3.37.70 \(\int \frac {3^{2/x} (x^4)^{-2/x} (-2 x^2+(-8 x+2 x^2) \log (x)+2 x \log (x) \log (\frac {x^4}{3})+3^{-1/x} (x^4)^{\frac {1}{x}} (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log (\frac {x^4}{3})))}{x \log ^3(x)} \, dx\)

Optimal. Leaf size=25 \[ \left (20-\frac {3^{\frac {1}{x}} x \left (x^4\right )^{-1/x}}{\log (x)}\right )^2 \]

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Rubi [F]  time = 5.52, 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^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3^(2/x)*(-2*x^2 + (-8*x + 2*x^2)*Log[x] + 2*x*Log[x]*Log[x^4/3] + ((x^4)^x^(-1)*(40*x*Log[x] + (160 - 40*
x)*Log[x]^2 - 40*Log[x]^2*Log[x^4/3]))/3^x^(-1)))/(x*(x^4)^(2/x)*Log[x]^3),x]

[Out]

-2*Defer[Int][(9^x^(-1)*x)/((x^4)^(2/x)*Log[x]^3), x] - 8*Defer[Int][9^x^(-1)/((x^4)^(2/x)*Log[x]^2), x] + 2*D
efer[Int][(9^x^(-1)*x)/((x^4)^(2/x)*Log[x]^2), x] + 40*Defer[Int][3^x^(-1)/((x^4)^x^(-1)*Log[x]^2), x] - 40*De
fer[Int][3^x^(-1)/((x^4)^x^(-1)*Log[x]), x] + 160*Defer[Int][3^x^(-1)/(x*(x^4)^x^(-1)*Log[x]), x] + 2*Defer[In
t][(9^x^(-1)*Log[x^4/3])/((x^4)^(2/x)*Log[x]^2), x] - 40*Defer[Int][(3^x^(-1)*Log[x^4/3])/(x*(x^4)^x^(-1)*Log[
x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2\ 3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (-x+\log (x) \left (-4+x+\log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx\\ &=2 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (-x+\log (x) \left (-4+x+\log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx\\ &=2 \int \left (\frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{\log ^3(x)}-\frac {20\ 3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{x \log ^2(x)}\right ) \, dx\\ &=2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{\log ^3(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{x \log ^2(x)} \, dx\\ &=2 \int \left (\frac {3^{2/x} \left (x^4\right )^{-2/x} (-x-4 \log (x)+x \log (x))}{\log ^3(x)}+\frac {3^{2/x} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)}\right ) \, dx-40 \int \left (\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} (-x-4 \log (x)+x \log (x))}{x \log ^2(x)}+\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)}\right ) \, dx\\ &=2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} (-x-4 \log (x)+x \log (x))}{\log ^3(x)} \, dx+2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} (-x-4 \log (x)+x \log (x))}{x \log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=2 \int \left (-\frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^3(x)}+\frac {3^{2/x} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)}\right ) \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \left (-\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)}+\frac {3^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-1/x}}{x \log (x)}\right ) \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=-\left (2 \int \frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {3^{2/x} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-1/x}}{x \log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=-\left (2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {9^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \left (\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)}-\frac {4\ 3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)}\right ) \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=2 \int \left (-\frac {4\ 9^{\frac {1}{x}} \left (x^4\right )^{-2/x}}{\log ^2(x)}+\frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^2(x)}\right ) \, dx-2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx+160 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)} \, dx\\ &=-\left (2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-8 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx+160 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.25, size = 39, normalized size = 1.56 \begin {gather*} \frac {3^{\frac {1}{x}} x \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-40 \left (x^4\right )^{\frac {1}{x}} \log (x)\right )}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3^(2/x)*(-2*x^2 + (-8*x + 2*x^2)*Log[x] + 2*x*Log[x]*Log[x^4/3] + ((x^4)^x^(-1)*(40*x*Log[x] + (160
- 40*x)*Log[x]^2 - 40*Log[x]^2*Log[x^4/3]))/3^x^(-1)))/(x*(x^4)^(2/x)*Log[x]^3),x]

[Out]

(3^x^(-1)*x*(3^x^(-1)*x - 40*(x^4)^x^(-1)*Log[x]))/((x^4)^(2/x)*Log[x]^2)

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fricas [B]  time = 0.74, size = 43, normalized size = 1.72 \begin {gather*} -\frac {{\left (40 \, x e^{\left (-\frac {\log \relax (3) - 4 \, \log \relax (x)}{x}\right )} \log \relax (x) - x^{2}\right )} e^{\left (\frac {2 \, {\left (\log \relax (3) - 4 \, \log \relax (x)\right )}}{x}\right )}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*log(x)^2*log(1/3*x^4)+(-40*x+160)*log(x)^2+40*x*log(x))*exp(log(1/3*x^4)/x)+2*x*log(x)*log(1/3
*x^4)+(2*x^2-8*x)*log(x)-2*x^2)/x/log(x)^3/exp(log(1/3*x^4)/x)^2,x, algorithm="fricas")

[Out]

