3.50.89 \(\int \frac {e^{\frac {2 (-2+x \log (\frac {3-2 x+e^{\frac {4}{621+50 x+x^2}} x}{x}))}{x}} (4627692-4653774 x-824496 x^2-51188 x^3-1388 x^4-14 x^5+e^{\frac {4}{621+50 x+x^2}} (1542564 x+248400 x^2+14568 x^3+384 x^4+4 x^5))}{1156923 x^2-584982 x^3-112974 x^4-7184 x^5-197 x^6-2 x^7+e^{\frac {4}{621+50 x+x^2}} (385641 x^3+62100 x^4+3742 x^5+100 x^6+x^7)} \, dx\)

Optimal. Leaf size=32 \[ -1+e^{-4/x} \left (-2+e^{\frac {4}{-4+(25+x)^2}}+\frac {3}{x}\right )^2 \]

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Rubi [A]  time = 8.45, antiderivative size = 34, normalized size of antiderivative = 1.06, number of steps used = 1, number of rules used = 1, integrand size = 174, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {6706} \begin {gather*} \frac {e^{-4/x} \left (e^{\frac {4}{x^2+50 x+621}} x-2 x+3\right )^2}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*(-2 + x*Log[(3 - 2*x + E^(4/(621 + 50*x + x^2))*x)/x]))/x)*(4627692 - 4653774*x - 824496*x^2 - 5118
8*x^3 - 1388*x^4 - 14*x^5 + E^(4/(621 + 50*x + x^2))*(1542564*x + 248400*x^2 + 14568*x^3 + 384*x^4 + 4*x^5)))/
(1156923*x^2 - 584982*x^3 - 112974*x^4 - 7184*x^5 - 197*x^6 - 2*x^7 + E^(4/(621 + 50*x + x^2))*(385641*x^3 + 6
2100*x^4 + 3742*x^5 + 100*x^6 + x^7)),x]

[Out]

(3 - 2*x + E^(4/(621 + 50*x + x^2))*x)^2/(E^(4/x)*x^2)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{-4/x} \left (3-2 x+e^{\frac {4}{621+50 x+x^2}} x\right )^2}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 33, normalized size = 1.03 \begin {gather*} \frac {e^{-4/x} \left (3+\left (-2+e^{\frac {4}{621+50 x+x^2}}\right ) x\right )^2}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(-2 + x*Log[(3 - 2*x + E^(4/(621 + 50*x + x^2))*x)/x]))/x)*(4627692 - 4653774*x - 824496*x^2
- 51188*x^3 - 1388*x^4 - 14*x^5 + E^(4/(621 + 50*x + x^2))*(1542564*x + 248400*x^2 + 14568*x^3 + 384*x^4 + 4*x
^5)))/(1156923*x^2 - 584982*x^3 - 112974*x^4 - 7184*x^5 - 197*x^6 - 2*x^7 + E^(4/(621 + 50*x + x^2))*(385641*x
^3 + 62100*x^4 + 3742*x^5 + 100*x^6 + x^7)),x]

[Out]

(3 + (-2 + E^(4/(621 + 50*x + x^2)))*x)^2/(E^(4/x)*x^2)

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fricas [A]  time = 0.83, size = 35, normalized size = 1.09 \begin {gather*} e^{\left (\frac {2 \, {\left (x \log \left (\frac {x e^{\left (\frac {4}{x^{2} + 50 \, x + 621}\right )} - 2 \, x + 3}{x}\right ) - 2\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+384*x^4+14568*x^3+248400*x^2+1542564*x)*exp(4/(x^2+50*x+621))-14*x^5-1388*x^4-51188*x^3-8244
96*x^2-4653774*x+4627692)*exp((x*log((x*exp(4/(x^2+50*x+621))+3-2*x)/x)-2)/x)^2/((x^7+100*x^6+3742*x^5+62100*x
^4+385641*x^3)*exp(4/(x^2+50*x+621))-2*x^7-197*x^6-7184*x^5-112974*x^4-584982*x^3+1156923*x^2),x, algorithm="f
ricas")

