3.6.77 \(\int \frac {24 e^{4+2 x}+8 e^x x^2+(4 e^x x^2+e^{2 x} (12 e^4 x+21 x^2-3 e x^2)) \log (\frac {e^{-2 x} (16 x^2+e^x (96 e^4 x+168 x^2-24 e x^2)+e^{2 x} (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)))}{9 x^2})}{(4 x^2+e^x (12 e^4 x+21 x^2-3 e x^2)) \log ^2(\frac {e^{-2 x} (16 x^2+e^x (96 e^4 x+168 x^2-24 e x^2)+e^{2 x} (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)))}{9 x^2})} \, dx\)

Optimal. Leaf size=34 \[ \frac {e^x}{\log \left (\left (3-e+\frac {4 \left (e^4+x+\frac {e^{-x} x}{3}\right )}{x}\right )^2\right )} \]

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Rubi [B]  time = 1.05, antiderivative size = 132, normalized size of antiderivative = 3.88, number of steps used = 2, number of rules used = 2, integrand size = 253, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6688, 2288} \begin {gather*} \frac {e^x \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right ) \log \left (\frac {e^{-2 x} \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}{\left (3 (7-e) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (3 (7-e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24*E^(4 + 2*x) + 8*E^x*x^2 + (4*E^x*x^2 + E^(2*x)*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^
4*x + 168*x^2 - 24*E*x^2) + E^(2*x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(
2*x)*x^2)])/((4*x^2 + E^x*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^4*x + 168*x^2 - 24*E*x^2) + E
^(2*x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(2*x)*x^2)]^2),x]

[Out]

(E^x*(12*E^(4 + x) + 4*x + 21*E^x*x - 3*E^(1 + x)*x)*Log[(12*E^(4 + x) + 4*x + 21*E^x*x - 3*E^(1 + x)*x)^2/(9*
E^(2*x)*x^2)])/((12*E^(4 + x) + 4*x + 3*(7 - E)*E^x*x)*Log[(12*E^(4 + x) + 4*x + 3*(7 - E)*E^x*x)^2/(9*E^(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 (24 e^{4+x}+8 x^2+x \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right ) \log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )\right )}{x \left (12 e^{4+x}+4 x+21 \left (1-\frac {e}{7}\right ) e^x x\right ) \log ^2\left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 \left (1-\frac {e}{7}\right ) e^x x\right )^2}{9 x^2}\right )} \, dx\\ &=\frac {e^x \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right ) \log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )}{\left (12 e^{4+x}+4 x+3 (7-e) e^x x\right ) \log ^2\left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+3 (7-e) e^x x\right )^2}{9 x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 46, normalized size = 1.35 \begin {gather*} \frac {e^x}{\log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*E^(4 + 2*x) + 8*E^x*x^2 + (4*E^x*x^2 + E^(2*x)*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*
(96*E^4*x + 168*x^2 - 24*E*x^2) + E^(2*x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/
(9*E^(2*x)*x^2)])/((4*x^2 + E^x*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^4*x + 168*x^2 - 24*E*x^
2) + E^(2*x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(2*x)*x^2)]^2),x]

[Out]

E^x/Log[(12*E^(4 + x) + 4*x + 21*E^x*x - 3*E^(1 + x)*x)^2/(9*E^(2*x)*x^2)]

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fricas [B]  time = 0.63, size = 81, normalized size = 2.38 \begin {gather*} \frac {e^{x}}{\log \left (\frac {{\left (16 \, x^{2} + 9 \, {\left (x^{2} e^{2} - 14 \, x^{2} e + 49 \, x^{2} - 8 \, x e^{5} + 56 \, x e^{4} + 16 \, e^{8}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (x^{2} e - 7 \, x^{2} - 4 \, x e^{4}\right )} e^{x}\right )} e^{\left (-2 \, x\right )}}{9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*log(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*
x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/e
xp(x)^2/x^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)+4*x^2)/log(1/9*((144*e
xp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)
+168*x^2)*exp(x)+16*x^2)/exp(x)^2/x^2)^2,x, algorithm="fricas")

[Out]

e^x/log(1/9*(16*x^2 + 9*(x^2*e^2 - 14*x^2*e + 49*x^2 - 8*x*e^5 + 56*x*e^4 + 16*e^8)*e^(2*x) - 24*(x^2*e - 7*x^
2 - 4*x*e^4)*e^x)*e^(-2*x)/x^2)

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giac [B]  time = 26.08, size = 110, normalized size = 3.24 \begin {gather*} -\frac {e^{x}}{2 \, x + 2 \, \log \relax (3) - \log \left (441 \, x^{2} e^{\left (2 \, x\right )} + 9 \, x^{2} e^{\left (2 \, x + 2\right )} - 126 \, x^{2} e^{\left (2 \, x + 1\right )} - 24 \, x^{2} e^{\left (x + 1\right )} + 168 \, x^{2} e^{x} + 16 \, x^{2} - 72 \, x e^{\left (2 \, x + 5\right )} + 504 \, x e^{\left (2 \, x + 4\right )} + 96 \, x e^{\left (x + 4\right )} + 144 \, e^{\left (2 \, x + 8\right )}\right ) + 2 \, \log \left (x \mathrm {sgn}\relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*log(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*
x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/e
xp(x)^2/x^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)+4*x^2)/log(1/9*((144*e
xp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)
+168*x^2)*exp(x)+16*x^2)/exp(x)^2/x^2)^2,x, algorithm="giac")

