3.50.10 \(\int \frac {e^x (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7)+(e^x (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8)+e^x (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7) \log (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4})) \log (-x+\log (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}))}{(x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+(-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7) \log (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4})) \log ^2(-x+\log (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}))} \, dx\)

Optimal. Leaf size=34 \[ \frac {e^x}{\log \left (-x+\log \left (x+\frac {1}{9} \left (2+\frac {3}{-1-2 x+x^2}\right )^2\right )\right )} \]

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Rubi [A]  time = 1.43, antiderivative size = 51, normalized size of antiderivative = 1.50, number of steps used = 2, number of rules used = 2, integrand size = 398, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {6688, 2288} \begin {gather*} \frac {e^x}{\log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (-x^2+2 x+1\right )^2}\right )-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-4 + 111*x - 48*x^2 - 125*x^3 + 3*x^4 + 111*x^5 - 59*x^6 + 9*x^7) + (E^x*(x + 3*x^2 + 57*x^3 + 113*x
^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8) + E^x*(-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9*x^7)*
Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*
x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]])/((x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^5
- 57*x^6 + 50*x^7 - 9*x^8 + (-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9*x^7)*Log[(1 + x + 56*x
^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*
x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]]^2),x]

[Out]

E^x/Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9*(1 + 2*x - x^2)^2)]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7-\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \left (x-\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right )\right )}{\left (1+3 x+57 x^2+113 x^3-84 x^4-57 x^5+50 x^6-9 x^7\right ) \left (x-\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right )} \, dx\\ &=\frac {e^x}{\log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (1+2 x-x^2\right )^2}\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 49, normalized size = 1.44 \begin {gather*} \frac {e^x}{\log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-4 + 111*x - 48*x^2 - 125*x^3 + 3*x^4 + 111*x^5 - 59*x^6 + 9*x^7) + (E^x*(x + 3*x^2 + 57*x^3 +
113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8) + E^x*(-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9
*x^7)*Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x
+ 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]])/((x + 3*x^2 + 57*x^3 + 113*x^4 - 8
4*x^5 - 57*x^6 + 50*x^7 - 9*x^8 + (-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9*x^7)*Log[(1 + x
+ 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3
- 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]]^2),x]

[Out]

E^x/Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9*(-1 - 2*x + x^2)^2)]]

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fricas [A]  time = 0.65, size = 56, normalized size = 1.65 \begin {gather*} \frac {e^{x}}{\log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^
4-36*x^3+18*x^2+36*x+9))+(-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*log(log((9*x^5-32*x^4+2*
x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4-125*x^3-48*x^2+111*x-4)*exp(x))/(
(9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*
x+9))-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/log(log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3
+18*x^2+36*x+9))-x)^2,x, algorithm="fricas")

[Out]

e^x/log(-x + log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1)))

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giac [B]  time = 20.86, size = 262, normalized size = 7.71 \begin {gather*} \frac {x e^{x} - e^{x} \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )}{x \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right ) - \log \left (9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1\right ) \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right ) + \log \left (9 \, x^{4} - 36 \, x^{3} + 18 \, x^{2} + 36 \, x + 9\right ) \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^
4-36*x^3+18*x^2+36*x+9))+(-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*log(log((9*x^5-32*x^4+2*
x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4-125*x^3-48*x^2+111*x-4)*exp(x))/(
(9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*
x+9))-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/log(log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3
+18*x^2+36*x+9))-x)^2,x, algorithm="giac")

[Out]

(x*e^x - e^x*log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1)))/(x*log(-x + l
og(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1))) - log(9*x^5 - 32*x^4 + 2*x^
3 + 56*x^2 + x + 1)*log(-x + log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1)
)) + log(9*x^4 - 36*x^3 + 18*x^2 + 36*x + 9)*log(-x + log(1/9*(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1)/(x^4 -
4*x^3 + 2*x^2 + 4*x + 1))))

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maple [C]  time = 0.24, size = 257, normalized size = 7.56

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{\ln \left (\ln \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )-2 \ln \left (x^{2}-2 x -1\right )+\frac {i \pi \,\mathrm {csgn}\left (i \left (x^{2}-2 x -1\right )^{2}\right ) \left (-\mathrm {csgn}\left (i \left (x^{2}-2 x -1\right )^{2}\right )+\mathrm {csgn}\left (i \left (x^{2}-2 x -1\right )\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right )+\mathrm {csgn}\left (i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right )+\mathrm {csgn}\left (\frac {i}{\left (x^{2}-2 x -1\right )^{2}}\right )\right )}{2}-x \right )}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*ln((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^
3+18*x^2+36*x+9))+(-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*ln(ln((9*x^5-32*x^4+2*x^3+56*x^
2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4-125*x^3-48*x^2+111*x-4)*exp(x))/((9*x^7-50
*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*ln((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-9*x^
8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/ln(ln((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x
+9))-x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(x)/ln(ln(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)-2*ln(x^2-2*x-1)+1/2*I*Pi*csgn(I*(x^2-2*x-1)^2)*(-csgn(I*
(x^2-2*x-1)^2)+csgn(I*(x^2-2*x-1)))^2-1/2*I*Pi*csgn(I*(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)/(x^2-2*x-1)^2)
*(-csgn(I*(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)/(x^2-2*x-1)^2)+csgn(I*(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x
+1/9)))*(-csgn(I*(x^5-32/9*x^4+2/9*x^3+56/9*x^2+1/9*x+1/9)/(x^2-2*x-1)^2)+csgn(I/(x^2-2*x-1)^2))-x)

