3.41.4 \(\int \frac {e^{-\frac {4+x+x^4}{x^3}} (3 x^4+(36+42 x+6 x^2+3 x^3-3 x^5) \log (2+2 x))}{x^3+x^4} \, dx\)

Optimal. Leaf size=23 \[ 3 e^{-x-\frac {4+x}{x^3}} x \log (2+2 x) \]

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Rubi [B]  time = 1.75, antiderivative size = 51, normalized size of antiderivative = 2.22, number of steps used = 9, number of rules used = 5, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {1593, 6741, 6742, 2288, 2554} \begin {gather*} -\frac {3 e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \left (-x^4+2 x+12\right ) \log (2 x+2)}{\left (-\frac {12}{x^4}-\frac {2}{x^3}+1\right ) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*x^4 + (36 + 42*x + 6*x^2 + 3*x^3 - 3*x^5)*Log[2 + 2*x])/(E^((4 + x + x^4)/x^3)*(x^3 + x^4)),x]

[Out]

(-3*E^(-4/x^3 - x^(-2) - x)*(12 + 2*x - x^4)*Log[2 + 2*x])/((1 - 12/x^4 - 2/x^3)*x^3)

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {4+x+x^4}{x^3}} \left (3 x^4+\left (36+42 x+6 x^2+3 x^3-3 x^5\right ) \log (2+2 x)\right )}{x^3 (1+x)} \, dx\\ &=\int \frac {e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \left (3 x^4+\left (36+42 x+6 x^2+3 x^3-3 x^5\right ) \log (2+2 x)\right )}{x^3 (1+x)} \, dx\\ &=\int \left (\frac {3 e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} x}{1+x}+\frac {3 e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \left (12+2 x+x^3-x^4\right ) \log (2+2 x)}{x^3}\right ) \, dx\\ &=3 \int \frac {e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} x}{1+x} \, dx+3 \int \frac {e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \left (12+2 x+x^3-x^4\right ) \log (2+2 x)}{x^3} \, dx\\ &=-\frac {3 e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \left (12+2 x-x^4\right ) \log (2+2 x)}{\left (1-\frac {12}{x^4}-\frac {2}{x^3}\right ) x^3}+3 \int \left (e^{-\frac {4}{x^3}-\frac {1}{x^2}-x}+\frac {e^{-\frac {4}{x^3}-\frac {1}{x^2}-x}}{-1-x}\right ) \, dx-3 \int \frac {e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} x}{1+x} \, dx\\ &=-\frac {3 e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \left (12+2 x-x^4\right ) \log (2+2 x)}{\left (1-\frac {12}{x^4}-\frac {2}{x^3}\right ) x^3}+3 \int e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \, dx-3 \int \left (e^{-\frac {4}{x^3}-\frac {1}{x^2}-x}+\frac {e^{-\frac {4}{x^3}-\frac {1}{x^2}-x}}{-1-x}\right ) \, dx+3 \int \frac {e^{-\frac {4}{x^3}-\frac {1}{x^2}-x}}{-1-x} \, dx\\ &=-\frac {3 e^{-\frac {4}{x^3}-\frac {1}{x^2}-x} \left (12+2 x-x^4\right ) \log (2+2 x)}{\left (1-\frac {12}{x^4}-\frac {2}{x^3}\right ) x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 22, normalized size = 0.96 \begin {gather*} 3 e^{-\frac {4+x+x^4}{x^3}} x \log (2 (1+x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^4 + (36 + 42*x + 6*x^2 + 3*x^3 - 3*x^5)*Log[2 + 2*x])/(E^((4 + x + x^4)/x^3)*(x^3 + x^4)),x]

[Out]

(3*x*Log[2*(1 + x)])/E^((4 + x + x^4)/x^3)

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fricas [A]  time = 0.57, size = 21, normalized size = 0.91 \begin {gather*} 3 \, x e^{\left (-\frac {x^{4} + x + 4}{x^{3}}\right )} \log \left (2 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^5+3*x^3+6*x^2+42*x+36)*log(2*x+2)+3*x^4)/(x^4+x^3)/exp((x^4+x+4)/x^3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.24, size = 21, normalized size = 0.91 \begin {gather*} 3 \, x e^{\left (-\frac {x^{4} + x + 4}{x^{3}}\right )} \log \left (2 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^5+3*x^3+6*x^2+42*x+36)*log(2*x+2)+3*x^4)/(x^4+x^3)/exp((x^4+x+4)/x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 22, normalized size = 0.96

method	result	size
risch	\(3 x \,{\mathrm e}^{-\frac {x^{4}+x +4}{x^{3}}} \ln \left (2 x +2\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^5+3*x^3+6*x^2+42*x+36)*ln(2*x+2)+3*x^4)/(x^4+x^3)/exp((x^4+x+4)/x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.53, size = 28, normalized size = 1.22 \begin {gather*} 3 \, {\left (x \log \relax (2) + x \log \left (x + 1\right )\right )} e^{\left (-x - \frac {1}{x^{2}} - \frac {4}{x^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^5+3*x^3+6*x^2+42*x+36)*log(2*x+2)+3*x^4)/(x^4+x^3)/exp((x^4+x+4)/x^3),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {x^4+x+4}{x^3}}\,\left (3\,x^4+\ln \left (2\,x+2\right )\,\left (-3\,x^5+3\,x^3+6\,x^2+42\,x+36\right )\right )}{x^4+x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x + x^4 + 4)/x^3)*(3*x^4 + log(2*x + 2)*(42*x + 6*x^2 + 3*x^3 - 3*x^5 + 36)))/(x^3 + x^4),x)

[Out]

int((exp(-(x + x^4 + 4)/x^3)*(3*x^4 + log(2*x + 2)*(42*x + 6*x^2 + 3*x^3 - 3*x^5 + 36)))/(x^3 + x^4), x)

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sympy [A]  time = 0.48, size = 20, normalized size = 0.87 \begin {gather*} 3 x e^{- \frac {x^{4} + x + 4}{x^{3}}} \log {\left (2 x + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**5+3*x**3+6*x**2+42*x+36)*ln(2*x+2)+3*x**4)/(x**4+x**3)/exp((x**4+x+4)/x**3),x)

[Out]

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

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