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

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

[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))

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Rubi [F]
time = 0.47, 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 {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 \end {gather*}

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*}

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Mathematica [A]
time = 0.02, size = 71, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}

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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Mathics [A]
time = 2.38, size = 60, normalized size = 0.85 \begin {gather*} \frac {\left (1+\sqrt {2}\right ) \text {Log}\left [-1+\left (-1-\sqrt {2}\right ) x-\sqrt {2} x^2+x^7\right ]}{2}+\frac {\left (1-\sqrt {2}\right ) \text {Log}\left [-1+\left (-1+\sqrt {2}\right ) x+\sqrt {2} x^2+x^7\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [A]
time = 0.04, size = 61, normalized size = 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	\(\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right )}{2}+\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right ) \sqrt {2}}{2}-\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right ) \sqrt {2}}{2}+\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right )}{2}\)	\(102\)

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,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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (56) = 112\).
time = 0.32, size = 137, normalized size = 1.93 \begin {gather*} \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 {gather*}

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 = 0.12, size = 76, normalized size = 1.07 \begin {gather*} \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 {gather*}

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 = 0.01, size = 113, normalized size = 1.59 \begin {gather*} \frac {-7 \sqrt {2} \ln \left |x^{7}+x^{2} \sqrt {2}+x \sqrt {2}-x-1\right |+7 \sqrt {2} \ln \left |x^{7}-x^{2} \sqrt {2}-x \sqrt {2}-x-1\right |}{14}+\frac {\ln \left |x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x+1\right |}{2} \end {gather*}

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)*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))

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Mupad [B]
time = 0.28, size = 103, normalized size = 1.45 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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