-(40*x*e^(-(log(3) - 4*log(x))/x)*log(x) - x^2)*e^(2*(log(3) - 4*log(x))/x)/log(x)^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (x \log \left (\frac {1}{3} \, x^{4}\right ) \log \relax (x) - 20 \, {\left ({\left (x - 4\right )} \log \relax (x)^{2} + \log \left (\frac {1}{3} \, x^{4}\right ) \log \relax (x)^{2} - x \log \relax (x)\right )} \left (\frac {1}{3} \, x^{4}\right )^{\left (\frac {1}{x}\right )} - x^{2} + {\left (x^{2} - 4 \, x\right )} \log \relax (x)\right )}}{\left (\frac {1}{3} \, x^{4}\right )^{\frac {2}{x}} x \log \relax (x)^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*log(x)^2*log(1/3*x^4)+(-40*x+160)*log(x)^2+40*x*log(x))*exp(log(1/3*x^4)/x)+2*x*log(x)*log(1/3
*x^4)+(2*x^2-8*x)*log(x)-2*x^2)/x/log(x)^3/exp(log(1/3*x^4)/x)^2,x, algorithm="giac")

[Out]

integrate(2*(x*log(1/3*x^4)*log(x) - 20*((x - 4)*log(x)^2 + log(1/3*x^4)*log(x)^2 - x*log(x))*(1/3*x^4)^(1/x)
- x^2 + (x^2 - 4*x)*log(x))/((1/3*x^4)^(2/x)*x*log(x)^3), x)

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maple [A]  time = 0.24, size = 42, normalized size = 1.68

method	result	size
default	\(-\frac {40 x \,{\mathrm e}^{-\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \relax (x )}+\frac {x^{2} {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \relax (x )^{2}}\)	\(42\)
risch	\(-\frac {40 x 3^{\frac {1}{x}} x^{-\frac {4}{x}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i x^{2}\right )^{3}-2 \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )-\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{3}\right )^{3}-\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{4}\right )^{3}\right )}{2 x}}}{\ln \relax (x )}+\frac {x^{2} 3^{\frac {2}{x}} x^{-\frac {8}{x}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i x^{2}\right )^{3}-2 \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )-\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{3}\right )^{3}-\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{4}\right )^{3}\right )}{x}}}{\ln \relax (x )^{2}}\)	\(399\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40*ln(x)^2*ln(1/3*x^4)+(-40*x+160)*ln(x)^2+40*x*ln(x))*exp(ln(1/3*x^4)/x)+2*x*ln(x)*ln(1/3*x^4)+(2*x^2-
8*x)*ln(x)-2*x^2)/x/ln(x)^3/exp(ln(1/3*x^4)/x)^2,x,method=_RETURNVERBOSE)

[Out]

-40*x/ln(x)/exp(ln(1/3*x^4)/x)+x^2*exp(-2*ln(1/3*x^4)/x)/ln(x)^2

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maxima [B]  time = 0.52, size = 46, normalized size = 1.84 \begin {gather*} \frac {x^{2} e^{\left (\frac {2 \, \log \relax (3)}{x} - \frac {8 \, \log \relax (x)}{x}\right )} - 40 \, x e^{\left (\frac {\log \relax (3)}{x} - \frac {4 \, \log \relax (x)}{x}\right )} \log \relax (x)}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*log(x)^2*log(1/3*x^4)+(-40*x+160)*log(x)^2+40*x*log(x))*exp(log(1/3*x^4)/x)+2*x*log(x)*log(1/3
*x^4)+(2*x^2-8*x)*log(x)-2*x^2)/x/log(x)^3/exp(log(1/3*x^4)/x)^2,x, algorithm="maxima")

[Out]

(x^2*e^(2*log(3)/x - 8*log(x)/x) - 40*x*e^(log(3)/x - 4*log(x)/x)*log(x))/log(x)^2

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mupad [B]  time = 2.53, size = 41, normalized size = 1.64 \begin {gather*} \frac {3^{1/x}\,x\,\left (3^{1/x}\,x-40\,\ln \relax (x)\,{\left (x^4\right )}^{1/x}\right )}{{\ln \relax (x)}^2\,{\left (x^4\right )}^{2/x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*log(x^4/3))/x)*(log(x)*(8*x - 2*x^2) + 2*x^2 + exp(log(x^4/3)/x)*(40*log(x^4/3)*log(x)^2 - 40*x*
log(x) + log(x)^2*(40*x - 160)) - 2*x*log(x^4/3)*log(x)))/(x*log(x)^3),x)

[Out]

(3^(1/x)*x*(3^(1/x)*x - 40*log(x)*(x^4)^(1/x)))/(log(x)^2*(x^4)^(2/x))

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sympy [B]  time = 0.43, size = 44, normalized size = 1.76 \begin {gather*} \frac {x^{2} e^{- \frac {2 \left (4 \log {\relax (x )} - \log {\relax (3 )}\right )}{x}} \log {\relax (x )} - 40 x e^{- \frac {4 \log {\relax (x )} - \log {\relax (3 )}}{x}} \log {\relax (x )}^{2}}{\log {\relax (x )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*ln(x)**2*ln(1/3*x**4)+(-40*x+160)*ln(x)**2+40*x*ln(x))*exp(ln(1/3*x**4)/x)+2*x*ln(x)*ln(1/3*x*
*4)+(2*x**2-8*x)*ln(x)-2*x**2)/x/ln(x)**3/exp(ln(1/3*x**4)/x)**2,x)

[Out]

(x**2*exp(-2*(4*log(x) - log(3))/x)*log(x) - 40*x*exp(-(4*log(x) - log(3))/x)*log(x)**2)/log(x)**3

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