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maxima [A]  time = 0.68, size = 49, normalized size = 1.44 \begin {gather*} \frac {e^{x}}{\log \left (-x - 2 \, \log \relax (3) + \log \left (9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1\right ) - 2 \, \log \left (x^{2} - 2 \, x - 1\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*exp(x)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^
4-36*x^3+18*x^2+36*x+9))+(-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)*exp(x))*log(log((9*x^5-32*x^4+2*
x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*x+9))-x)+(9*x^7-59*x^6+111*x^5+3*x^4-125*x^3-48*x^2+111*x-4)*exp(x))/(
(9*x^7-50*x^6+57*x^5+84*x^4-113*x^3-57*x^2-3*x-1)*log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3+18*x^2+36*
x+9))-9*x^8+50*x^7-57*x^6-84*x^5+113*x^4+57*x^3+3*x^2+x)/log(log((9*x^5-32*x^4+2*x^3+56*x^2+x+1)/(9*x^4-36*x^3
+18*x^2+36*x+9))-x)^2,x, algorithm="maxima")

[Out]

e^x/log(-x - 2*log(3) + log(9*x^5 - 32*x^4 + 2*x^3 + 56*x^2 + x + 1) - 2*log(x^2 - 2*x - 1))

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mupad [B]  time = 7.09, size = 57, normalized size = 1.68 \begin {gather*} \frac {{\mathrm {e}}^x}{\ln \left (\ln \left (\frac {9\,x^5-32\,x^4+2\,x^3+56\,x^2+x+1}{9\,x^4-36\,x^3+18\,x^2+36\,x+9}\right )-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log((x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5 + 1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9)) - x)*(exp(x)*(x +
3*x^2 + 57*x^3 + 113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8) - exp(x)*log((x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^
5 + 1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9))*(3*x + 57*x^2 + 113*x^3 - 84*x^4 - 57*x^5 + 50*x^6 - 9*x^7 + 1))
+ exp(x)*(111*x - 48*x^2 - 125*x^3 + 3*x^4 + 111*x^5 - 59*x^6 + 9*x^7 - 4))/(log(log((x + 56*x^2 + 2*x^3 - 32*
x^4 + 9*x^5 + 1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9)) - x)^2*(x - log((x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5 +
1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9))*(3*x + 57*x^2 + 113*x^3 - 84*x^4 - 57*x^5 + 50*x^6 - 9*x^7 + 1) + 3*x
^2 + 57*x^3 + 113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8)),x)

[Out]

exp(x)/log(log((x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5 + 1)/(36*x + 18*x^2 - 36*x^3 + 9*x^4 + 9)) - x)

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sympy [B]  time = 10.69, size = 51, normalized size = 1.50 \begin {gather*} \frac {e^{x}}{\log {\left (- x + \log {\left (\frac {9 x^{5} - 32 x^{4} + 2 x^{3} + 56 x^{2} + x + 1}{9 x^{4} - 36 x^{3} + 18 x^{2} + 36 x + 9} \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((9*x**7-50*x**6+57*x**5+84*x**4-113*x**3-57*x**2-3*x-1)*exp(x)*ln((9*x**5-32*x**4+2*x**3+56*x**2+x
+1)/(9*x**4-36*x**3+18*x**2+36*x+9))+(-9*x**8+50*x**7-57*x**6-84*x**5+113*x**4+57*x**3+3*x**2+x)*exp(x))*ln(ln
((9*x**5-32*x**4+2*x**3+56*x**2+x+1)/(9*x**4-36*x**3+18*x**2+36*x+9))-x)+(9*x**7-59*x**6+111*x**5+3*x**4-125*x
**3-48*x**2+111*x-4)*exp(x))/((9*x**7-50*x**6+57*x**5+84*x**4-113*x**3-57*x**2-3*x-1)*ln((9*x**5-32*x**4+2*x**
3+56*x**2+x+1)/(9*x**4-36*x**3+18*x**2+36*x+9))-9*x**8+50*x**7-57*x**6-84*x**5+113*x**4+57*x**3+3*x**2+x)/ln(l
n((9*x**5-32*x**4+2*x**3+56*x**2+x+1)/(9*x**4-36*x**3+18*x**2+36*x+9))-x)**2,x)

[Out]

exp(x)/log(-x + log((9*x**5 - 32*x**4 + 2*x**3 + 56*x**2 + x + 1)/(9*x**4 - 36*x**3 + 18*x**2 + 36*x + 9)))

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