3.58.16 \(\int \frac {25 x^6+10 x^7+x^8+e^x (-25 x^6-10 x^7-x^8)+(-150 x^6-70 x^7-8 x^8+e^3 (-150 x^5-70 x^6-8 x^7)+e^x (150 x^5+70 x^6+8 x^7)) \log (-e^3+e^x-x)}{(-e^3+e^x-x) \log ^2(-e^3+e^x-x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {(-5-x)^2 x^6}{\log \left (-e^3+e^x-x\right )} \]

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Rubi [F]  time = 5.60, 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 {25 x^6+10 x^7+x^8+e^x \left (-25 x^6-10 x^7-x^8\right )+\left (-150 x^6-70 x^7-8 x^8+e^3 \left (-150 x^5-70 x^6-8 x^7\right )+e^x \left (150 x^5+70 x^6+8 x^7\right )\right ) \log \left (-e^3+e^x-x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25*x^6 + 10*x^7 + x^8 + E^x*(-25*x^6 - 10*x^7 - x^8) + (-150*x^6 - 70*x^7 - 8*x^8 + E^3*(-150*x^5 - 70*x^
6 - 8*x^7) + E^x*(150*x^5 + 70*x^6 + 8*x^7))*Log[-E^3 + E^x - x])/((-E^3 + E^x - x)*Log[-E^3 + E^x - x]^2),x]

[Out]

