3.45.27 \(\int \frac {1}{144} e^{\frac {1}{36} (17 e^{x/4}-17 x+e^x (-9 e^{x/4}+9 x))} (-68+17 e^{x/4}+e^x (36-45 e^{x/4}+36 x)) \, dx\)

Optimal. Leaf size=24 \[ e^{\frac {1}{4} \left (-\frac {17}{9}+e^x\right ) \left (-e^{x/4}+x\right )} \]

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Rubi [F]  time = 0.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{144} \exp \left (\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )\right ) \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((17*E^(x/4) - 17*x + E^x*(-9*E^(x/4) + 9*x))/36)*(-68 + 17*E^(x/4) + E^x*(36 - 45*E^(x/4) + 36*x)))/14
4,x]

[Out]

(-17*Defer[Subst][Defer[Int][E^(-1/36*((-17 + 9*E^(4*x))*(E^x - 4*x))), x], x, x/4])/9 + (17*Defer[Subst][Defe
r[Int][E^(-1/36*((-17 + 9*E^(4*x))*(E^x - 4*x)) + x), x], x, x/4])/36 + Defer[Subst][Defer[Int][E^(-1/36*((-17
+ 9*E^(4*x))*(E^x - 4*x)) + 4*x), x], x, x/4] - (5*Defer[Subst][Defer[Int][E^(-1/36*((-17 + 9*E^(4*x))*(E^x -
4*x)) + 5*x), x], x, x/4])/4 + 4*Defer[Subst][Defer[Int][E^(-1/36*((-17 + 9*E^(4*x))*(E^x - 4*x)) + 4*x)*x, x
], x, x/4]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{144} \int \exp \left (\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )\right ) \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx\\ &=\frac {1}{36} \operatorname {Subst}\left (\int e^{-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )} \left (-68+17 e^x+e^{4 x} \left (36-45 e^x+144 x\right )\right ) \, dx,x,\frac {x}{4}\right )\\ &=\frac {1}{36} \operatorname {Subst}\left (\int \left (-68 e^{-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )}+17 \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+x\right )-9 \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+4 x\right ) \left (-4+5 e^x-16 x\right )\right ) \, dx,x,\frac {x}{4}\right )\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+4 x\right ) \left (-4+5 e^x-16 x\right ) \, dx,x,\frac {x}{4}\right )\right )+\frac {17}{36} \operatorname {Subst}\left (\int \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+x\right ) \, dx,x,\frac {x}{4}\right )-\frac {17}{9} \operatorname {Subst}\left (\int e^{-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )} \, dx,x,\frac {x}{4}\right )\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \left (-4 \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+4 x\right )+5 \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+5 x\right )-16 \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+4 x\right ) x\right ) \, dx,x,\frac {x}{4}\right )\right )+\frac {17}{36} \operatorname {Subst}\left (\int \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+x\right ) \, dx,x,\frac {x}{4}\right )-\frac {17}{9} \operatorname {Subst}\left (\int e^{-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )} \, dx,x,\frac {x}{4}\right )\\ &=\frac {17}{36} \operatorname {Subst}\left (\int \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+x\right ) \, dx,x,\frac {x}{4}\right )-\frac {5}{4} \operatorname {Subst}\left (\int \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+5 x\right ) \, dx,x,\frac {x}{4}\right )-\frac {17}{9} \operatorname {Subst}\left (\int e^{-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )} \, dx,x,\frac {x}{4}\right )+4 \operatorname {Subst}\left (\int \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+4 x\right ) x \, dx,x,\frac {x}{4}\right )+\operatorname {Subst}\left (\int \exp \left (-\frac {1}{36} \left (-17+9 e^{4 x}\right ) \left (e^x-4 x\right )+4 x\right ) \, dx,x,\frac {x}{4}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.48, size = 24, normalized size = 1.00 \begin {gather*} e^{-\frac {1}{36} \left (-17+9 e^x\right ) \left (e^{x/4}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((17*E^(x/4) - 17*x + E^x*(-9*E^(x/4) + 9*x))/36)*(-68 + 17*E^(x/4) + E^x*(36 - 45*E^(x/4) + 36*x
)))/144,x]

