3.98.45 \(\int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7)+(90-100 x^3+40 x^4-4 x^5+e^{-5+x} (-90+100 x^3-40 x^4+4 x^5)) \log (-1+e^{-5+x})+(-50 x+20 x^2-2 x^3+e^{-5+x} (50 x-20 x^2+2 x^3)) \log ^2(-1+e^{-5+x})}{-25 x^4+10 x^5-x^6+e^{-5+x} (25 x^4-10 x^5+x^6)+(-50 x^2+20 x^3-2 x^4+e^{-5+x} (50 x^2-20 x^3+2 x^4)) \log (-1+e^{-5+x})+(-25+10 x-x^2+e^{-5+x} (25-10 x+x^2)) \log ^2(-1+e^{-5+x})} \, dx\)

Optimal. Leaf size=25 \[ x \left (x+\frac {18}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )}\right ) \]

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Rubi [F]  time = 4.91, 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 {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-90*x^2 + 36*x^3 - 50*x^5 + 20*x^6 - 2*x^7 + E^(-5 + x)*(90*x + 72*x^2 - 36*x^3 + 50*x^5 - 20*x^6 + 2*x^7
) + (90 - 100*x^3 + 40*x^4 - 4*x^5 + E^(-5 + x)*(-90 + 100*x^3 - 40*x^4 + 4*x^5))*Log[-1 + E^(-5 + x)] + (-50*
x + 20*x^2 - 2*x^3 + E^(-5 + x)*(50*x - 20*x^2 + 2*x^3))*Log[-1 + E^(-5 + x)]^2)/(-25*x^4 + 10*x^5 - x^6 + E^(
-5 + x)*(25*x^4 - 10*x^5 + x^6) + (-50*x^2 + 20*x^3 - 2*x^4 + E^(-5 + x)*(50*x^2 - 20*x^3 + 2*x^4))*Log[-1 + E
^(-5 + x)] + (-25 + 10*x - x^2 + E^(-5 + x)*(25 - 10*x + x^2))*Log[-1 + E^(-5 + x)]^2),x]

[Out]