-25*Defer[Int][x^6/Log[-E^3 + E^x - x]^2, x] - 10*Defer[Int][x^7/Log[-E^3 + E^x - x]^2, x] - Defer[Int][x^8/Lo
g[-E^3 + E^x - x]^2, x] - Defer[Int][x^9/((-E^3 + E^x - x)*Log[-E^3 + E^x - x]^2), x] - 25*(1 - E^3)*Defer[Int
][x^6/((E^3 - E^x + x)*Log[-E^3 + E^x - x]^2), x] + 5*(3 + 2*E^3)*Defer[Int][x^7/((E^3 - E^x + x)*Log[-E^3 + E
^x - x]^2), x] + (9 + E^3)*Defer[Int][x^8/((E^3 - E^x + x)*Log[-E^3 + E^x - x]^2), x] + 150*Defer[Int][x^5/Log
[-E^3 + E^x - x], x] + 70*Defer[Int][x^6/Log[-E^3 + E^x - x], x] + 8*Defer[Int][x^7/Log[-E^3 + E^x - x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^5 (5+x) \left (\left (-1+e^x\right ) x (5+x)-2 \left (-e^3+e^x-x\right ) (15+4 x) \log \left (-e^3+e^x-x\right )\right )}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx\\ &=\int \left (-\frac {x^6 (5+x)^2 \left (-1+e^3+x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )}-\frac {x^5 (5+x) \left (5 x+x^2-30 \log \left (-e^3+e^x-x\right )-8 x \log \left (-e^3+e^x-x\right )\right )}{\log ^2\left (-e^3+e^x-x\right )}\right ) \, dx\\ &=-\int \frac {x^6 (5+x)^2 \left (-1+e^3+x\right )}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\int \frac {x^5 (5+x) \left (5 x+x^2-30 \log \left (-e^3+e^x-x\right )-8 x \log \left (-e^3+e^x-x\right )\right )}{\log ^2\left (-e^3+e^x-x\right )} \, dx\\ &=-\int \left (\frac {x^9}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )}-\frac {25 \left (-1+e^3\right ) x^6}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )}-\frac {5 \left (3+2 e^3\right ) x^7}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )}-\frac {\left (9+e^3\right ) x^8}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )}\right ) \, dx-\int \frac {x^5 (5+x) \left (x (5+x)-2 (15+4 x) \log \left (-e^3+e^x-x\right )\right )}{\log ^2\left (-e^3+e^x-x\right )} \, dx\\ &=-\left (\left (-9-e^3\right ) \int \frac {x^8}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx\right )-\left (25 \left (1-e^3\right )\right ) \int \frac {x^6}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx+\left (5 \left (3+2 e^3\right )\right ) \int \frac {x^7}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\int \left (\frac {x^6 (5+x)^2}{\log ^2\left (-e^3+e^x-x\right )}-\frac {2 x^5 (5+x) (15+4 x)}{\log \left (-e^3+e^x-x\right )}\right ) \, dx-\int \frac {x^9}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx\\ &=2 \int \frac {x^5 (5+x) (15+4 x)}{\log \left (-e^3+e^x-x\right )} \, dx-\left (-9-e^3\right ) \int \frac {x^8}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\left (25 \left (1-e^3\right )\right ) \int \frac {x^6}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx+\left (5 \left (3+2 e^3\right )\right ) \int \frac {x^7}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\int \frac {x^9}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\int \frac {x^6 (5+x)^2}{\log ^2\left (-e^3+e^x-x\right )} \, dx\\ &=2 \int \left (\frac {75 x^5}{\log \left (-e^3+e^x-x\right )}+\frac {35 x^6}{\log \left (-e^3+e^x-x\right )}+\frac {4 x^7}{\log \left (-e^3+e^x-x\right )}\right ) \, dx-\left (-9-e^3\right ) \int \frac {x^8}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\left (25 \left (1-e^3\right )\right ) \int \frac {x^6}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx+\left (5 \left (3+2 e^3\right )\right ) \int \frac {x^7}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\int \left (\frac {25 x^6}{\log ^2\left (-e^3+e^x-x\right )}+\frac {10 x^7}{\log ^2\left (-e^3+e^x-x\right )}+\frac {x^8}{\log ^2\left (-e^3+e^x-x\right )}\right ) \, dx-\int \frac {x^9}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx\\ &=8 \int \frac {x^7}{\log \left (-e^3+e^x-x\right )} \, dx-10 \int \frac {x^7}{\log ^2\left (-e^3+e^x-x\right )} \, dx-25 \int \frac {x^6}{\log ^2\left (-e^3+e^x-x\right )} \, dx+70 \int \frac {x^6}{\log \left (-e^3+e^x-x\right )} \, dx+150 \int \frac {x^5}{\log \left (-e^3+e^x-x\right )} \, dx-\left (-9-e^3\right ) \int \frac {x^8}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\left (25 \left (1-e^3\right )\right ) \int \frac {x^6}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx+\left (5 \left (3+2 e^3\right )\right ) \int \frac {x^7}{\left (e^3-e^x+x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx-\int \frac {x^8}{\log ^2\left (-e^3+e^x-x\right )} \, dx-\int \frac {x^9}{\left (-e^3+e^x-x\right ) \log ^2\left (-e^3+e^x-x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.66, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^6 (5+x)^2}{\log \left (-e^3+e^x-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25*x^6 + 10*x^7 + x^8 + E^x*(-25*x^6 - 10*x^7 - x^8) + (-150*x^6 - 70*x^7 - 8*x^8 + E^3*(-150*x^5 -
70*x^6 - 8*x^7) + E^x*(150*x^5 + 70*x^6 + 8*x^7))*Log[-E^3 + E^x - x])/((-E^3 + E^x - x)*Log[-E^3 + E^x - x]^
2),x]

[Out]

(x^6*(5 + x)^2)/Log[-E^3 + E^x - x]

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fricas [A]  time = 0.53, size = 28, normalized size = 1.08 \begin {gather*} \frac {x^{8} + 10 \, x^{7} + 25 \, x^{6}}{\log \left (-x - e^{3} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8*x^8-70*x^7-150*x^6)*log(exp(x)-exp(
3)-x)+(-x^8-10*x^7-25*x^6)*exp(x)+x^8+10*x^7+25*x^6)/(exp(x)-exp(3)-x)/log(exp(x)-exp(3)-x)^2,x, algorithm="fr
icas")

[Out]

(x^8 + 10*x^7 + 25*x^6)/log(-x - e^3 + e^x)

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giac [A]  time = 0.20, size = 28, normalized size = 1.08 \begin {gather*} \frac {x^{8} + 10 \, x^{7} + 25 \, x^{6}}{\log \left (-x - e^{3} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8*x^8-70*x^7-150*x^6)*log(exp(x)-exp(
3)-x)+(-x^8-10*x^7-25*x^6)*exp(x)+x^8+10*x^7+25*x^6)/(exp(x)-exp(3)-x)/log(exp(x)-exp(3)-x)^2,x, algorithm="gi
ac")

