3.3.13 \(\int \frac {e^x (-117-9 x)+e^x (-117+117 x+9 x^2) \log (x)-9 x \log ^2(x)}{x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=21 \[ 9 \left (\frac {e^x (13+x)}{x \log (x)}-\log (x)\right ) \]

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Rubi [A]  time = 0.42, antiderivative size = 29, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {6742, 2288} \begin {gather*} \frac {9 e^x \left (x^2 \log (x)+13 x \log (x)\right )}{x^2 \log ^2(x)}-9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-117 - 9*x) + E^x*(-117 + 117*x + 9*x^2)*Log[x] - 9*x*Log[x]^2)/(x^2*Log[x]^2),x]

[Out]

-9*Log[x] + (9*E^x*(13*x*Log[x] + x^2*Log[x]))/(x^2*Log[x]^2)

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {9}{x}+\frac {9 e^x \left (-13-x-13 \log (x)+13 x \log (x)+x^2 \log (x)\right )}{x^2 \log ^2(x)}\right ) \, dx\\ &=-9 \log (x)+9 \int \frac {e^x \left (-13-x-13 \log (x)+13 x \log (x)+x^2 \log (x)\right )}{x^2 \log ^2(x)} \, dx\\ &=-9 \log (x)+\frac {9 e^x \left (13 x \log (x)+x^2 \log (x)\right )}{x^2 \log ^2(x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 20, normalized size = 0.95 \begin {gather*} \frac {9 e^x (13+x)}{x \log (x)}-9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-117 - 9*x) + E^x*(-117 + 117*x + 9*x^2)*Log[x] - 9*x*Log[x]^2)/(x^2*Log[x]^2),x]

[Out]

(9*E^x*(13 + x))/(x*Log[x]) - 9*Log[x]

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fricas [A]  time = 0.57, size = 23, normalized size = 1.10 \begin {gather*} -\frac {9 \, {\left (x \log \relax (x)^{2} - {\left (x + 13\right )} e^{x}\right )}}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*log(x)^2+(9*x^2+117*x-117)*exp(x)*log(x)+(-9*x-117)*exp(x))/x^2/log(x)^2,x, algorithm="fricas"
)

[Out]

-9*(x*log(x)^2 - (x + 13)*e^x)/(x*log(x))

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giac [A]  time = 0.33, size = 25, normalized size = 1.19 \begin {gather*} -\frac {9 \, {\left (x \log \relax (x)^{2} - x e^{x} - 13 \, e^{x}\right )}}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*log(x)^2+(9*x^2+117*x-117)*exp(x)*log(x)+(-9*x-117)*exp(x))/x^2/log(x)^2,x, algorithm="giac")

[Out]

-9*(x*log(x)^2 - x*e^x - 13*e^x)/(x*log(x))

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maple [A]  time = 0.04, size = 20, normalized size = 0.95

method	result	size
risch	\(\frac {9 \,{\mathrm e}^{x} \left (x +13\right )}{\ln \relax (x ) x}-9 \ln \relax (x )\)	\(20\)
default	\(\frac {9 \,{\mathrm e}^{x} x +117 \,{\mathrm e}^{x}}{\ln \relax (x ) x}-9 \ln \relax (x )\)	\(24\)
norman	\(\frac {9 \,{\mathrm e}^{x} x +117 \,{\mathrm e}^{x}}{\ln \relax (x ) x}-9 \ln \relax (x )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*x*ln(x)^2+(9*x^2+117*x-117)*exp(x)*ln(x)+(-9*x-117)*exp(x))/x^2/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

9*exp(x)*(x+13)/ln(x)/x-9*ln(x)

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maxima [A]  time = 0.91, size = 19, normalized size = 0.90 \begin {gather*} \frac {9 \, {\left (x + 13\right )} e^{x}}{x \log \relax (x)} - 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*log(x)^2+(9*x^2+117*x-117)*exp(x)*log(x)+(-9*x-117)*exp(x))/x^2/log(x)^2,x, algorithm="maxima"
)

[Out]

9*(x + 13)*e^x/(x*log(x)) - 9*log(x)

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mupad [B]  time = 0.38, size = 23, normalized size = 1.10 \begin {gather*} \frac {117\,{\mathrm {e}}^x+9\,x\,{\mathrm {e}}^x}{x\,\ln \relax (x)}-9\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x*log(x)^2 + exp(x)*(9*x + 117) - exp(x)*log(x)*(117*x + 9*x^2 - 117))/(x^2*log(x)^2),x)

[Out]

(117*exp(x) + 9*x*exp(x))/(x*log(x)) - 9*log(x)

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sympy [A]  time = 0.27, size = 17, normalized size = 0.81 \begin {gather*} - 9 \log {\relax (x )} + \frac {\left (9 x + 117\right ) e^{x}}{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x*ln(x)**2+(9*x**2+117*x-117)*exp(x)*ln(x)+(-9*x-117)*exp(x))/x**2/ln(x)**2,x)

[Out]

-9*log(x) + (9*x + 117)*exp(x)/(x*log(x))

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