[Out]

e^(2*(x*log((x*e^(4/(x^2 + 50*x + 621)) - 2*x + 3)/x) - 2)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (7 \, x^{5} + 694 \, x^{4} + 25594 \, x^{3} + 412248 \, x^{2} - 2 \, {\left (x^{5} + 96 \, x^{4} + 3642 \, x^{3} + 62100 \, x^{2} + 385641 \, x\right )} e^{\left (\frac {4}{x^{2} + 50 \, x + 621}\right )} + 2326887 \, x - 2313846\right )} e^{\left (\frac {2 \, {\left (x \log \left (\frac {x e^{\left (\frac {4}{x^{2} + 50 \, x + 621}\right )} - 2 \, x + 3}{x}\right ) - 2\right )}}{x}\right )}}{2 \, x^{7} + 197 \, x^{6} + 7184 \, x^{5} + 112974 \, x^{4} + 584982 \, x^{3} - 1156923 \, x^{2} - {\left (x^{7} + 100 \, x^{6} + 3742 \, x^{5} + 62100 \, x^{4} + 385641 \, x^{3}\right )} e^{\left (\frac {4}{x^{2} + 50 \, x + 621}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+384*x^4+14568*x^3+248400*x^2+1542564*x)*exp(4/(x^2+50*x+621))-14*x^5-1388*x^4-51188*x^3-8244
96*x^2-4653774*x+4627692)*exp((x*log((x*exp(4/(x^2+50*x+621))+3-2*x)/x)-2)/x)^2/((x^7+100*x^6+3742*x^5+62100*x
^4+385641*x^3)*exp(4/(x^2+50*x+621))-2*x^7-197*x^6-7184*x^5-112974*x^4-584982*x^3+1156923*x^2),x, algorithm="g
iac")

[Out]

integrate(2*(7*x^5 + 694*x^4 + 25594*x^3 + 412248*x^2 - 2*(x^5 + 96*x^4 + 3642*x^3 + 62100*x^2 + 385641*x)*e^(
4/(x^2 + 50*x + 621)) + 2326887*x - 2313846)*e^(2*(x*log((x*e^(4/(x^2 + 50*x + 621)) - 2*x + 3)/x) - 2)/x)/(2*
x^7 + 197*x^6 + 7184*x^5 + 112974*x^4 + 584982*x^3 - 1156923*x^2 - (x^7 + 100*x^6 + 3742*x^5 + 62100*x^4 + 385
641*x^3)*e^(4/(x^2 + 50*x + 621))), x)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (4 x^{5}+384 x^{4}+14568 x^{3}+248400 x^{2}+1542564 x \right ) {\mathrm e}^{\frac {4}{x^{2}+50 x +621}}-14 x^{5}-1388 x^{4}-51188 x^{3}-824496 x^{2}-4653774 x +4627692\right ) {\mathrm e}^{\frac {2 x \ln \left (\frac {x \,{\mathrm e}^{\frac {4}{x^{2}+50 x +621}}+3-2 x}{x}\right )-4}{x}}}{\left (x^{7}+100 x^{6}+3742 x^{5}+62100 x^{4}+385641 x^{3}\right ) {\mathrm e}^{\frac {4}{x^{2}+50 x +621}}-2 x^{7}-197 x^{6}-7184 x^{5}-112974 x^{4}-584982 x^{3}+1156923 x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^5+384*x^4+14568*x^3+248400*x^2+1542564*x)*exp(4/(x^2+50*x+621))-14*x^5-1388*x^4-51188*x^3-824496*x^2
-4653774*x+4627692)*exp((x*ln((x*exp(4/(x^2+50*x+621))+3-2*x)/x)-2)/x)^2/((x^7+100*x^6+3742*x^5+62100*x^4+3856
41*x^3)*exp(4/(x^2+50*x+621))-2*x^7-197*x^6-7184*x^5-112974*x^4-584982*x^3+1156923*x^2),x)