[Out]

E^(-1/36*((-17 + 9*E^x)*(E^(x/4) - x)))

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fricas [A]  time = 0.74, size = 22, normalized size = 0.92 \begin {gather*} e^{\left (\frac {1}{4} \, x e^{x} - \frac {17}{36} \, x - \frac {1}{4} \, e^{\left (\frac {5}{4} \, x\right )} + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*e
xp(1/4*x)-17/36*x),x, algorithm="fricas")

[Out]

e^(1/4*x*e^x - 17/36*x - 1/4*e^(5/4*x) + 17/36*e^(1/4*x))

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giac [A]  time = 0.26, size = 22, normalized size = 0.92 \begin {gather*} e^{\left (\frac {1}{4} \, x e^{x} - \frac {17}{36} \, x - \frac {1}{4} \, e^{\left (\frac {5}{4} \, x\right )} + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*e
xp(1/4*x)-17/36*x),x, algorithm="giac")

[Out]

e^(1/4*x*e^x - 17/36*x - 1/4*e^(5/4*x) + 17/36*e^(1/4*x))

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maple [A]  time = 0.09, size = 18, normalized size = 0.75

method	result	size
risch	\({\mathrm e}^{\frac {\left (x -{\mathrm e}^{\frac {x}{4}}\right ) \left (9 \,{\mathrm e}^{x}-17\right )}{36}}\)	\(18\)
norman	\({\mathrm e}^{\frac {\left (-9 \,{\mathrm e}^{\frac {x}{4}}+9 x \right ) {\mathrm e}^{x}}{36}+\frac {17 \,{\mathrm e}^{\frac {x}{4}}}{36}-\frac {17 x}{36}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4
*x)-17/36*x),x,method=_RETURNVERBOSE)

[Out]

exp(1/36*(x-exp(1/4*x))*(9*exp(x)-17))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{144} \, \int {\left (9 \, {\left (4 \, x - 5 \, e^{\left (\frac {1}{4} \, x\right )} + 4\right )} e^{x} + 17 \, e^{\left (\frac {1}{4} \, x\right )} - 68\right )} e^{\left (\frac {1}{4} \, {\left (x - e^{\left (\frac {1}{4} \, x\right )}\right )} e^{x} - \frac {17}{36} \, x + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*e
xp(1/4*x)-17/36*x),x, algorithm="maxima")

[Out]

1/144*integrate((9*(4*x - 5*e^(1/4*x) + 4)*e^x + 17*e^(1/4*x) - 68)*e^(1/4*(x - e^(1/4*x))*e^x - 17/36*x + 17/
36*e^(1/4*x)), x)

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mupad [B]  time = 3.30, size = 27, normalized size = 1.12 \begin {gather*} {\mathrm {e}}^{\frac {17\,{\mathrm {e}}^{x/4}}{36}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{4}}\,{\mathrm {e}}^{-\frac {17\,x}{36}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{x/4}\,{\mathrm {e}}^x}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((17*exp(x/4))/36 - (17*x)/36 + (exp(x)*(9*x - 9*exp(x/4)))/36)*(17*exp(x/4) + exp(x)*(36*x - 45*exp(x
/4) + 36) - 68))/144,x)

[Out]

exp((17*exp(x/4))/36)*exp((x*exp(x))/4)*exp(-(17*x)/36)*exp(-(exp(x/4)*exp(x))/4)

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sympy [A]  time = 0.27, size = 27, normalized size = 1.12 \begin {gather*} e^{- \frac {17 x}{36} + \left (\frac {x}{4} - \frac {e^{\frac {x}{4}}}{4}\right ) e^{x} + \frac {17 e^{\frac {x}{4}}}{36}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*e
xp(1/4*x)-17/36*x),x)

[Out]

exp(-17*x/36 + (x/4 - exp(x/4)/4)*exp(x) + 17*exp(x/4)/36)

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