3.1.51 \(\int \frac {e^{-1+x} (15625-15625 x)+(-31250 e^{-1+x}-31250 x) \log (\frac {x}{e^{-1+x}+x})}{e^{-1+x} x^3+x^4} \, dx\)

Optimal. Leaf size=17 \[ \frac {15625 \log \left (\frac {x}{e^{-1+x}+x}\right )}{x^2} \]

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Rubi [F]  time = 1.38, 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 {e^{-1+x} (15625-15625 x)+\left (-31250 e^{-1+x}-31250 x\right ) \log \left (\frac {x}{e^{-1+x}+x}\right )}{e^{-1+x} x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-1 + x)*(15625 - 15625*x) + (-31250*E^(-1 + x) - 31250*x)*Log[x/(E^(-1 + x) + x)])/(E^(-1 + x)*x^3 + x
^4),x]

[Out]

(15625*(1 + x)^2)/(2*x^2) + (15625*Log[x/(E^x + E*x)])/x^2 - 15625*Defer[Int][E^x/(x^3*(E^x + E*x)), x] - 1562
5*E*Defer[Int][1/(x^2*(E^x + E*x)), x] + 15625*Defer[Int][E^x/(x^2*(E^x + E*x)), x] + 15625*E*Defer[Int][1/(x*
(E^x + E*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e \left (e^{-1+x} (15625-15625 x)+\left (-31250 e^{-1+x}-31250 x\right ) \log \left (\frac {x}{e^{-1+x}+x}\right )\right )}{x^3 \left (e^x+e x\right )} \, dx\\ &=e \int \frac {e^{-1+x} (15625-15625 x)+\left (-31250 e^{-1+x}-31250 x\right ) \log \left (\frac {x}{e^{-1+x}+x}\right )}{x^3 \left (e^x+e x\right )} \, dx\\ &=e \int \left (\frac {15625 (-1+x)}{x^2 \left (e^x+e x\right )}-\frac {15625 \left (1+x+2 \log \left (\frac {x}{e^x+e x}\right )\right )}{e x^3}\right ) \, dx\\ &=-\left (15625 \int \frac {1+x+2 \log \left (\frac {x}{e^x+e x}\right )}{x^3} \, dx\right )+(15625 e) \int \frac {-1+x}{x^2 \left (e^x+e x\right )} \, dx\\ &=-\left (15625 \int \left (\frac {1+x}{x^3}+\frac {2 \log \left (\frac {x}{e^x+e x}\right )}{x^3}\right ) \, dx\right )+(15625 e) \int \left (-\frac {1}{x^2 \left (e^x+e x\right )}+\frac {1}{x \left (e^x+e x\right )}\right ) \, dx\\ &=-\left (15625 \int \frac {1+x}{x^3} \, dx\right )-31250 \int \frac {\log \left (\frac {x}{e^x+e x}\right )}{x^3} \, dx-(15625 e) \int \frac {1}{x^2 \left (e^x+e x\right )} \, dx+(15625 e) \int \frac {1}{x \left (e^x+e x\right )} \, dx\\ &=\frac {15625 (1+x)^2}{2 x^2}+\frac {15625 \log \left (\frac {x}{e^x+e x}\right )}{x^2}-15625 \int \frac {e^x (1-x)}{x^3 \left (e^x+e x\right )} \, dx-(15625 e) \int \frac {1}{x^2 \left (e^x+e x\right )} \, dx+(15625 e) \int \frac {1}{x \left (e^x+e x\right )} \, dx\\ &=\frac {15625 (1+x)^2}{2 x^2}+\frac {15625 \log \left (\frac {x}{e^x+e x}\right )}{x^2}-15625 \int \left (\frac {e^x}{x^3 \left (e^x+e x\right )}-\frac {e^x}{x^2 \left (e^x+e x\right )}\right ) \, dx-(15625 e) \int \frac {1}{x^2 \left (e^x+e x\right )} \, dx+(15625 e) \int \frac {1}{x \left (e^x+e x\right )} \, dx\\ &=\frac {15625 (1+x)^2}{2 x^2}+\frac {15625 \log \left (\frac {x}{e^x+e x}\right )}{x^2}-15625 \int \frac {e^x}{x^3 \left (e^x+e x\right )} \, dx+15625 \int \frac {e^x}{x^2 \left (e^x+e x\right )} \, dx-(15625 e) \int \frac {1}{x^2 \left (e^x+e x\right )} \, dx+(15625 e) \int \frac {1}{x \left (e^x+e x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 25, normalized size = 1.47 \begin {gather*} -15625 \left (-\frac {1}{x^2}-\frac {\log \left (\frac {x}{e^x+e x}\right )}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-1 + x)*(15625 - 15625*x) + (-31250*E^(-1 + x) - 31250*x)*Log[x/(E^(-1 + x) + x)])/(E^(-1 + x)*x
^3 + x^4),x]