[Out]

-e^x/(2*x + 2*log(3) - log(441*x^2*e^(2*x) + 9*x^2*e^(2*x + 2) - 126*x^2*e^(2*x + 1) - 24*x^2*e^(x + 1) + 168*
x^2*e^x + 16*x^2 - 72*x*e^(2*x + 5) + 504*x*e^(2*x + 4) + 96*x*e^(x + 4) + 144*e^(2*x + 8)) + 2*log(x*sgn(x)))

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maple [C]  time = 0.77, size = 758, normalized size = 22.29

method	result	size
risch	\(\frac {2 i {\mathrm e}^{x}}{\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )\right )^{2} \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )\right ) \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )-\pi \,\mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+\pi \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )^{3}+4 i \ln \left (x \,{\mathrm e}^{x +1}-4 \,{\mathrm e}^{4+x}+\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )-4 i \ln \left ({\mathrm e}^{x}\right )-4 i \ln \relax (x )}\)	\(758\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*ln(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(
4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/exp(x)^2
/x^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)+4*x^2)/ln(1/9*((144*exp(4)^2+
(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2
)*exp(x)+16*x^2)/exp(x)^2/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

2*I*exp(x)/(Pi*csgn(I/x^2)*csgn(I*exp(-2*x)*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)*csgn(I/x^2*(-x*exp(x
+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2*exp(-2*x))-Pi*csgn(I/x^2)*csgn(I/x^2*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/
3)*x)^2*exp(-2*x))^2-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3-Pi*csgn(I*exp(x)
)^2*csgn(I*exp(2*x))+2*Pi*csgn(I*exp(x))*csgn(I*exp(2*x))^2+Pi*csgn(I*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*
x))^2*csgn(I*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)+2*Pi*csgn(I*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)
*x))*csgn(I*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)^2+Pi*csgn(I*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*
x)^2)^3+Pi*csgn(I*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)*csgn(I*exp(-2*x))*csgn(I*exp(-2*x)*(-x*exp(x+1
)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)-Pi*csgn(I*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)*csgn(I*exp(-2*x)*(-
x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)^2-Pi*csgn(I*exp(-2*x))*csgn(I*exp(-2*x)*(-x*exp(x+1)+4*exp(4+x)-(-
7*exp(x)-4/3)*x)^2)^2-Pi*csgn(I*exp(2*x))^3+Pi*csgn(I*exp(-2*x)*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)^
3-Pi*csgn(I*exp(-2*x)*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2)*csgn(I/x^2*(-x*exp(x+1)+4*exp(4+x)-(-7*exp
(x)-4/3)*x)^2*exp(-2*x))^2+Pi*csgn(I/x^2*(-x*exp(x+1)+4*exp(4+x)-(-7*exp(x)-4/3)*x)^2*exp(-2*x))^3+4*I*ln(x*ex
p(x+1)-4*exp(4+x)+(-7*exp(x)-4/3)*x)-4*I*ln(exp(x))-4*I*ln(x))

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maxima [A]  time = 1.09, size = 34, normalized size = 1.00 \begin {gather*} -\frac {e^{x}}{2 \, {\left (x + \log \relax (3) - \log \left (3 \, {\left (x {\left (e - 7\right )} - 4 \, e^{4}\right )} e^{x} - 4 \, x\right ) + \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*log(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*
x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/e
xp(x)^2/x^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)+4*x^2)/log(1/9*((144*e
xp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)
+168*x^2)*exp(x)+16*x^2)/exp(x)^2/x^2)^2,x, algorithm="maxima")

[Out]

-1/2*e^x/(x + log(3) - log(3*(x*(e - 7) - 4*e^4)*e^x - 4*x) + log(x))

