3.31.38 \(\int \frac {1}{4} (24+e^{\frac {1}{4} (16+\frac {3}{e^4 x})} (-4+\frac {3}{e^4 x})+8 x+e^x (4+4 x)) \, dx\)

Optimal. Leaf size=29 \[ x-\left (-5+e^{4+\frac {3}{4 e^4 x}}-e^x-x\right ) x \]

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Rubi [A]  time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.45, number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {12, 2288, 2176, 2194} \begin {gather*} x^2-e^{\frac {\frac {3}{x}+16 e^4}{4 e^4}} x+6 x-e^x+e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24 + E^((16 + 3/(E^4*x))/4)*(-4 + 3/(E^4*x)) + 8*x + E^x*(4 + 4*x))/4,x]

[Out]

-E^x + 6*x - E^((16*E^4 + 3/x)/(4*E^4))*x + x^2 + E^x*(1 + x)

Rule 12

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (24+e^{\frac {1}{4} \left (16+\frac {3}{e^4 x}\right )} \left (-4+\frac {3}{e^4 x}\right )+8 x+e^x (4+4 x)\right ) \, dx\\ &=6 x+x^2+\frac {1}{4} \int e^{\frac {1}{4} \left (16+\frac {3}{e^4 x}\right )} \left (-4+\frac {3}{e^4 x}\right ) \, dx+\frac {1}{4} \int e^x (4+4 x) \, dx\\ &=6 x-e^{\frac {16 e^4+\frac {3}{x}}{4 e^4}} x+x^2+e^x (1+x)-\int e^x \, dx\\ &=-e^x+6 x-e^{\frac {16 e^4+\frac {3}{x}}{4 e^4}} x+x^2+e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 29, normalized size = 1.00 \begin {gather*} 6 x-e^{4+\frac {3}{4 e^4 x}} x+e^x x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 + E^((16 + 3/(E^4*x))/4)*(-4 + 3/(E^4*x)) + 8*x + E^x*(4 + 4*x))/4,x]

[Out]

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

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fricas [A]  time = 0.63, size = 29, normalized size = 1.00 \begin {gather*} x^{2} + x e^{x} - x e^{\left (\frac {{\left (16 \, x e^{4} + 3\right )} e^{\left (-4\right )}}{4 \, x}\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(log(3/x)-4)-4)*exp(1/4*exp(log(3/x)-4)+4)+1/4*(4*x+4)*exp(x)+2*x+6,x, algorithm="fricas")

[Out]

x^2 + x*e^x - x*e^(1/4*(16*x*e^4 + 3)*e^(-4)/x) + 6*x

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giac [A]  time = 0.35, size = 24, normalized size = 0.83 \begin {gather*} x^{2} + x e^{x} - x e^{\left (\frac {3 \, e^{\left (-4\right )}}{4 \, x} + 4\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(log(3/x)-4)-4)*exp(1/4*exp(log(3/x)-4)+4)+1/4*(4*x+4)*exp(x)+2*x+6,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 25, normalized size = 0.86

method	result	size
norman	\(x^{2}+{\mathrm e}^{x} x +6 x -x \,{\mathrm e}^{\frac {3 \,{\mathrm e}^{-4}}{4 x}+4}\)	\(25\)
risch	\(x^{2}+{\mathrm e}^{x} x +6 x -x \,{\mathrm e}^{\frac {3 \,{\mathrm e}^{-4}+16 x}{4 x}}\)	\(29\)
default	\(x^{2}+6 x +{\mathrm e}^{x} x -{\mathrm e}^{\frac {{\mathrm e}^{-4+\ln \left (\frac {3}{x}\right )+\ln \relax (x )}}{4 x}+4} x\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(exp(ln(3/x)-4)-4)*exp(1/4*exp(ln(3/x)-4)+4)+1/4*(4*x+4)*exp(x)+2*x+6,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 24, normalized size = 0.83 \begin {gather*} x^{2} + x e^{x} - x e^{\left (\frac {3 \, e^{\left (-4\right )}}{4 \, x} + 4\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(log(3/x)-4)-4)*exp(1/4*exp(log(3/x)-4)+4)+1/4*(4*x+4)*exp(x)+2*x+6,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.81, size = 22, normalized size = 0.76 \begin {gather*} x\,\left ({\mathrm {e}}^x-{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{-4}}{4\,x}+4}+6\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.23, size = 24, normalized size = 0.83 \begin {gather*} x^{2} + x e^{x} - x e^{4 + \frac {3}{4 x e^{4}}} + 6 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(ln(3/x)-4)-4)*exp(1/4*exp(ln(3/x)-4)+4)+1/4*(4*x+4)*exp(x)+2*x+6,x)

[Out]

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

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