3.12.77 \(\int \frac {-80+15 x+(80-20 x) \log (4-x)}{e^{22} (-4 x^5+x^6)+e^{22} (8 x^5-2 x^6) \log (4-x)+e^{22} (-4 x^5+x^6) \log ^2(4-x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {5}{e^{22} x^4 (-1+\log (4-x))} \]

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Rubi [F]  time = 0.73, 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 {-80+15 x+(80-20 x) \log (4-x)}{e^{22} \left (-4 x^5+x^6\right )+e^{22} \left (8 x^5-2 x^6\right ) \log (4-x)+e^{22} \left (-4 x^5+x^6\right ) \log ^2(4-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-80 + 15*x + (80 - 20*x)*Log[4 - x])/(E^22*(-4*x^5 + x^6) + E^22*(8*x^5 - 2*x^6)*Log[4 - x] + E^22*(-4*x^
5 + x^6)*Log[4 - x]^2),x]

[Out]

-5/(256*E^22*(1 - Log[4 - x])) + (5*Defer[Int][1/(x^4*(-1 + Log[4 - x])^2), x])/(4*E^22) + (5*Defer[Int][1/(x^
3*(-1 + Log[4 - x])^2), x])/(16*E^22) + (5*Defer[Int][1/(x^2*(-1 + Log[4 - x])^2), x])/(64*E^22) + (5*Defer[In
t][1/(x*(-1 + Log[4 - x])^2), x])/(256*E^22) - (20*Defer[Int][1/(x^5*(-1 + Log[4 - x])), x])/E^22

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {80-15 x+20 (-4+x) \log (4-x)}{e^{22} (4-x) x^5 (1-\log (4-x))^2} \, dx\\ &=\frac {\int \frac {80-15 x+20 (-4+x) \log (4-x)}{(4-x) x^5 (1-\log (4-x))^2} \, dx}{e^{22}}\\ &=\frac {\int \left (-\frac {5}{(-4+x) x^4 (-1+\log (4-x))^2}-\frac {20}{x^5 (-1+\log (4-x))}\right ) \, dx}{e^{22}}\\ &=-\frac {5 \int \frac {1}{(-4+x) x^4 (-1+\log (4-x))^2} \, dx}{e^{22}}-\frac {20 \int \frac {1}{x^5 (-1+\log (4-x))} \, dx}{e^{22}}\\ &=-\frac {5 \int \left (\frac {1}{256 (-4+x) (-1+\log (4-x))^2}-\frac {1}{4 x^4 (-1+\log (4-x))^2}-\frac {1}{16 x^3 (-1+\log (4-x))^2}-\frac {1}{64 x^2 (-1+\log (4-x))^2}-\frac {1}{256 x (-1+\log (4-x))^2}\right ) \, dx}{e^{22}}-\frac {20 \int \frac {1}{x^5 (-1+\log (4-x))} \, dx}{e^{22}}\\ &=-\frac {5 \int \frac {1}{(-4+x) (-1+\log (4-x))^2} \, dx}{256 e^{22}}+\frac {5 \int \frac {1}{x (-1+\log (4-x))^2} \, dx}{256 e^{22}}+\frac {5 \int \frac {1}{x^2 (-1+\log (4-x))^2} \, dx}{64 e^{22}}+\frac {5 \int \frac {1}{x^3 (-1+\log (4-x))^2} \, dx}{16 e^{22}}+\frac {5 \int \frac {1}{x^4 (-1+\log (4-x))^2} \, dx}{4 e^{22}}-\frac {20 \int \frac {1}{x^5 (-1+\log (4-x))} \, dx}{e^{22}}\\ &=\frac {5 \int \frac {1}{x (-1+\log (4-x))^2} \, dx}{256 e^{22}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{x (-1+\log (x))^2} \, dx,x,4-x\right )}{256 e^{22}}+\frac {5 \int \frac {1}{x^2 (-1+\log (4-x))^2} \, dx}{64 e^{22}}+\frac {5 \int \frac {1}{x^3 (-1+\log (4-x))^2} \, dx}{16 e^{22}}+\frac {5 \int \frac {1}{x^4 (-1+\log (4-x))^2} \, dx}{4 e^{22}}-\frac {20 \int \frac {1}{x^5 (-1+\log (4-x))} \, dx}{e^{22}}\\ &=\frac {5 \int \frac {1}{x (-1+\log (4-x))^2} \, dx}{256 e^{22}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,-1+\log (4-x)\right )}{256 e^{22}}+\frac {5 \int \frac {1}{x^2 (-1+\log (4-x))^2} \, dx}{64 e^{22}}+\frac {5 \int \frac {1}{x^3 (-1+\log (4-x))^2} \, dx}{16 e^{22}}+\frac {5 \int \frac {1}{x^4 (-1+\log (4-x))^2} \, dx}{4 e^{22}}-\frac {20 \int \frac {1}{x^5 (-1+\log (4-x))} \, dx}{e^{22}}\\ &=-\frac {5}{256 e^{22} (1-\log (4-x))}+\frac {5 \int \frac {1}{x (-1+\log (4-x))^2} \, dx}{256 e^{22}}+\frac {5 \int \frac {1}{x^2 (-1+\log (4-x))^2} \, dx}{64 e^{22}}+\frac {5 \int \frac {1}{x^3 (-1+\log (4-x))^2} \, dx}{16 e^{22}}+\frac {5 \int \frac {1}{x^4 (-1+\log (4-x))^2} \, dx}{4 e^{22}}-\frac {20 \int \frac {1}{x^5 (-1+\log (4-x))} \, dx}{e^{22}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} \frac {5}{e^{22} x^4 (-1+\log (4-x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-80 + 15*x + (80 - 20*x)*Log[4 - x])/(E^22*(-4*x^5 + x^6) + E^22*(8*x^5 - 2*x^6)*Log[4 - x] + E^22*
(-4*x^5 + x^6)*Log[4 - x]^2),x]