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mupad [B]  time = 1.58, size = 326, normalized size = 9.59 \begin {gather*} \frac {{\mathrm {e}}^x+\frac {x\,{\mathrm {e}}^x\,\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )\,\left (4\,x+12\,{\mathrm {e}}^{x+4}-3\,x\,{\mathrm {e}}^{x+1}+21\,x\,{\mathrm {e}}^x\right )}{8\,\left (3\,{\mathrm {e}}^{x+4}+x^2\right )}}{\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )}-\frac {x^2\,{\mathrm {e}}^{-4}}{6}+\frac {x^3\,{\mathrm {e}}^{-4}}{6}-{\mathrm {e}}^x\,\left (\frac {x}{2}-\frac {x^2\,{\mathrm {e}}^{-4}\,\left (\mathrm {e}-7\right )}{8}\right )-\frac {x^4\,{\mathrm {e}}^{-8}\,\left (\mathrm {e}-7\right )}{24}+\frac {{\mathrm {e}}^{-8}\,\left (2\,x^7\,\mathrm {e}+8\,x^5\,{\mathrm {e}}^4-x^8\,\mathrm {e}-12\,x^6\,{\mathrm {e}}^4+4\,x^7\,{\mathrm {e}}^4-14\,x^7+7\,x^8\right )}{72\,\left (2\,x\,{\mathrm {e}}^4-x^2\,{\mathrm {e}}^4\right )\,\left ({\mathrm {e}}^x+\frac {x^2\,{\mathrm {e}}^{-4}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*exp(2*x)*exp(4) + 8*x^2*exp(x) + log((exp(-2*x)*((exp(2*x)*(144*exp(8) + exp(4)*(504*x - 72*x*exp(1))
- 126*x^2*exp(1) + 9*x^2*exp(2) + 441*x^2))/9 + (exp(x)*(96*x*exp(4) - 24*x^2*exp(1) + 168*x^2))/9 + (16*x^2)/
9))/x^2)*(exp(2*x)*(12*x*exp(4) - 3*x^2*exp(1) + 21*x^2) + 4*x^2*exp(x)))/(log((exp(-2*x)*((exp(2*x)*(144*exp(
8) + exp(4)*(504*x - 72*x*exp(1)) - 126*x^2*exp(1) + 9*x^2*exp(2) + 441*x^2))/9 + (exp(x)*(96*x*exp(4) - 24*x^
2*exp(1) + 168*x^2))/9 + (16*x^2)/9))/x^2)^2*(exp(x)*(12*x*exp(4) - 3*x^2*exp(1) + 21*x^2) + 4*x^2)),x)

[Out]

(exp(x) + (x*exp(x)*log((exp(-2*x)*((exp(2*x)*(144*exp(8) + exp(4)*(504*x - 72*x*exp(1)) - 126*x^2*exp(1) + 9*
x^2*exp(2) + 441*x^2))/9 + (exp(x)*(96*x*exp(4) - 24*x^2*exp(1) + 168*x^2))/9 + (16*x^2)/9))/x^2)*(4*x + 12*ex
p(x + 4) - 3*x*exp(x + 1) + 21*x*exp(x)))/(8*(3*exp(x + 4) + x^2)))/log((exp(-2*x)*((exp(2*x)*(144*exp(8) + ex
p(4)*(504*x - 72*x*exp(1)) - 126*x^2*exp(1) + 9*x^2*exp(2) + 441*x^2))/9 + (exp(x)*(96*x*exp(4) - 24*x^2*exp(1
) + 168*x^2))/9 + (16*x^2)/9))/x^2) - (x^2*exp(-4))/6 + (x^3*exp(-4))/6 - exp(x)*(x/2 - (x^2*exp(-4)*(exp(1) -
7))/8) - (x^4*exp(-8)*(exp(1) - 7))/24 + (exp(-8)*(2*x^7*exp(1) + 8*x^5*exp(4) - x^8*exp(1) - 12*x^6*exp(4) +
4*x^7*exp(4) - 14*x^7 + 7*x^8))/(72*(2*x*exp(4) - x^2*exp(4))*(exp(x) + (x^2*exp(-4))/3))

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sympy [B]  time = 0.47, size = 92, normalized size = 2.71 \begin {gather*} \frac {e^{x}}{\log {\left (\frac {\left (\frac {16 x^{2}}{9} + \frac {\left (- 24 e x^{2} + 168 x^{2} + 96 x e^{4}\right ) e^{x}}{9} + \frac {\left (- 126 e x^{2} + 9 x^{2} e^{2} + 441 x^{2} + \left (- 72 e x + 504 x\right ) e^{4} + 144 e^{8}\right ) e^{2 x}}{9}\right ) e^{- 2 x}}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x*exp(4)-3*x**2*exp(1)+21*x**2)*exp(x)**2+4*exp(x)*x**2)*ln(1/9*((144*exp(4)**2+(-72*x*exp(1)+
504*x)*exp(4)+9*x**2*exp(1)**2-126*x**2*exp(1)+441*x**2)*exp(x)**2+(96*x*exp(4)-24*x**2*exp(1)+168*x**2)*exp(x
)+16*x**2)/exp(x)**2/x**2)+24*exp(4)*exp(x)**2+8*exp(x)*x**2)/((12*x*exp(4)-3*x**2*exp(1)+21*x**2)*exp(x)+4*x*
*2)/ln(1/9*((144*exp(4)**2+(-72*x*exp(1)+504*x)*exp(4)+9*x**2*exp(1)**2-126*x**2*exp(1)+441*x**2)*exp(x)**2+(9
6*x*exp(4)-24*x**2*exp(1)+168*x**2)*exp(x)+16*x**2)/exp(x)**2/x**2)**2,x)

[Out]

exp(x)/log((16*x**2/9 + (-24*E*x**2 + 168*x**2 + 96*x*exp(4))*exp(x)/9 + (-126*E*x**2 + 9*x**2*exp(2) + 441*x*
*2 + (-72*E*x + 504*x)*exp(4) + 144*exp(8))*exp(2*x)/9)*exp(-2*x)/x**2)

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