3.5.9 \(\int e^{162+80 x} (400+e^{-162-80 x}) \, dx\)

Optimal. Leaf size=16 \[ 2+5 e^{-x+81 (2+x)}+x \]

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2248, 43} \begin {gather*} x+5 e^{80 x+162} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(162 + 80*x)*(400 + E^(-162 - 80*x)),x]

[Out]

5*E^(162 + 80*x) + x

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2248

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\frac {1}{80} \operatorname {Subst}\left (\int \frac {400+x}{x^2} \, dx,x,e^{-162-80 x}\right )\right )\\ &=-\left (\frac {1}{80} \operatorname {Subst}\left (\int \left (\frac {400}{x^2}+\frac {1}{x}\right ) \, dx,x,e^{-162-80 x}\right )\right )\\ &=5 e^{162+80 x}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.69 \begin {gather*} 5 e^{162+80 x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(162 + 80*x)*(400 + E^(-162 - 80*x)),x]

[Out]

5*E^(162 + 80*x) + x

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fricas [A]  time = 0.75, size = 10, normalized size = 0.62 \begin {gather*} x + 5 \, e^{\left (80 \, x + 162\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-80*x-162)+400)/exp(-80*x-162),x, algorithm="fricas")

[Out]

x + 5*e^(80*x + 162)

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giac [A]  time = 0.40, size = 10, normalized size = 0.62 \begin {gather*} x + 5 \, e^{\left (80 \, x + 162\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-80*x-162)+400)/exp(-80*x-162),x, algorithm="giac")

[Out]

x + 5*e^(80*x + 162)

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maple [A]  time = 0.04, size = 11, normalized size = 0.69

method	result	size
risch	\(x +5 \,{\mathrm e}^{80 x +162}\)	\(11\)
norman	\(\left (5+x \,{\mathrm e}^{-80 x -162}\right ) {\mathrm e}^{80 x +162}\)	\(20\)
derivativedivides	\(5 \,{\mathrm e}^{80 x +162}-\frac {\ln \left ({\mathrm e}^{-80 x -162}\right )}{80}\)	\(21\)
default	\(5 \,{\mathrm e}^{80 x +162}-\frac {\ln \left ({\mathrm e}^{-80 x -162}\right )}{80}\)	\(21\)
meijerg	\(-\frac {{\mathrm e}^{80 x -80 x \,{\mathrm e}^{162}-162} \left (1-{\mathrm e}^{80 x \,{\mathrm e}^{162} \left (1-{\mathrm e}^{-162}\right )}\right )}{80 \left (1-{\mathrm e}^{-162}\right )}-5 \,{\mathrm e}^{80 x -80 x \,{\mathrm e}^{162}} \left (1-{\mathrm e}^{80 x \,{\mathrm e}^{162}}\right )\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-80*x-162)+400)/exp(-80*x-162),x,method=_RETURNVERBOSE)

[Out]

x+5*exp(80*x+162)

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maxima [A]  time = 0.43, size = 11, normalized size = 0.69 \begin {gather*} x + 5 \, e^{\left (80 \, x + 162\right )} + \frac {81}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-80*x-162)+400)/exp(-80*x-162),x, algorithm="maxima")

[Out]

x + 5*e^(80*x + 162) + 81/40

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mupad [B]  time = 0.05, size = 10, normalized size = 0.62 \begin {gather*} x+5\,{\mathrm {e}}^{80\,x}\,{\mathrm {e}}^{162} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(80*x + 162)*(exp(- 80*x - 162) + 400),x)

[Out]

x + 5*exp(80*x)*exp(162)

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sympy [A]  time = 0.13, size = 8, normalized size = 0.50 \begin {gather*} x + 5 e^{80 x + 162} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-80*x-162)+400)/exp(-80*x-162),x)

[Out]

x + 5*exp(80*x + 162)

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