[Out]

int(((4*x^5+384*x^4+14568*x^3+248400*x^2+1542564*x)*exp(4/(x^2+50*x+621))-14*x^5-1388*x^4-51188*x^3-824496*x^2
-4653774*x+4627692)*exp((x*ln((x*exp(4/(x^2+50*x+621))+3-2*x)/x)-2)/x)^2/((x^7+100*x^6+3742*x^5+62100*x^4+3856
41*x^3)*exp(4/(x^2+50*x+621))-2*x^7-197*x^6-7184*x^5-112974*x^4-584982*x^3+1156923*x^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {{\left (2 \, {\left (2 \, x - 3\right )} e^{\left (-\frac {1}{x + 27} + \frac {1}{x + 23}\right )} - x e^{\left (-\frac {2}{x + 27} + \frac {2}{x + 23}\right )}\right )} e^{\left (-\frac {4}{x}\right )}}{x} + 2 \, \int \frac {{\left (14 \, x^{2} - 33 \, x + 18\right )} e^{\left (-\frac {4}{x}\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+384*x^4+14568*x^3+248400*x^2+1542564*x)*exp(4/(x^2+50*x+621))-14*x^5-1388*x^4-51188*x^3-8244
96*x^2-4653774*x+4627692)*exp((x*log((x*exp(4/(x^2+50*x+621))+3-2*x)/x)-2)/x)^2/((x^7+100*x^6+3742*x^5+62100*x
^4+385641*x^3)*exp(4/(x^2+50*x+621))-2*x^7-197*x^6-7184*x^5-112974*x^4-584982*x^3+1156923*x^2),x, algorithm="m
axima")

[Out]

-(2*(2*x - 3)*e^(-1/(x + 27) + 1/(x + 23)) - x*e^(-2/(x + 27) + 2/(x + 23)))*e^(-4/x)/x + 2*integrate((14*x^2
- 33*x + 18)*e^(-4/x)/x^4, x)

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mupad [B]  time = 3.95, size = 95, normalized size = 2.97 \begin {gather*} 4\,{\mathrm {e}}^{-\frac {4}{x}}-4\,{\mathrm {e}}^{\frac {4}{x^2+50\,x+621}-\frac {4}{x}}+{\mathrm {e}}^{\frac {8}{x^2+50\,x+621}-\frac {4}{x}}-\frac {12\,{\mathrm {e}}^{-\frac {4}{x}}}{x}+\frac {9\,{\mathrm {e}}^{-\frac {4}{x}}}{x^2}+\frac {6\,{\mathrm {e}}^{\frac {4}{x^2+50\,x+621}-\frac {4}{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*(x*log((x*exp(4/(50*x + x^2 + 621)) - 2*x + 3)/x) - 2))/x)*(4653774*x - exp(4/(50*x + x^2 + 621))*
(1542564*x + 248400*x^2 + 14568*x^3 + 384*x^4 + 4*x^5) + 824496*x^2 + 51188*x^3 + 1388*x^4 + 14*x^5 - 4627692)
)/(584982*x^3 - 1156923*x^2 - exp(4/(50*x + x^2 + 621))*(385641*x^3 + 62100*x^4 + 3742*x^5 + 100*x^6 + x^7) +
112974*x^4 + 7184*x^5 + 197*x^6 + 2*x^7),x)

[Out]

4*exp(-4/x) - 4*exp(4/(50*x + x^2 + 621) - 4/x) + exp(8/(50*x + x^2 + 621) - 4/x) - (12*exp(-4/x))/x + (9*exp(
-4/x))/x^2 + (6*exp(4/(50*x + x^2 + 621) - 4/x))/x

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**5+384*x**4+14568*x**3+248400*x**2+1542564*x)*exp(4/(x**2+50*x+621))-14*x**5-1388*x**4-51188*x
**3-824496*x**2-4653774*x+4627692)*exp((x*ln((x*exp(4/(x**2+50*x+621))+3-2*x)/x)-2)/x)**2/((x**7+100*x**6+3742
*x**5+62100*x**4+385641*x**3)*exp(4/(x**2+50*x+621))-2*x**7-197*x**6-7184*x**5-112974*x**4-584982*x**3+1156923
*x**2),x)

[Out]

Timed out

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