3.59.87 \(\int \frac {1}{4} e^{x^2-x^{12}-e^x x^{12}+x^{13}} (10 x-60 x^{11}+65 x^{12}+e^x (-60 x^{11}-5 x^{12})) \, dx\)

Optimal. Leaf size=23 \[ \frac {5}{4} e^{x^2-\left (1+e^x-x\right ) x^{12}} \]

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Rubi [A]  time = 0.31, antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12, 6706} \begin {gather*} \frac {5}{4} e^{x^{13}-e^x x^{12}-x^{12}+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(x^2 - x^12 - E^x*x^12 + x^13)*(10*x - 60*x^11 + 65*x^12 + E^x*(-60*x^11 - 5*x^12)))/4,x]

[Out]

(5*E^(x^2 - x^12 - E^x*x^12 + x^13))/4

Rule 12

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

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 {1}{4} \int e^{x^2-x^{12}-e^x x^{12}+x^{13}} \left (10 x-60 x^{11}+65 x^{12}+e^x \left (-60 x^{11}-5 x^{12}\right )\right ) \, dx\\ &=\frac {5}{4} e^{x^2-x^{12}-e^x x^{12}+x^{13}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 23, normalized size = 1.00 \begin {gather*} \frac {5}{4} e^{x^2-\left (1+e^x\right ) x^{12}+x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2 - x^12 - E^x*x^12 + x^13)*(10*x - 60*x^11 + 65*x^12 + E^x*(-60*x^11 - 5*x^12)))/4,x]

[Out]

(5*E^(x^2 - (1 + E^x)*x^12 + x^13))/4

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fricas [A]  time = 0.67, size = 22, normalized size = 0.96 \begin {gather*} \frac {5}{4} \, e^{\left (x^{13} - x^{12} e^{x} - x^{12} + x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-5*x^12-60*x^11)*exp(x)+65*x^12-60*x^11+10*x)*exp(-x^12*exp(x)+x^13-x^12+x^2),x, algorithm="fr
icas")

[Out]

5/4*e^(x^13 - x^12*e^x - x^12 + x^2)

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giac [A]  time = 0.16, size = 22, normalized size = 0.96 \begin {gather*} \frac {5}{4} \, e^{\left (x^{13} - x^{12} e^{x} - x^{12} + x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-5*x^12-60*x^11)*exp(x)+65*x^12-60*x^11+10*x)*exp(-x^12*exp(x)+x^13-x^12+x^2),x, algorithm="gi
ac")

[Out]

5/4*e^(x^13 - x^12*e^x - x^12 + x^2)

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maple [A]  time = 0.06, size = 25, normalized size = 1.09

method	result	size
risch	\(\frac {5 \,{\mathrm e}^{-x^{2} \left ({\mathrm e}^{x} x^{10}-x^{11}+x^{10}-1\right )}}{4}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((-5*x^12-60*x^11)*exp(x)+65*x^12-60*x^11+10*x)*exp(-x^12*exp(x)+x^13-x^12+x^2),x,method=_RETURNVERBOS
E)

[Out]

5/4*exp(-x^2*(exp(x)*x^10-x^11+x^10-1))

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maxima [A]  time = 0.45, size = 22, normalized size = 0.96 \begin {gather*} \frac {5}{4} \, e^{\left (x^{13} - x^{12} e^{x} - x^{12} + x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-5*x^12-60*x^11)*exp(x)+65*x^12-60*x^11+10*x)*exp(-x^12*exp(x)+x^13-x^12+x^2),x, algorithm="ma
xima")

[Out]

5/4*e^(x^13 - x^12*e^x - x^12 + x^2)

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mupad [B]  time = 4.17, size = 24, normalized size = 1.04 \begin {gather*} \frac {5\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^{13}}\,{\mathrm {e}}^{-x^{12}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x^{12}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2 - x^12*exp(x) - x^12 + x^13)*(10*x - exp(x)*(60*x^11 + 5*x^12) - 60*x^11 + 65*x^12))/4,x)

[Out]

(5*exp(x^2)*exp(x^13)*exp(-x^12*exp(x))*exp(-x^12))/4

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sympy [A]  time = 0.21, size = 20, normalized size = 0.87 \begin {gather*} \frac {5 e^{x^{13} - x^{12} e^{x} - x^{12} + x^{2}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-5*x**12-60*x**11)*exp(x)+65*x**12-60*x**11+10*x)*exp(-x**12*exp(x)+x**13-x**12+x**2),x)

[Out]

5*exp(x**13 - x**12*exp(x) - x**12 + x**2)/4

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