[Out]

5/(E^22*x^4*(-1 + Log[4 - x]))

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fricas [A]  time = 0.93, size = 24, normalized size = 1.33 \begin {gather*} \frac {5}{x^{4} e^{22} \log \left (-x + 4\right ) - x^{4} e^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x+80)*log(-x+4)+15*x-80)/((x^6-4*x^5)*exp(11)^2*log(-x+4)^2+(-2*x^6+8*x^5)*exp(11)^2*log(-x+4)
+(x^6-4*x^5)*exp(11)^2),x, algorithm="fricas")

[Out]

5/(x^4*e^22*log(-x + 4) - x^4*e^22)

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giac [A]  time = 1.04, size = 24, normalized size = 1.33 \begin {gather*} \frac {5}{x^{4} e^{22} \log \left (-x + 4\right ) - x^{4} e^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x+80)*log(-x+4)+15*x-80)/((x^6-4*x^5)*exp(11)^2*log(-x+4)^2+(-2*x^6+8*x^5)*exp(11)^2*log(-x+4)
+(x^6-4*x^5)*exp(11)^2),x, algorithm="giac")

[Out]

5/(x^4*e^22*log(-x + 4) - x^4*e^22)

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maple [A]  time = 0.15, size = 18, normalized size = 1.00

method	result	size
risch	\(\frac {5 \,{\mathrm e}^{-22}}{\left (\ln \left (-x +4\right )-1\right ) x^{4}}\)	\(18\)
norman	\(\frac {5 \,{\mathrm e}^{-22}}{\left (\ln \left (-x +4\right )-1\right ) x^{4}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*x+80)*ln(-x+4)+15*x-80)/((x^6-4*x^5)*exp(11)^2*ln(-x+4)^2+(-2*x^6+8*x^5)*exp(11)^2*ln(-x+4)+(x^6-4*x
^5)*exp(11)^2),x,method=_RETURNVERBOSE)

[Out]

5/(ln(-x+4)-1)*exp(-22)/x^4

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maxima [A]  time = 0.47, size = 24, normalized size = 1.33 \begin {gather*} \frac {5}{x^{4} e^{22} \log \left (-x + 4\right ) - x^{4} e^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x+80)*log(-x+4)+15*x-80)/((x^6-4*x^5)*exp(11)^2*log(-x+4)^2+(-2*x^6+8*x^5)*exp(11)^2*log(-x+4)
+(x^6-4*x^5)*exp(11)^2),x, algorithm="maxima")

[Out]

5/(x^4*e^22*log(-x + 4) - x^4*e^22)

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mupad [B]  time = 0.39, size = 17, normalized size = 0.94 \begin {gather*} \frac {5\,{\mathrm {e}}^{-22}}{x^4\,\left (\ln \left (4-x\right )-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4 - x)*(20*x - 80) - 15*x + 80)/(exp(22)*(4*x^5 - x^6) - exp(22)*log(4 - x)*(8*x^5 - 2*x^6) + exp(22)
*log(4 - x)^2*(4*x^5 - x^6)),x)

[Out]

(5*exp(-22))/(x^4*(log(4 - x) - 1))

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sympy [A]  time = 0.13, size = 19, normalized size = 1.06 \begin {gather*} \frac {5}{x^{4} e^{22} \log {\left (4 - x \right )} - x^{4} e^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x+80)*ln(-x+4)+15*x-80)/((x**6-4*x**5)*exp(11)**2*ln(-x+4)**2+(-2*x**6+8*x**5)*exp(11)**2*ln(-
x+4)+(x**6-4*x**5)*exp(11)**2),x)

[Out]

5/(x**4*exp(22)*log(4 - x) - x**4*exp(22))

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