x^2 - 198*Defer[Int][(x^2 + Log[-1 + E^(-5 + x)])^(-2), x] - 18*E^5*Defer[Int][1/((-E^5 + E^x)*(x^2 + Log[-1 +
E^(-5 + x)])^2), x] - 990*Defer[Int][1/((-5 + x)*(x^2 + Log[-1 + E^(-5 + x)])^2), x] - 90*E^5*Defer[Int][1/((
-E^5 + E^x)*(-5 + x)*(x^2 + Log[-1 + E^(-5 + x)])^2), x] - 36*Defer[Int][x/(x^2 + Log[-1 + E^(-5 + x)])^2, x]
- 90*Defer[Int][1/((-5 + x)^2*(x^2 + Log[-1 + E^(-5 + x)])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-x \left (-e^5 x \left (45-18 x+25 x^3-10 x^4+x^5\right )+e^x \left (45+36 x-18 x^2+25 x^4-10 x^5+x^6\right )\right )-\left (-e^5+e^x\right ) \left (-45+50 x^3-20 x^4+2 x^5\right ) \log \left (-1+e^{-5+x}\right )-\left (-e^5+e^x\right ) (-5+x)^2 x \log ^2\left (-1+e^{-5+x}\right )\right )}{\left (e^5-e^x\right ) (5-x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx\\ &=2 \int \frac {-x \left (-e^5 x \left (45-18 x+25 x^3-10 x^4+x^5\right )+e^x \left (45+36 x-18 x^2+25 x^4-10 x^5+x^6\right )\right )-\left (-e^5+e^x\right ) \left (-45+50 x^3-20 x^4+2 x^5\right ) \log \left (-1+e^{-5+x}\right )-\left (-e^5+e^x\right ) (-5+x)^2 x \log ^2\left (-1+e^{-5+x}\right )}{\left (e^5-e^x\right ) (5-x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx\\ &=2 \int \left (\frac {9 e^5 x}{\left (e^5-e^x\right ) (-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}+\frac {45 x+36 x^2-18 x^3+25 x^5-10 x^6+x^7-45 \log \left (-1+e^{-5+x}\right )+50 x^3 \log \left (-1+e^{-5+x}\right )-20 x^4 \log \left (-1+e^{-5+x}\right )+2 x^5 \log \left (-1+e^{-5+x}\right )+25 x \log ^2\left (-1+e^{-5+x}\right )-10 x^2 \log ^2\left (-1+e^{-5+x}\right )+x^3 \log ^2\left (-1+e^{-5+x}\right )}{(-5+x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {45 x+36 x^2-18 x^3+25 x^5-10 x^6+x^7-45 \log \left (-1+e^{-5+x}\right )+50 x^3 \log \left (-1+e^{-5+x}\right )-20 x^4 \log \left (-1+e^{-5+x}\right )+2 x^5 \log \left (-1+e^{-5+x}\right )+25 x \log ^2\left (-1+e^{-5+x}\right )-10 x^2 \log ^2\left (-1+e^{-5+x}\right )+x^3 \log ^2\left (-1+e^{-5+x}\right )}{(-5+x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx+\left (18 e^5\right ) \int \frac {x}{\left (e^5-e^x\right ) (-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx\\ &=2 \int \frac {x \left (45+36 x-18 x^2+25 x^4-10 x^5+x^6\right )+\left (-45+50 x^3-20 x^4+2 x^5\right ) \log \left (-1+e^{-5+x}\right )+(-5+x)^2 x \log ^2\left (-1+e^{-5+x}\right )}{(5-x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx+\left (18 e^5\right ) \int \left (-\frac {1}{\left (-e^5+e^x\right ) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}-\frac {5}{\left (-e^5+e^x\right ) (-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}\right ) \, dx\\ &=2 \int \left (x-\frac {9 x (1+2 x)}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}-\frac {45}{(-5+x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )}\right ) \, dx-\left (18 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-\left (90 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) (-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx\\ &=x^2-18 \int \frac {x (1+2 x)}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-90 \int \frac {1}{(-5+x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )} \, dx-\left (18 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-\left (90 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) (-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx\\ &=x^2-18 \int \left (\frac {11}{\left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}+\frac {55}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}+\frac {2 x}{\left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2}\right ) \, dx-90 \int \frac {1}{(-5+x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )} \, dx-\left (18 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-\left (90 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) (-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx\\ &=x^2-36 \int \frac {x}{\left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-90 \int \frac {1}{(-5+x)^2 \left (x^2+\log \left (-1+e^{-5+x}\right )\right )} \, dx-198 \int \frac {1}{\left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-990 \int \frac {1}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-\left (18 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx-\left (90 e^5\right ) \int \frac {1}{\left (-e^5+e^x\right ) (-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 32, normalized size = 1.28 \begin {gather*} 2 \left (\frac {x^2}{2}+\frac {9 x}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-90*x^2 + 36*x^3 - 50*x^5 + 20*x^6 - 2*x^7 + E^(-5 + x)*(90*x + 72*x^2 - 36*x^3 + 50*x^5 - 20*x^6 +
2*x^7) + (90 - 100*x^3 + 40*x^4 - 4*x^5 + E^(-5 + x)*(-90 + 100*x^3 - 40*x^4 + 4*x^5))*Log[-1 + E^(-5 + x)] +
(-50*x + 20*x^2 - 2*x^3 + E^(-5 + x)*(50*x - 20*x^2 + 2*x^3))*Log[-1 + E^(-5 + x)]^2)/(-25*x^4 + 10*x^5 - x^6
+ E^(-5 + x)*(25*x^4 - 10*x^5 + x^6) + (-50*x^2 + 20*x^3 - 2*x^4 + E^(-5 + x)*(50*x^2 - 20*x^3 + 2*x^4))*Log[
-1 + E^(-5 + x)] + (-25 + 10*x - x^2 + E^(-5 + x)*(25 - 10*x + x^2))*Log[-1 + E^(-5 + x)]^2),x]

[Out]

2*(x^2/2 + (9*x)/((-5 + x)*(x^2 + Log[-1 + E^(-5 + x)])))

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fricas [B]  time = 1.11, size = 52, normalized size = 2.08 \begin {gather*} \frac {x^{5} - 5 \, x^{4} + {\left (x^{3} - 5 \, x^{2}\right )} \log \left (e^{\left (x - 5\right )} - 1\right ) + 18 \, x}{x^{3} - 5 \, x^{2} + {\left (x - 5\right )} \log \left (e^{\left (x - 5\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-20*x^2+50*x)*exp(x-5)-2*x^3+20*x^2-50*x)*log(exp(x-5)-1)^2+((4*x^5-40*x^4+100*x^3-90)*exp(x
-5)-4*x^5+40*x^4-100*x^3+90)*log(exp(x-5)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(x-5)-2*x^7+20*x^6-50
*x^5+36*x^3-90*x^2)/(((x^2-10*x+25)*exp(x-5)-x^2+10*x-25)*log(exp(x-5)-1)^2+((2*x^4-20*x^3+50*x^2)*exp(x-5)-2*
x^4+20*x^3-50*x^2)*log(exp(x-5)-1)+(x^6-10*x^5+25*x^4)*exp(x-5)-x^6+10*x^5-25*x^4),x, algorithm="fricas")

