3.15.45 \(\int \frac {e^{\frac {-4205 x+e^{4 x} x}{4205}} (-4205-4205 x+e^{4 x} (x+4 x^2))}{4205 x^2 \log (5)} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{-x+\frac {e^{4 x} x}{4205}}-x}{x \log (5)} \]

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Rubi [B]  time = 0.14, antiderivative size = 62, normalized size of antiderivative = 2.21, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 2288} \begin {gather*} \frac {e^{\frac {e^{4 x} x-4205 x}{4205}} \left (4205 x-e^{4 x} \left (4 x^2+x\right )\right )}{x^2 \left (-4 e^{4 x} x-e^{4 x}+4205\right ) \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-4205*x + E^(4*x)*x)/4205)*(-4205 - 4205*x + E^(4*x)*(x + 4*x^2)))/(4205*x^2*Log[5]),x]

[Out]

(E^((-4205*x + E^(4*x)*x)/4205)*(4205*x - E^(4*x)*(x + 4*x^2)))/(x^2*(4205 - E^(4*x) - 4*E^(4*x)*x)*Log[5])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{\frac {-4205 x+e^{4 x} x}{4205}} \left (-4205-4205 x+e^{4 x} \left (x+4 x^2\right )\right )}{x^2} \, dx}{4205 \log (5)}\\ &=\frac {e^{\frac {-4205 x+e^{4 x} x}{4205}} \left (4205 x-e^{4 x} \left (x+4 x^2\right )\right )}{x^2 \left (4205-e^{4 x}-4 e^{4 x} x\right ) \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 23, normalized size = 0.82 \begin {gather*} \frac {e^{\left (-1+\frac {e^{4 x}}{4205}\right ) x}}{x \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-4205*x + E^(4*x)*x)/4205)*(-4205 - 4205*x + E^(4*x)*(x + 4*x^2)))/(4205*x^2*Log[5]),x]

[Out]

E^((-1 + E^(4*x)/4205)*x)/(x*Log[5])

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fricas [A]  time = 0.56, size = 20, normalized size = 0.71 \begin {gather*} \frac {e^{\left (\frac {1}{4205} \, x e^{\left (4 \, x\right )} - x\right )}}{x \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4205*((4*x^2+x)*exp(x)^4-4205*x-4205)*exp(1/4205*x*exp(x)^4-x)/x^2/log(5),x, algorithm="fricas")

[Out]

e^(1/4205*x*e^(4*x) - x)/(x*log(5))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (4 \, x^{2} + x\right )} e^{\left (4 \, x\right )} - 4205 \, x - 4205\right )} e^{\left (\frac {1}{4205} \, x e^{\left (4 \, x\right )} - x\right )}}{4205 \, x^{2} \log \relax (5)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4205*((4*x^2+x)*exp(x)^4-4205*x-4205)*exp(1/4205*x*exp(x)^4-x)/x^2/log(5),x, algorithm="giac")

[Out]

integrate(1/4205*((4*x^2 + x)*e^(4*x) - 4205*x - 4205)*e^(1/4205*x*e^(4*x) - x)/(x^2*log(5)), x)

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maple [A]  time = 0.05, size = 19, normalized size = 0.68

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {x \left ({\mathrm e}^{4 x}-4205\right )}{4205}}}{\ln \relax (5) x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4205*((4*x^2+x)*exp(x)^4-4205*x-4205)*exp(1/4205*x*exp(x)^4-x)/x^2/ln(5),x,method=_RETURNVERBOSE)

[Out]

1/ln(5)/x*exp(1/4205*x*(exp(4*x)-4205))

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maxima [A]  time = 0.83, size = 20, normalized size = 0.71 \begin {gather*} \frac {e^{\left (\frac {1}{4205} \, x e^{\left (4 \, x\right )} - x\right )}}{x \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4205*((4*x^2+x)*exp(x)^4-4205*x-4205)*exp(1/4205*x*exp(x)^4-x)/x^2/log(5),x, algorithm="maxima")

[Out]

e^(1/4205*x*e^(4*x) - x)/(x*log(5))

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mupad [B]  time = 1.03, size = 20, normalized size = 0.71 \begin {gather*} \frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{4\,x}}{4205}}}{x\,\ln \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x*exp(4*x))/4205 - x)*(4205*x - exp(4*x)*(x + 4*x^2) + 4205))/(4205*x^2*log(5)),x)

[Out]

(exp(-x)*exp((x*exp(4*x))/4205))/(x*log(5))

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sympy [A]  time = 0.18, size = 15, normalized size = 0.54 \begin {gather*} \frac {e^{\frac {x e^{4 x}}{4205} - x}}{x \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4205*((4*x**2+x)*exp(x)**4-4205*x-4205)*exp(1/4205*x*exp(x)**4-x)/x**2/ln(5),x)

[Out]

exp(x*exp(4*x)/4205 - x)/(x*log(5))

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