3.4.90 \(\int \frac {-1+x+e^5 (-4+e^5) x}{x} \, dx\)

Optimal. Leaf size=24 \[ x+e^5 \left (-4+e^5\right ) x-\log \left (\frac {2 x}{4+\log (2)}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.71, 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 = {6, 43} \begin {gather*} \left (1-4 e^5+e^{10}\right ) x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x + E^5*(-4 + E^5)*x)/x,x]

[Out]

(1 - 4*E^5 + E^10)*x - Log[x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+\left (1+e^5 \left (-4+e^5\right )\right ) x}{x} \, dx\\ &=\int \left (1-4 e^5+e^{10}-\frac {1}{x}\right ) \, dx\\ &=\left (1-4 e^5+e^{10}\right ) x-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.71 \begin {gather*} x-4 e^5 x+e^{10} x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x + E^5*(-4 + E^5)*x)/x,x]

[Out]

x - 4*E^5*x + E^10*x - Log[x]

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fricas [A]  time = 0.60, size = 15, normalized size = 0.62 \begin {gather*} x e^{10} - 4 \, x e^{5} + x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(log(exp(5)-4)+5)+x-1)/x,x, algorithm="fricas")

[Out]

x*e^10 - 4*x*e^5 + x - log(x)

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giac [A]  time = 0.34, size = 17, normalized size = 0.71 \begin {gather*} x e^{\left (\log \left (e^{5} - 4\right ) + 5\right )} + x - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(log(exp(5)-4)+5)+x-1)/x,x, algorithm="giac")

[Out]

x*e^(log(e^5 - 4) + 5) + x - log(abs(x))

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maple [A]  time = 0.05, size = 16, normalized size = 0.67

method	result	size
risch	\(-4 x \,{\mathrm e}^{5}+x \,{\mathrm e}^{10}+x -\ln \relax (x )\)	\(16\)
default	\(x \,{\mathrm e}^{\ln \left ({\mathrm e}^{5}-4\right )+5}+x -\ln \relax (x )\)	\(17\)
norman	\(\left ({\mathrm e}^{10}-4 \,{\mathrm e}^{5}+1\right ) x -\ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(ln(exp(5)-4)+5)+x-1)/x,x,method=_RETURNVERBOSE)

[Out]

-4*x*exp(5)+x*exp(10)+x-ln(x)

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maxima [A]  time = 0.46, size = 15, normalized size = 0.62 \begin {gather*} x {\left (e^{10} - 4 \, e^{5} + 1\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(log(exp(5)-4)+5)+x-1)/x,x, algorithm="maxima")

[Out]

x*(e^10 - 4*e^5 + 1) - log(x)

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mupad [B]  time = 0.06, size = 15, normalized size = 0.62 \begin {gather*} x\,\left ({\mathrm {e}}^{10}-4\,{\mathrm {e}}^5+1\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x*exp(log(exp(5) - 4) + 5) - 1)/x,x)

[Out]

x*(exp(10) - 4*exp(5) + 1) - log(x)

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sympy [A]  time = 0.08, size = 14, normalized size = 0.58 \begin {gather*} x \left (- 4 e^{5} + 1 + e^{10}\right ) - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(ln(exp(5)-4)+5)+x-1)/x,x)

[Out]

x*(-4*exp(5) + 1 + exp(10)) - log(x)

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