3.78.92 \(\int \frac {-6 e^x+e^x (-11+3 x) \log (121-66 x+9 x^2)}{e^{25} (-11+3 x) \log ^2(121-66 x+9 x^2)} \, dx\)

Optimal. Leaf size=20 \[ \frac {e^{-25+x}}{\log \left ((2+3 (3-x))^2\right )} \]

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Rubi [A]  time = 0.18, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 3, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {12, 6688, 2288} \begin {gather*} \frac {e^{x-25}}{\log \left ((11-3 x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*E^x + E^x*(-11 + 3*x)*Log[121 - 66*x + 9*x^2])/(E^25*(-11 + 3*x)*Log[121 - 66*x + 9*x^2]^2),x]

[Out]

E^(-25 + x)/Log[(11 - 3*x)^2]

Rule 12

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

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-6 e^x+e^x (-11+3 x) \log \left (121-66 x+9 x^2\right )}{(-11+3 x) \log ^2\left (121-66 x+9 x^2\right )} \, dx}{e^{25}}\\ &=\frac {\int \frac {e^x \left (\frac {6}{11-3 x}+\log \left ((11-3 x)^2\right )\right )}{\log ^2\left ((11-3 x)^2\right )} \, dx}{e^{25}}\\ &=\frac {e^{-25+x}}{\log \left ((11-3 x)^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.80 \begin {gather*} \frac {e^{-25+x}}{\log \left ((11-3 x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*E^x + E^x*(-11 + 3*x)*Log[121 - 66*x + 9*x^2])/(E^25*(-11 + 3*x)*Log[121 - 66*x + 9*x^2]^2),x]

[Out]

E^(-25 + x)/Log[(11 - 3*x)^2]

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fricas [A]  time = 0.63, size = 18, normalized size = 0.90 \begin {gather*} \frac {e^{\left (x - 25\right )}}{\log \left (9 \, x^{2} - 66 \, x + 121\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-11+3*x)*exp(x)*log(9*x^2-66*x+121)-6*exp(x))/(-11+3*x)/exp(25)/log(9*x^2-66*x+121)^2,x, algorithm
="fricas")

[Out]

e^(x - 25)/log(9*x^2 - 66*x + 121)

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giac [A]  time = 0.21, size = 18, normalized size = 0.90 \begin {gather*} \frac {e^{\left (x - 25\right )}}{\log \left (9 \, x^{2} - 66 \, x + 121\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-11+3*x)*exp(x)*log(9*x^2-66*x+121)-6*exp(x))/(-11+3*x)/exp(25)/log(9*x^2-66*x+121)^2,x, algorithm
="giac")

[Out]

e^(x - 25)/log(9*x^2 - 66*x + 121)

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maple [A]  time = 0.27, size = 18, normalized size = 0.90

method	result	size
default	\(\frac {{\mathrm e}^{-25} {\mathrm e}^{x}}{\ln \left (\left (-11+3 x \right )^{2}\right )}\)	\(18\)
norman	\(\frac {{\mathrm e}^{-25} {\mathrm e}^{x}}{\ln \left (9 x^{2}-66 x +121\right )}\)	\(21\)
risch	\(\frac {2 i {\mathrm e}^{x -25}}{\pi \mathrm {csgn}\left (i \left (x -\frac {11}{3}\right )\right )^{2} \mathrm {csgn}\left (i \left (x -\frac {11}{3}\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (x -\frac {11}{3}\right )\right ) \mathrm {csgn}\left (i \left (x -\frac {11}{3}\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x -\frac {11}{3}\right )^{2}\right )^{3}+4 i \ln \left (x -\frac {11}{3}\right )}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-11+3*x)*exp(x)*ln(9*x^2-66*x+121)-6*exp(x))/(-11+3*x)/exp(25)/ln(9*x^2-66*x+121)^2,x,method=_RETURNVERB
OSE)

[Out]

1/exp(25)*exp(x)/ln((-11+3*x)^2)

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maxima [A]  time = 0.41, size = 14, normalized size = 0.70 \begin {gather*} \frac {e^{\left (x - 25\right )}}{2 \, \log \left (3 \, x - 11\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-11+3*x)*exp(x)*log(9*x^2-66*x+121)-6*exp(x))/(-11+3*x)/exp(25)/log(9*x^2-66*x+121)^2,x, algorithm
="maxima")

[Out]

1/2*e^(x - 25)/log(3*x - 11)

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mupad [B]  time = 5.92, size = 18, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {e}}^{-25}\,{\mathrm {e}}^x}{\ln \left (9\,x^2-66\,x+121\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-25)*(6*exp(x) - exp(x)*log(9*x^2 - 66*x + 121)*(3*x - 11)))/(log(9*x^2 - 66*x + 121)^2*(3*x - 11)),
x)

[Out]

(exp(-25)*exp(x))/log(9*x^2 - 66*x + 121)

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sympy [A]  time = 0.30, size = 17, normalized size = 0.85 \begin {gather*} \frac {e^{x}}{e^{25} \log {\left (9 x^{2} - 66 x + 121 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-11+3*x)*exp(x)*ln(9*x**2-66*x+121)-6*exp(x))/(-11+3*x)/exp(25)/ln(9*x**2-66*x+121)**2,x)

[Out]

exp(-25)*exp(x)/log(9*x**2 - 66*x + 121)

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