[Out]

(x^5 - 5*x^4 + (x^3 - 5*x^2)*log(e^(x - 5) - 1) + 18*x)/(x^3 - 5*x^2 + (x - 5)*log(e^(x - 5) - 1))

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giac [B]  time = 0.81, size = 83, normalized size = 3.32 \begin {gather*} \frac {x^{5} - 5 \, x^{4} + x^{3} \log \left (-e^{5} + e^{x}\right ) - 5 \, x^{3} - 5 \, x^{2} \log \left (-e^{5} + e^{x}\right ) + 25 \, x^{2} + 18 \, x}{x^{3} - 5 \, x^{2} + x \log \left (-e^{5} + e^{x}\right ) - 5 \, x - 5 \, \log \left (-e^{5} + e^{x}\right ) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-20*x^2+50*x)*exp(x-5)-2*x^3+20*x^2-50*x)*log(exp(x-5)-1)^2+((4*x^5-40*x^4+100*x^3-90)*exp(x
-5)-4*x^5+40*x^4-100*x^3+90)*log(exp(x-5)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(x-5)-2*x^7+20*x^6-50
*x^5+36*x^3-90*x^2)/(((x^2-10*x+25)*exp(x-5)-x^2+10*x-25)*log(exp(x-5)-1)^2+((2*x^4-20*x^3+50*x^2)*exp(x-5)-2*
x^4+20*x^3-50*x^2)*log(exp(x-5)-1)+(x^6-10*x^5+25*x^4)*exp(x-5)-x^6+10*x^5-25*x^4),x, algorithm="giac")

[Out]

(x^5 - 5*x^4 + x^3*log(-e^5 + e^x) - 5*x^3 - 5*x^2*log(-e^5 + e^x) + 25*x^2 + 18*x)/(x^3 - 5*x^2 + x*log(-e^5
+ e^x) - 5*x - 5*log(-e^5 + e^x) + 25)

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maple [A]  time = 0.04, size = 26, normalized size = 1.04

method	result	size
risch	\(x^{2}+\frac {18 x}{\left (x -5\right ) \left (\ln \left ({\mathrm e}^{x -5}-1\right )+x^{2}\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3-20*x^2+50*x)*exp(x-5)-2*x^3+20*x^2-50*x)*ln(exp(x-5)-1)^2+((4*x^5-40*x^4+100*x^3-90)*exp(x-5)-4*x
^5+40*x^4-100*x^3+90)*ln(exp(x-5)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(x-5)-2*x^7+20*x^6-50*x^5+36*
x^3-90*x^2)/(((x^2-10*x+25)*exp(x-5)-x^2+10*x-25)*ln(exp(x-5)-1)^2+((2*x^4-20*x^3+50*x^2)*exp(x-5)-2*x^4+20*x^
3-50*x^2)*ln(exp(x-5)-1)+(x^6-10*x^5+25*x^4)*exp(x-5)-x^6+10*x^5-25*x^4),x,method=_RETURNVERBOSE)

[Out]

x^2+18*x/(x-5)/(ln(exp(x-5)-1)+x^2)

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maxima [B]  time = 0.67, size = 68, normalized size = 2.72 \begin {gather*} \frac {x^{5} - 5 \, x^{4} - 5 \, x^{3} + 25 \, x^{2} + {\left (x^{3} - 5 \, x^{2}\right )} \log \left (-e^{5} + e^{x}\right ) + 18 \, x}{x^{3} - 5 \, x^{2} + {\left (x - 5\right )} \log \left (-e^{5} + e^{x}\right ) - 5 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-20*x^2+50*x)*exp(x-5)-2*x^3+20*x^2-50*x)*log(exp(x-5)-1)^2+((4*x^5-40*x^4+100*x^3-90)*exp(x
-5)-4*x^5+40*x^4-100*x^3+90)*log(exp(x-5)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(x-5)-2*x^7+20*x^6-50
*x^5+36*x^3-90*x^2)/(((x^2-10*x+25)*exp(x-5)-x^2+10*x-25)*log(exp(x-5)-1)^2+((2*x^4-20*x^3+50*x^2)*exp(x-5)-2*
x^4+20*x^3-50*x^2)*log(exp(x-5)-1)+(x^6-10*x^5+25*x^4)*exp(x-5)-x^6+10*x^5-25*x^4),x, algorithm="maxima")

[Out]

(x^5 - 5*x^4 - 5*x^3 + 25*x^2 + (x^3 - 5*x^2)*log(-e^5 + e^x) + 18*x)/(x^3 - 5*x^2 + (x - 5)*log(-e^5 + e^x) -
5*x + 25)

