3.20.78 \(\int \frac {-6561 x^2+e^{x^4} (-6561 x^2+4 x^4)+(-1+e^{x^4} (-1-162 x)-162 x) \log (1+e^{x^4})+(-1-e^{x^4}) \log ^2(1+e^{x^4})}{6561 x^2+6561 e^{x^4} x^2+(162 x+162 e^{x^4} x) \log (1+e^{x^4})+(1+e^{x^4}) \log ^2(1+e^{x^4})} \, dx\)

Optimal. Leaf size=23 \[ -13-x-\frac {1}{81+\frac {\log \left (1+e^{x^4}\right )}{x}} \]

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Rubi [F]  time = 1.80, 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 {-6561 x^2+e^{x^4} \left (-6561 x^2+4 x^4\right )+\left (-1+e^{x^4} (-1-162 x)-162 x\right ) \log \left (1+e^{x^4}\right )+\left (-1-e^{x^4}\right ) \log ^2\left (1+e^{x^4}\right )}{6561 x^2+6561 e^{x^4} x^2+\left (162 x+162 e^{x^4} x\right ) \log \left (1+e^{x^4}\right )+\left (1+e^{x^4}\right ) \log ^2\left (1+e^{x^4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6561*x^2 + E^x^4*(-6561*x^2 + 4*x^4) + (-1 + E^x^4*(-1 - 162*x) - 162*x)*Log[1 + E^x^4] + (-1 - E^x^4)*L
og[1 + E^x^4]^2)/(6561*x^2 + 6561*E^x^4*x^2 + (162*x + 162*E^x^4*x)*Log[1 + E^x^4] + (1 + E^x^4)*Log[1 + E^x^4
]^2),x]

[Out]

-x + Defer[Int][(-81*x - Log[1 + E^x^4])^(-1), x] + 81*Defer[Int][x/(81*x + Log[1 + E^x^4])^2, x] + 4*Defer[In
t][x^4/(81*x + Log[1 + E^x^4])^2, x] - 4*Defer[Int][x^4/((1 + E^x^4)*(81*x + Log[1 + E^x^4])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (-6561+e^{x^4} \left (-6561+4 x^2\right )\right )-\left (1+e^{x^4}\right ) (1+162 x) \log \left (1+e^{x^4}\right )-\left (1+e^{x^4}\right ) \log ^2\left (1+e^{x^4}\right )}{\left (1+e^{x^4}\right ) \left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx\\ &=\int \left (-\frac {4 x^4}{\left (1+e^{x^4}\right ) \left (81 x+\log \left (1+e^{x^4}\right )\right )^2}+\frac {-6561 x^2+4 x^4-\log \left (1+e^{x^4}\right )-162 x \log \left (1+e^{x^4}\right )-\log ^2\left (1+e^{x^4}\right )}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {x^4}{\left (1+e^{x^4}\right ) \left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx\right )+\int \frac {-6561 x^2+4 x^4-\log \left (1+e^{x^4}\right )-162 x \log \left (1+e^{x^4}\right )-\log ^2\left (1+e^{x^4}\right )}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx\\ &=-\left (4 \int \frac {x^4}{\left (1+e^{x^4}\right ) \left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx\right )+\int \left (-1+\frac {1}{-81 x-\log \left (1+e^{x^4}\right )}+\frac {x \left (81+4 x^3\right )}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2}\right ) \, dx\\ &=-x-4 \int \frac {x^4}{\left (1+e^{x^4}\right ) \left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx+\int \frac {1}{-81 x-\log \left (1+e^{x^4}\right )} \, dx+\int \frac {x \left (81+4 x^3\right )}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx\\ &=-x-4 \int \frac {x^4}{\left (1+e^{x^4}\right ) \left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx+\int \frac {1}{-81 x-\log \left (1+e^{x^4}\right )} \, dx+\int \left (\frac {81 x}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2}+\frac {4 x^4}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2}\right ) \, dx\\ &=-x+4 \int \frac {x^4}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx-4 \int \frac {x^4}{\left (1+e^{x^4}\right ) \left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx+81 \int \frac {x}{\left (81 x+\log \left (1+e^{x^4}\right )\right )^2} \, dx+\int \frac {1}{-81 x-\log \left (1+e^{x^4}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 21, normalized size = 0.91 \begin {gather*} -x-\frac {x}{81 x+\log \left (1+e^{x^4}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6561*x^2 + E^x^4*(-6561*x^2 + 4*x^4) + (-1 + E^x^4*(-1 - 162*x) - 162*x)*Log[1 + E^x^4] + (-1 - E^
x^4)*Log[1 + E^x^4]^2)/(6561*x^2 + 6561*E^x^4*x^2 + (162*x + 162*E^x^4*x)*Log[1 + E^x^4] + (1 + E^x^4)*Log[1 +
E^x^4]^2),x]

[Out]

-x - x/(81*x + Log[1 + E^x^4])

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fricas [A]  time = 0.71, size = 31, normalized size = 1.35 \begin {gather*} -\frac {81 \, x^{2} + x \log \left (e^{\left (x^{4}\right )} + 1\right ) + x}{81 \, x + \log \left (e^{\left (x^{4}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x^4)-1)*log(exp(x^4)+1)^2+((-162*x-1)*exp(x^4)-162*x-1)*log(exp(x^4)+1)+(4*x^4-6561*x^2)*exp(
x^4)-6561*x^2)/((exp(x^4)+1)*log(exp(x^4)+1)^2+(162*x*exp(x^4)+162*x)*log(exp(x^4)+1)+6561*x^2*exp(x^4)+6561*x
^2),x, algorithm="fricas")

