3.4.48 \(\int \frac {4^{\frac {1}{x}} (\frac {1}{1-4 x+6 x^2-4 x^3+x^4})^{\frac {1}{x}} (-4 x-3 x^2-x^3+(1-x^2) \log (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}))}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx\)

Optimal. Leaf size=26 \[ \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(1-x)^4}\right )^{\frac {1}{x}}}{1+x} \]

________________________________________________________________________________________

Rubi [F]  time = 4.70, 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 {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4^x^(-1)*((1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(-1))^x^(-1)*(-4*x - 3*x^2 - x^3 + (1 - x^2)*Log[4/(1 - 4*x + 6
*x^2 - 4*x^3 + x^4)]))/(-2*x^2 - 2*x^3 + 2*x^4 + 2*x^5),x]

[Out]

-Defer[Int][(4^x^(-1)*((-1 + x)^(-4))^x^(-1))/(-1 + x), x] - Log[4/(1 - x)^4]*Defer[Int][(2^(-1 + 2/x)*((-1 +
x)^(-4))^x^(-1))/x^2, x] + Log[4/(1 - x)^4]*Defer[Int][(2^(-1 + 2/x)*((-1 + x)^(-4))^x^(-1))/x, x] + Defer[Int
][(2^(1 + 2/x)*((-1 + x)^(-4))^x^(-1))/x, x] - Defer[Int][(2^(-1 + 2/x)*((-1 + x)^(-4))^x^(-1))/(1 + x)^2, x]
- Log[4/(1 - x)^4]*Defer[Int][(2^(-1 + 2/x)*((-1 + x)^(-4))^x^(-1))/(1 + x), x] - Defer[Int][(4^x^(-1)*((-1 +
x)^(-4))^x^(-1))/(1 + x), x] - 4*Defer[Int][Defer[Int][(2^(-1 + 2/x)*((-1 + x)^(-4))^x^(-1))/x^2, x]/(-1 + x),
x] + 4*Defer[Int][Defer[Int][(2^(-1 + 2/x)*((-1 + x)^(-4))^x^(-1))/x, x]/(-1 + x), x] - 4*Defer[Int][Defer[In
t][(2^(-1 + 2/x)*((-1 + x)^(-4))^x^(-1))/(1 + x), x]/(-1 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \left (x \left (4+3 x+x^2\right )+\left (-1+x^2\right ) \log \left (\frac {4}{(-1+x)^4}\right )\right )}{(1-x) x^2 (1+x)^2} \, dx\\ &=\int \left (\frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \left (-4-3 x-x^2\right )}{(-1+x) x (1+x)^2}-\frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \log \left (\frac {4}{(-1+x)^4}\right )}{x^2 (1+x)}\right ) \, dx\\ &=\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \left (-4-3 x-x^2\right )}{(-1+x) x (1+x)^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} \log \left (\frac {4}{(-1+x)^4}\right )}{x^2 (1+x)} \, dx\\ &=-\left (\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx\right )+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx+\int \left (\frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x}-\frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2}-\frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} x}{-1+x^2}\right ) \, dx+\int -\frac {4 \left (\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx+\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\right )}{-1+x} \, dx\\ &=-\left (4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx+\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx\right )-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}} x}{-1+x^2} \, dx\\ &=-\left (4 \int \left (\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x}+\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x}\right ) \, dx\right )-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \left (\frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x}+\frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x}\right ) \, dx\\ &=-\left (4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x} \, dx\right )-4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx-\int \frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {2^{2/x} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\\ &=-\left (4 \int \left (\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx}{-1+x}-\frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x}\right ) \, dx\right )-4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\\ &=-\left (4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx}{-1+x} \, dx\right )+4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx}{-1+x} \, dx-4 \int \frac {\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx}{-1+x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x^2} \, dx+\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\log \left (\frac {4}{(-1+x)^4}\right ) \int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{-1+x} \, dx+\int \frac {2^{1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{x} \, dx-\int \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{(1+x)^2} \, dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.69, size = 24, normalized size = 0.92 \begin {gather*} \frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(-1+x)^4}\right )^{\frac {1}{x}}}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4^x^(-1)*((1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(-1))^x^(-1)*(-4*x - 3*x^2 - x^3 + (1 - x^2)*Log[4/(1 - 4
*x + 6*x^2 - 4*x^3 + x^4)]))/(-2*x^2 - 2*x^3 + 2*x^4 + 2*x^5),x]