[Out]

(x^8 + 10*x^7 + 25*x^6)/log(-x - e^3 + e^x)

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maple [A]  time = 0.04, size = 26, normalized size = 1.00

method	result	size
risch	\(\frac {x^{6} \left (x^{2}+10 x +25\right )}{\ln \left ({\mathrm e}^{x}-{\mathrm e}^{3}-x \right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8*x^8-70*x^7-150*x^6)*ln(exp(x)-exp(3)-x)+(
-x^8-10*x^7-25*x^6)*exp(x)+x^8+10*x^7+25*x^6)/(exp(x)-exp(3)-x)/ln(exp(x)-exp(3)-x)^2,x,method=_RETURNVERBOSE)

[Out]

x^6*(x^2+10*x+25)/ln(exp(x)-exp(3)-x)

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maxima [A]  time = 0.41, size = 28, normalized size = 1.08 \begin {gather*} \frac {x^{8} + 10 \, x^{7} + 25 \, x^{6}}{\log \left (-x - e^{3} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^7+70*x^6+150*x^5)*exp(x)+(-8*x^7-70*x^6-150*x^5)*exp(3)-8*x^8-70*x^7-150*x^6)*log(exp(x)-exp(
3)-x)+(-x^8-10*x^7-25*x^6)*exp(x)+x^8+10*x^7+25*x^6)/(exp(x)-exp(3)-x)/log(exp(x)-exp(3)-x)^2,x, algorithm="ma
xima")

[Out]

(x^8 + 10*x^7 + 25*x^6)/log(-x - e^3 + e^x)

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mupad [B]  time = 0.41, size = 130, normalized size = 5.00 \begin {gather*} \frac {x^6\,{\left (x+5\right )}^2+\frac {2\,x^5\,\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^3-x\right )\,\left (4\,x^2+35\,x+75\right )\,\left (x+{\mathrm {e}}^3-{\mathrm {e}}^x\right )}{{\mathrm {e}}^x-1}}{\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^3-x\right )}-\frac {150\,x^5\,{\mathrm {e}}^3+70\,x^6\,{\mathrm {e}}^3+8\,x^7\,{\mathrm {e}}^3-150\,x^5+80\,x^6+62\,x^7+8\,x^8}{{\mathrm {e}}^x-1}+150\,x^5+70\,x^6+8\,x^7 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(25*x^6 - log(exp(x) - exp(3) - x)*(exp(3)*(150*x^5 + 70*x^6 + 8*x^7) - exp(x)*(150*x^5 + 70*x^6 + 8*x^7)
+ 150*x^6 + 70*x^7 + 8*x^8) + 10*x^7 + x^8 - exp(x)*(25*x^6 + 10*x^7 + x^8))/(log(exp(x) - exp(3) - x)^2*(x +
exp(3) - exp(x))),x)

[Out]

(x^6*(x + 5)^2 + (2*x^5*log(exp(x) - exp(3) - x)*(35*x + 4*x^2 + 75)*(x + exp(3) - exp(x)))/(exp(x) - 1))/log(
exp(x) - exp(3) - x) - (150*x^5*exp(3) + 70*x^6*exp(3) + 8*x^7*exp(3) - 150*x^5 + 80*x^6 + 62*x^7 + 8*x^8)/(ex
p(x) - 1) + 150*x^5 + 70*x^6 + 8*x^7

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sympy [A]  time = 0.19, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^{8} + 10 x^{7} + 25 x^{6}}{\log {\left (- x + e^{x} - e^{3} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**7+70*x**6+150*x**5)*exp(x)+(-8*x**7-70*x**6-150*x**5)*exp(3)-8*x**8-70*x**7-150*x**6)*ln(exp
(x)-exp(3)-x)+(-x**8-10*x**7-25*x**6)*exp(x)+x**8+10*x**7+25*x**6)/(exp(x)-exp(3)-x)/ln(exp(x)-exp(3)-x)**2,x)

[Out]

(x**8 + 10*x**7 + 25*x**6)/log(-x + exp(x) - exp(3))

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