[Out]

-15625*(-x^(-2) - Log[x/(E^x + E*x)]/x^2)

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fricas [A]  time = 0.54, size = 16, normalized size = 0.94 \begin {gather*} \frac {15625 \, \log \left (\frac {x}{x + e^{\left (x - 1\right )}}\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-31250*exp(x-1)-31250*x)*log(x/(exp(x-1)+x))+(-15625*x+15625)*exp(x-1))/(x^3*exp(x-1)+x^4),x, algo
rithm="fricas")

[Out]

15625*log(x/(x + e^(x - 1)))/x^2

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giac [A]  time = 0.29, size = 19, normalized size = 1.12 \begin {gather*} \frac {15625 \, {\left (\log \left (\frac {x}{x e + e^{x}}\right ) + 1\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-31250*exp(x-1)-31250*x)*log(x/(exp(x-1)+x))+(-15625*x+15625)*exp(x-1))/(x^3*exp(x-1)+x^4),x, algo
rithm="giac")

[Out]

15625*(log(x/(x*e + e^x)) + 1)/x^2

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maple [A]  time = 0.09, size = 17, normalized size = 1.00

method	result	size
norman	\(\frac {15625 \ln \left (\frac {x}{{\mathrm e}^{x -1}+x}\right )}{x^{2}}\)	\(17\)
risch	\(-\frac {15625 \ln \left ({\mathrm e}^{x -1}+x \right )}{x^{2}}+\frac {-\frac {15625 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x -1}+x}\right ) \mathrm {csgn}\left (\frac {i x}{{\mathrm e}^{x -1}+x}\right )}{2}+\frac {15625 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{{\mathrm e}^{x -1}+x}\right )^{2}}{2}+\frac {15625 i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x -1}+x}\right ) \mathrm {csgn}\left (\frac {i x}{{\mathrm e}^{x -1}+x}\right )^{2}}{2}-\frac {15625 i \pi \mathrm {csgn}\left (\frac {i x}{{\mathrm e}^{x -1}+x}\right )^{3}}{2}+15625 \ln \relax (x )}{x^{2}}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-31250*exp(x-1)-31250*x)*ln(x/(exp(x-1)+x))+(-15625*x+15625)*exp(x-1))/(x^3*exp(x-1)+x^4),x,method=_RETU
RNVERBOSE)

[Out]

15625*ln(x/(exp(x-1)+x))/x^2

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maxima [A]  time = 0.70, size = 19, normalized size = 1.12 \begin {gather*} -\frac {15625 \, {\left (\log \left (x e + e^{x}\right ) - \log \relax (x) - 1\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-31250*exp(x-1)-31250*x)*log(x/(exp(x-1)+x))+(-15625*x+15625)*exp(x-1))/(x^3*exp(x-1)+x^4),x, algo
rithm="maxima")

[Out]

-15625*(log(x*e + e^x) - log(x) - 1)/x^2

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mupad [B]  time = 0.48, size = 19, normalized size = 1.12 \begin {gather*} \frac {15625\,\left (\ln \left (\frac {x}{{\mathrm {e}}^x+x\,\mathrm {e}}\right )+1\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x - 1)*(15625*x - 15625) + log(x/(x + exp(x - 1)))*(31250*x + 31250*exp(x - 1)))/(x^3*exp(x - 1) + x
^4),x)

[Out]

(15625*(log(x/(exp(x) + x*exp(1))) + 1))/x^2

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sympy [A]  time = 0.31, size = 14, normalized size = 0.82 \begin {gather*} \frac {15625 \log {\left (\frac {x}{x + e^{x - 1}} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-31250*exp(x-1)-31250*x)*ln(x/(exp(x-1)+x))+(-15625*x+15625)*exp(x-1))/(x**3*exp(x-1)+x**4),x)

[Out]

15625*log(x/(x + exp(x - 1)))/x**2

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