqrt{2} x+x+1\right )-\left (\sqrt{2}-1\right ) \log \left (x^7+\sqrt{2} x^2+\left (\sqrt{2}-1\right ) x-1\right )\right ) \]

[Out]

((1 + Sqrt[2])*Log[1 + x + Sqrt[2]*x + Sqrt[2]*x^2 - x^7] - (-1 + Sqrt[2])*Log[-1 + (-1 + Sqrt[2])*x + Sqrt[2]
*x^2 + x^7])/2

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Rubi [F]  time = 0.754187, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(3 + 3*x - 4*x^2 - 4*x^3 - 7*x^6 + 4*x^7 + 10*x^8 + 7*x^13)/(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 - 2*x^8
+ x^14),x]

[Out]

Log[1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 - 2*x^8 + x^14]/2 + 2*Defer[Int][(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*
x^7 - 2*x^8 + x^14)^(-1), x] + 4*Defer[Int][x/(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 - 2*x^8 + x^14), x] + 2*D
efer[Int][x^2/(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 - 2*x^8 + x^14), x] + 12*Defer[Int][x^7/(1 + 2*x - x^2 -
4*x^3 - 2*x^4 - 2*x^7 - 2*x^8 + x^14), x] + 10*Defer[Int][x^8/(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 - 2*x^8 +
x^14), x]

Rubi steps

\begin{align*} \int \frac{3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx &=\frac{1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+\frac{1}{14} \int \frac{28+56 x+28 x^2+168 x^7+140 x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx\\ &=\frac{1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+\frac{1}{14} \int \frac{28 \left (1+2 x+x^2+6 x^7+5 x^8\right )}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx\\ &=\frac{1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+2 \int \frac{1+2 x+x^2+6 x^7+5 x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx\\ &=\frac{1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+2 \int \left (\frac{1}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac{2 x}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac{x^2}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac{6 x^7}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac{5 x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}\right ) \, dx\\ &=\frac{1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+2 \int \frac{1}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+2 \int \frac{x^2}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+4 \int \frac{x}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+10 \int \frac{x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+12 \int \frac{x^7}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx\\ \end{align*}

Mathematica [A]  time = 0.036176, size = 71, normalized size = 1. \[ \frac{1}{2} \left (\left (1+\sqrt{2}\right ) \log \left (-x^7+\sqrt{2} x^2+\sqrt{2} x+x+1\right )-\left (\sqrt{2}-1\right ) \log \left (x^7+\sqrt{2} x^2+\left (\sqrt{2}-1\right ) x-1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 3*x - 4*x^2 - 4*x^3 - 7*x^6 + 4*x^7 + 10*x^8 + 7*x^13)/(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 -
2*x^8 + x^14),x]

[Out]

((1 + Sqrt[2])*Log[1 + x + Sqrt[2]*x + Sqrt[2]*x^2 - x^7] - (-1 + Sqrt[2])*Log[-1 + (-1 + Sqrt[2])*x + Sqrt[2]
*x^2 + x^7])/2

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Maple [A]  time = 0.022, size = 102, normalized size = 1.4 \begin{align*}{\frac{\ln \left ({x}^{7}-{x}^{2}\sqrt{2}+ \left ( -\sqrt{2}-1 \right ) x-1 \right ) }{2}}+{\frac{\ln \left ({x}^{7}-{x}^{2}\sqrt{2}+ \left ( -\sqrt{2}-1 \right ) x-1 \right ) \sqrt{2}}{2}}+{\frac{\ln \left ( -1+{x}^{7}+x \left ( \sqrt{2}-1 \right ) +{x}^{2}\sqrt{2} \right ) }{2}}-{\frac{\ln \left ( -1+{x}^{7}+x \left ( \sqrt{2}-1 \right ) +{x}^{2}\sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7-2*x^4-4*x^3-x^2+2*x+1),x)

[Out]