[Out]

-(81*x^2 + x*log(e^(x^4) + 1) + x)/(81*x + log(e^(x^4) + 1))

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giac [A]  time = 0.57, size = 31, normalized size = 1.35 \begin {gather*} -\frac {81 \, x^{2} + x \log \left (e^{\left (x^{4}\right )} + 1\right ) + x}{81 \, x + \log \left (e^{\left (x^{4}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x^4)-1)*log(exp(x^4)+1)^2+((-162*x-1)*exp(x^4)-162*x-1)*log(exp(x^4)+1)+(4*x^4-6561*x^2)*exp(
x^4)-6561*x^2)/((exp(x^4)+1)*log(exp(x^4)+1)^2+(162*x*exp(x^4)+162*x)*log(exp(x^4)+1)+6561*x^2*exp(x^4)+6561*x
^2),x, algorithm="giac")

[Out]

-(81*x^2 + x*log(e^(x^4) + 1) + x)/(81*x + log(e^(x^4) + 1))

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maple [A]  time = 0.06, size = 21, normalized size = 0.91

method	result	size
risch	\(-x -\frac {x}{81 x +\ln \left ({\mathrm e}^{x^{4}}+1\right )}\)	\(21\)
norman	\(\frac {\frac {\ln \left ({\mathrm e}^{x^{4}}+1\right )}{81}-\ln \left ({\mathrm e}^{x^{4}}+1\right ) x -81 x^{2}}{81 x +\ln \left ({\mathrm e}^{x^{4}}+1\right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp(x^4)-1)*ln(exp(x^4)+1)^2+((-162*x-1)*exp(x^4)-162*x-1)*ln(exp(x^4)+1)+(4*x^4-6561*x^2)*exp(x^4)-656
1*x^2)/((exp(x^4)+1)*ln(exp(x^4)+1)^2+(162*x*exp(x^4)+162*x)*ln(exp(x^4)+1)+6561*x^2*exp(x^4)+6561*x^2),x,meth
od=_RETURNVERBOSE)

[Out]

-x-x/(81*x+ln(exp(x^4)+1))

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maxima [A]  time = 0.52, size = 31, normalized size = 1.35 \begin {gather*} -\frac {81 \, x^{2} + x \log \left (e^{\left (x^{4}\right )} + 1\right ) + x}{81 \, x + \log \left (e^{\left (x^{4}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x^4)-1)*log(exp(x^4)+1)^2+((-162*x-1)*exp(x^4)-162*x-1)*log(exp(x^4)+1)+(4*x^4-6561*x^2)*exp(
x^4)-6561*x^2)/((exp(x^4)+1)*log(exp(x^4)+1)^2+(162*x*exp(x^4)+162*x)*log(exp(x^4)+1)+6561*x^2*exp(x^4)+6561*x
^2),x, algorithm="maxima")

[Out]

-(81*x^2 + x*log(e^(x^4) + 1) + x)/(81*x + log(e^(x^4) + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\ln \left ({\mathrm {e}}^{x^4}+1\right )}^2\,\left ({\mathrm {e}}^{x^4}+1\right )+{\mathrm {e}}^{x^4}\,\left (6561\,x^2-4\,x^4\right )+\ln \left ({\mathrm {e}}^{x^4}+1\right )\,\left (162\,x+{\mathrm {e}}^{x^4}\,\left (162\,x+1\right )+1\right )+6561\,x^2}{{\ln \left ({\mathrm {e}}^{x^4}+1\right )}^2\,\left ({\mathrm {e}}^{x^4}+1\right )+6561\,x^2\,{\mathrm {e}}^{x^4}+\ln \left ({\mathrm {e}}^{x^4}+1\right )\,\left (162\,x+162\,x\,{\mathrm {e}}^{x^4}\right )+6561\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(exp(x^4) + 1)^2*(exp(x^4) + 1) + exp(x^4)*(6561*x^2 - 4*x^4) + log(exp(x^4) + 1)*(162*x + exp(x^4)*(
162*x + 1) + 1) + 6561*x^2)/(log(exp(x^4) + 1)^2*(exp(x^4) + 1) + 6561*x^2*exp(x^4) + log(exp(x^4) + 1)*(162*x
+ 162*x*exp(x^4)) + 6561*x^2),x)

[Out]

int(-(log(exp(x^4) + 1)^2*(exp(x^4) + 1) + exp(x^4)*(6561*x^2 - 4*x^4) + log(exp(x^4) + 1)*(162*x + exp(x^4)*(
162*x + 1) + 1) + 6561*x^2)/(log(exp(x^4) + 1)^2*(exp(x^4) + 1) + 6561*x^2*exp(x^4) + log(exp(x^4) + 1)*(162*x
+ 162*x*exp(x^4)) + 6561*x^2), x)

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sympy [A]  time = 0.29, size = 15, normalized size = 0.65 \begin {gather*} - x - \frac {x}{81 x + \log {\left (e^{x^{4}} + 1 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x**4)-1)*ln(exp(x**4)+1)**2+((-162*x-1)*exp(x**4)-162*x-1)*ln(exp(x**4)+1)+(4*x**4-6561*x**2)
*exp(x**4)-6561*x**2)/((exp(x**4)+1)*ln(exp(x**4)+1)**2+(162*x*exp(x**4)+162*x)*ln(exp(x**4)+1)+6561*x**2*exp(
x**4)+6561*x**2),x)

[Out]

-x - x/(81*x + log(exp(x**4) + 1))

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