[Out]

(2^(-1 + 2/x)*((-1 + x)^(-4))^x^(-1))/(1 + x)

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 33, normalized size = 1.27 \begin {gather*} \frac {\left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{\left (\frac {1}{x}\right )}}{2 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+1)*log(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(log(4/(x^4-4*x^3+6*x^2-4*x+1))/x)/(2*x^5
+2*x^4-2*x^3-2*x^2),x, algorithm="fricas")

[Out]

1/2*(4/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))^(1/x)/(x + 1)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{3} + 3 \, x^{2} + {\left (x^{2} - 1\right )} \log \left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + 4 \, x\right )} \left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{\left (\frac {1}{x}\right )}}{2 \, {\left (x^{5} + x^{4} - x^{3} - x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+1)*log(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(log(4/(x^4-4*x^3+6*x^2-4*x+1))/x)/(2*x^5
+2*x^4-2*x^3-2*x^2),x, algorithm="giac")

[Out]

integrate(-1/2*(x^3 + 3*x^2 + (x^2 - 1)*log(4/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) + 4*x)*(4/(x^4 - 4*x^3 + 6*x^2
- 4*x + 1))^(1/x)/(x^5 + x^4 - x^3 - x^2), x)

________________________________________________________________________________________

maple [A]  time = 0.27, size = 34, normalized size = 1.31

method	result	size
risch	\(\frac {\left (\frac {4}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )^{\frac {1}{x}}}{2 x +2}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+1)*ln(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(ln(4/(x^4-4*x^3+6*x^2-4*x+1))/x)/(2*x^5+2*x^4-2
*x^3-2*x^2),x,method=_RETURNVERBOSE)

[Out]

1/2/(x+1)*(4/(x^4-4*x^3+6*x^2-4*x+1))^(1/x)

________________________________________________________________________________________

maxima [A]  time = 0.78, size = 25, normalized size = 0.96 \begin {gather*} \frac {e^{\left (\frac {2 \, \log \relax (2)}{x} - \frac {4 \, \log \left (x - 1\right )}{x}\right )}}{2 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+1)*log(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(log(4/(x^4-4*x^3+6*x^2-4*x+1))/x)/(2*x^5
+2*x^4-2*x^3-2*x^2),x, algorithm="maxima")

[Out]

1/2*e^(2*log(2)/x - 4*log(x - 1)/x)/(x + 1)

________________________________________________________________________________________

mupad [B]  time = 0.86, size = 34, normalized size = 1.31 \begin {gather*} \frac {{\left (\frac {4}{x^4-4\,x^3+6\,x^2-4\,x+1}\right )}^{1/x}}{2\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(4/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))/x)*(4*x + log(4/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))*(x^2 - 1) + 3
*x^2 + x^3))/(2*x^2 + 2*x^3 - 2*x^4 - 2*x^5),x)

[Out]

(4/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))^(1/x)/(2*(x + 1))

________________________________________________________________________________________

sympy [A]  time = 0.36, size = 29, normalized size = 1.12 \begin {gather*} \frac {e^{\frac {\log {\left (\frac {4}{x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1} \right )}}{x}}}{2 x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+1)*ln(4/(x**4-4*x**3+6*x**2-4*x+1))-x**3-3*x**2-4*x)*exp(ln(4/(x**4-4*x**3+6*x**2-4*x+1))/x)
/(2*x**5+2*x**4-2*x**3-2*x**2),x)

[Out]

exp(log(4/(x**4 - 4*x**3 + 6*x**2 - 4*x + 1))/x)/(2*x + 2)

________________________________________________________________________________________