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mupad [B]  time = 6.32, size = 192, normalized size = 7.68 \begin {gather*} x^2-\frac {45}{x^3-\frac {19\,x^2}{2}+20\,x+\frac {25}{2}}-\frac {\frac {18\,\left (5\,x\,{\mathrm {e}}^{x-5}+4\,x^2\,{\mathrm {e}}^{x-5}-2\,x^3\,{\mathrm {e}}^{x-5}-5\,x^2+2\,x^3\right )}{{\left (x-5\right )}^2\,\left ({\mathrm {e}}^{x-5}-2\,x+2\,x\,{\mathrm {e}}^{x-5}\right )}-\frac {90\,\ln \left ({\mathrm {e}}^{-5}\,{\mathrm {e}}^x-1\right )\,\left ({\mathrm {e}}^{x-5}-1\right )}{{\left (x-5\right )}^2\,\left ({\mathrm {e}}^{x-5}-2\,x+2\,x\,{\mathrm {e}}^{x-5}\right )}}{\ln \left ({\mathrm {e}}^{-5}\,{\mathrm {e}}^x-1\right )+x^2}+\frac {90\,\left (-2\,x^3+9\,x^2+6\,x-5\right )}{\left (2\,x-{\mathrm {e}}^{x-5}\,\left (2\,x+1\right )\right )\,\left (2\,x+1\right )\,{\left (x-5\right )}^3\,\left (2\,x^2+x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(exp(x - 5) - 1)*(exp(x - 5)*(100*x^3 - 40*x^4 + 4*x^5 - 90) - 100*x^3 + 40*x^4 - 4*x^5 + 90) - log(ex
p(x - 5) - 1)^2*(50*x - exp(x - 5)*(50*x - 20*x^2 + 2*x^3) - 20*x^2 + 2*x^3) - 90*x^2 + 36*x^3 - 50*x^5 + 20*x
^6 - 2*x^7 + exp(x - 5)*(90*x + 72*x^2 - 36*x^3 + 50*x^5 - 20*x^6 + 2*x^7))/(log(exp(x - 5) - 1)*(exp(x - 5)*(
50*x^2 - 20*x^3 + 2*x^4) - 50*x^2 + 20*x^3 - 2*x^4) + exp(x - 5)*(25*x^4 - 10*x^5 + x^6) - 25*x^4 + 10*x^5 - x
^6 + log(exp(x - 5) - 1)^2*(10*x + exp(x - 5)*(x^2 - 10*x + 25) - x^2 - 25)),x)

[Out]

x^2 - 45/(20*x - (19*x^2)/2 + x^3 + 25/2) - ((18*(5*x*exp(x - 5) + 4*x^2*exp(x - 5) - 2*x^3*exp(x - 5) - 5*x^2
+ 2*x^3))/((x - 5)^2*(exp(x - 5) - 2*x + 2*x*exp(x - 5))) - (90*log(exp(-5)*exp(x) - 1)*(exp(x - 5) - 1))/((x
- 5)^2*(exp(x - 5) - 2*x + 2*x*exp(x - 5))))/(log(exp(-5)*exp(x) - 1) + x^2) + (90*(6*x + 9*x^2 - 2*x^3 - 5))
/((2*x - exp(x - 5)*(2*x + 1))*(2*x + 1)*(x - 5)^3*(x + 2*x^2 - 1))

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sympy [A]  time = 0.38, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + \frac {18 x}{x^{3} - 5 x^{2} + \left (x - 5\right ) \log {\left (e^{x - 5} - 1 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3-20*x**2+50*x)*exp(x-5)-2*x**3+20*x**2-50*x)*ln(exp(x-5)-1)**2+((4*x**5-40*x**4+100*x**3-90
)*exp(x-5)-4*x**5+40*x**4-100*x**3+90)*ln(exp(x-5)-1)+(2*x**7-20*x**6+50*x**5-36*x**3+72*x**2+90*x)*exp(x-5)-2
*x**7+20*x**6-50*x**5+36*x**3-90*x**2)/(((x**2-10*x+25)*exp(x-5)-x**2+10*x-25)*ln(exp(x-5)-1)**2+((2*x**4-20*x
**3+50*x**2)*exp(x-5)-2*x**4+20*x**3-50*x**2)*ln(exp(x-5)-1)+(x**6-10*x**5+25*x**4)*exp(x-5)-x**6+10*x**5-25*x
**4),x)

[Out]

x**2 + 18*x/(x**3 - 5*x**2 + (x - 5)*log(exp(x - 5) - 1))

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