1/2*ln(x^7-x^2*2^(1/2)+(-2^(1/2)-1)*x-1)+1/2*ln(x^7-x^2*2^(1/2)+(-2^(1/2)-1)*x-1)*2^(1/2)+1/2*ln(-1+x^7+x*(2^(
1/2)-1)+x^2*2^(1/2))-1/2*ln(-1+x^7+x*(2^(1/2)-1)+x^2*2^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{7 \, x^{13} + 10 \, x^{8} + 4 \, x^{7} - 7 \, x^{6} - 4 \, x^{3} - 4 \, x^{2} + 3 \, x + 3}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7-2*x^4-4*x^3-x^2+2*x+1),x, algorithm=
"maxima")

[Out]

integrate((7*x^13 + 10*x^8 + 4*x^7 - 7*x^6 - 4*x^3 - 4*x^2 + 3*x + 3)/(x^14 - 2*x^8 - 2*x^7 - 2*x^4 - 4*x^3 -
x^2 + 2*x + 1), x)

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Fricas [B]  time = 1.61445, size = 319, normalized size = 4.49 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\frac{x^{14} - 2 \, x^{8} - 2 \, x^{7} + 2 \, x^{4} + 4 \, x^{3} + 3 \, x^{2} - 2 \, \sqrt{2}{\left (x^{9} + x^{8} - x^{3} - 2 \, x^{2} - x\right )} + 2 \, x + 1}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1}\right ) + \frac{1}{2} \, \log \left (x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7-2*x^4-4*x^3-x^2+2*x+1),x, algorithm=
"fricas")

[Out]

1/2*sqrt(2)*log((x^14 - 2*x^8 - 2*x^7 + 2*x^4 + 4*x^3 + 3*x^2 - 2*sqrt(2)*(x^9 + x^8 - x^3 - 2*x^2 - x) + 2*x
+ 1)/(x^14 - 2*x^8 - 2*x^7 - 2*x^4 - 4*x^3 - x^2 + 2*x + 1)) + 1/2*log(x^14 - 2*x^8 - 2*x^7 - 2*x^4 - 4*x^3 -
x^2 + 2*x + 1)

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Sympy [A]  time = 0.204742, size = 76, normalized size = 1.07 \begin{align*} \left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \log{\left (x^{7} - \sqrt{2} x^{2} - 2 x \left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) - 1 \right )} + \left (\frac{1}{2} - \frac{\sqrt{2}}{2}\right ) \log{\left (x^{7} + \sqrt{2} x^{2} - 2 x \left (\frac{1}{2} - \frac{\sqrt{2}}{2}\right ) - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x**13+10*x**8+4*x**7-7*x**6-4*x**3-4*x**2+3*x+3)/(x**14-2*x**8-2*x**7-2*x**4-4*x**3-x**2+2*x+1),x
)

[Out]

(1/2 + sqrt(2)/2)*log(x**7 - sqrt(2)*x**2 - 2*x*(1/2 + sqrt(2)/2) - 1) + (1/2 - sqrt(2)/2)*log(x**7 + sqrt(2)*
x**2 - 2*x*(1/2 - sqrt(2)/2) - 1)

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Giac [A]  time = 1.13533, size = 132, normalized size = 1.86 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left ({\left | 14 \, x^{7} + 14 \, \sqrt{2} x^{2} + 14 \, x{\left (\sqrt{2} - 1\right )} - 14 \right |}\right ) + \frac{1}{2} \, \sqrt{2} \log \left ({\left | 14 \, x^{7} - 14 \, \sqrt{2} x^{2} - 14 \, x{\left (\sqrt{2} + 1\right )} - 14 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7-2*x^4-4*x^3-x^2+2*x+1),x, algorithm=
"giac")

[Out]

-1/2*sqrt(2)*log(abs(14*x^7 + 14*sqrt(2)*x^2 + 14*x*(sqrt(2) - 1) - 14)) + 1/2*sqrt(2)*log(abs(14*x^7 - 14*sqr
t(2)*x^2 - 14*x*(sqrt(2) + 1) - 14)) + 1/2*log(abs(x^14 - 2*x^8 - 2*x^7 - 2*x^4 - 4*x^3 - x^2 + 2*x + 1))