\(\int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx\) [284]

Optimal result

Integrand size = 71, antiderivative size = 71 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (1+x+\sqrt {2} x+\sqrt {2} x^2-x^7\right )-\left (-1+\sqrt {2}\right ) \log \left (-1+\left (-1+\sqrt {2}\right ) x+\sqrt {2} x^2+x^7\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \]

[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*} \text {integral}& = \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] (verified)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (1+x+\sqrt {2} x+\sqrt {2} x^2-x^7\right )-\left (-1+\sqrt {2}\right ) \log \left (-1+\left (-1+\sqrt {2}\right ) x+\sqrt {2} x^2+x^7\right )\right ) \]

[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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86

method	result	size
default	\(\left (\frac {1}{2}+\frac {\sqrt {2}}{2}\right ) \ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right )+\left (-\frac {\sqrt {2}}{2}+\frac {1}{2}\right ) \ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right )\)	\(61\)
risch	\(\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right )}{2}+\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right ) \sqrt {2}}{2}-\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right ) \sqrt {2}}{2}+\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right )}{2}\)	\(102\)

[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,method=_RETURNVER
BOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (56) = 112\).

Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.93 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{14} - 2 \, x^{8} - 2 \, x^{7} + 2 \, x^{4} + 4 \, x^{3} + 3 \, x^{2} - 2 \, \sqrt {2} {\left (x^{9} + x^{8} - x^{3} - 2 \, x^{2} - x\right )} + 2 \, x + 1}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1}\right ) + \frac {1}{2} \, \log \left (x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} - \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) - 1 \right )} + \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} + \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) - 1 \right )} \]

[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)

Maxima [F]

\[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\int { \frac {7 \, x^{13} + 10 \, x^{8} + 4 \, x^{7} - 7 \, x^{6} - 4 \, x^{3} - 4 \, x^{2} + 3 \, x + 3}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} + \sqrt {2} x^{2} + \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} - \sqrt {2} x^{2} - \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1 \right |}\right ) \]

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

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.45 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2}-\frac {\sqrt {2}\,\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\sqrt {2}\,\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2} \]

[In]

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

[Out]

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