3.32.26 \(\int \frac {-e^{\frac {3 x^3+x^4+e^5 (6-4 x-2 x^2)}{x}} x+e^{\frac {3 x^3+x^4+e^5 (6-4 x-2 x^2)}{x}} (6 x^3+3 x^4+e^5 (-6-2 x^2)) \log (2 x)}{x^2 \log ^2(2 x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {e^{(3+x) \left (x^2-\frac {e^5 (-2+2 x)}{x}\right )}}{\log (2 x)} \]

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Rubi [B]  time = 1.34, antiderivative size = 79, normalized size of antiderivative = 2.63, number of steps used = 2, number of rules used = 2, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6688, 2288} \begin {gather*} -\frac {e^{x^2 (x+3)-2 e^5 \left (x-\frac {3}{x}+2\right )} \left (3 x^3 (x+2)-2 e^5 \left (x^2+3\right )\right )}{x^2 \left (-x^2+2 e^5 \left (\frac {3}{x^2}+1\right )-2 (x+3) x\right ) \log (2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-(E^((3*x^3 + x^4 + E^5*(6 - 4*x - 2*x^2))/x)*x) + E^((3*x^3 + x^4 + E^5*(6 - 4*x - 2*x^2))/x)*(6*x^3 + 3
*x^4 + E^5*(-6 - 2*x^2))*Log[2*x])/(x^2*Log[2*x]^2),x]

[Out]

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

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^{e^5 \left (-4+\frac {6}{x}-2 x\right )+x^2 (3+x)} \left (-x+\left (3 x^3 (2+x)-2 e^5 \left (3+x^2\right )\right ) \log (2 x)\right )}{x^2 \log ^2(2 x)} \, dx\\ &=-\frac {e^{x^2 (3+x)-2 e^5 \left (2-\frac {3}{x}+x\right )} \left (3 x^3 (2+x)-2 e^5 \left (3+x^2\right )\right )}{x^2 \left (2 e^5 \left (1+\frac {3}{x^2}\right )-x^2-2 x (3+x)\right ) \log (2 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 31, normalized size = 1.03 \begin {gather*} \frac {e^{e^5 \left (-4+\frac {6}{x}-2 x\right )+x^2 (3+x)}}{\log (2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^((3*x^3 + x^4 + E^5*(6 - 4*x - 2*x^2))/x)*x) + E^((3*x^3 + x^4 + E^5*(6 - 4*x - 2*x^2))/x)*(6*x
^3 + 3*x^4 + E^5*(-6 - 2*x^2))*Log[2*x])/(x^2*Log[2*x]^2),x]

[Out]

E^(E^5*(-4 + 6/x - 2*x) + x^2*(3 + x))/Log[2*x]

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fricas [A]  time = 0.80, size = 33, normalized size = 1.10 \begin {gather*} \frac {e^{\left (\frac {x^{4} + 3 \, x^{3} - 2 \, {\left (x^{2} + 2 \, x - 3\right )} e^{5}}{x}\right )}}{\log \left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-6)*exp(5)+3*x^4+6*x^3)*exp(((-2*x^2-4*x+6)*exp(5)+x^4+3*x^3)/x)*log(2*x)-x*exp(((-2*x^2-4*
x+6)*exp(5)+x^4+3*x^3)/x))/x^2/log(2*x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.47, size = 37, normalized size = 1.23 \begin {gather*} \frac {e^{\left (\frac {x^{4} + 3 \, x^{3} - 2 \, x^{2} e^{5} - 4 \, x e^{5} + 6 \, e^{5}}{x}\right )}}{\log \left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-6)*exp(5)+3*x^4+6*x^3)*exp(((-2*x^2-4*x+6)*exp(5)+x^4+3*x^3)/x)*log(2*x)-x*exp(((-2*x^2-4*
x+6)*exp(5)+x^4+3*x^3)/x))/x^2/log(2*x)^2,x, algorithm="giac")

[Out]

e^((x^4 + 3*x^3 - 2*x^2*e^5 - 4*x*e^5 + 6*e^5)/x)/log(2*x)

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maple [A]  time = 0.04, size = 32, normalized size = 1.07

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {\left (3+x \right ) \left (-x^{3}+2 x \,{\mathrm e}^{5}-2 \,{\mathrm e}^{5}\right )}{x}}}{\ln \left (2 x \right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^2-6)*exp(5)+3*x^4+6*x^3)*exp(((-2*x^2-4*x+6)*exp(5)+x^4+3*x^3)/x)*ln(2*x)-x*exp(((-2*x^2-4*x+6)*ex
p(5)+x^4+3*x^3)/x))/x^2/ln(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(-(3+x)*(-x^3+2*x*exp(5)-2*exp(5))/x)/ln(2*x)

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maxima [A]  time = 0.74, size = 42, normalized size = 1.40 \begin {gather*} \frac {e^{\left (x^{3} + 3 \, x^{2} - 2 \, x e^{5} + \frac {6 \, e^{5}}{x}\right )}}{e^{\left (4 \, e^{5}\right )} \log \relax (2) + e^{\left (4 \, e^{5}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2-6)*exp(5)+3*x^4+6*x^3)*exp(((-2*x^2-4*x+6)*exp(5)+x^4+3*x^3)/x)*log(2*x)-x*exp(((-2*x^2-4*
x+6)*exp(5)+x^4+3*x^3)/x))/x^2/log(2*x)^2,x, algorithm="maxima")

[Out]

e^(x^3 + 3*x^2 - 2*x*e^5 + 6*e^5/x)/(e^(4*e^5)*log(2) + e^(4*e^5)*log(x))

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mupad [B]  time = 2.02, size = 37, normalized size = 1.23 \begin {gather*} \frac {{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^5}{x}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^5}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{3\,x^2}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^5}}{\ln \relax (2)+\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*exp((3*x^3 - exp(5)*(4*x + 2*x^2 - 6) + x^4)/x) - log(2*x)*exp((3*x^3 - exp(5)*(4*x + 2*x^2 - 6) + x^4
)/x)*(6*x^3 - exp(5)*(2*x^2 + 6) + 3*x^4))/(x^2*log(2*x)^2),x)

[Out]

(exp((6*exp(5))/x)*exp(-4*exp(5))*exp(x^3)*exp(3*x^2)*exp(-2*x*exp(5)))/(log(2) + log(x))

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sympy [A]  time = 0.35, size = 29, normalized size = 0.97 \begin {gather*} \frac {e^{\frac {x^{4} + 3 x^{3} + \left (- 2 x^{2} - 4 x + 6\right ) e^{5}}{x}}}{\log {\left (2 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**2-6)*exp(5)+3*x**4+6*x**3)*exp(((-2*x**2-4*x+6)*exp(5)+x**4+3*x**3)/x)*ln(2*x)-x*exp(((-2*x
**2-4*x+6)*exp(5)+x**4+3*x**3)/x))/x**2/ln(2*x)**2,x)

[Out]

exp((x**4 + 3*x**3 + (-2*x**2 - 4*x + 6)*exp(5))/x)/log(2*x)

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