3.57.33 \(\int \frac {32 e^x \log (261+4 e^x)}{261+4 e^x} \, dx\)

Optimal. Leaf size=12 \[ 4 \log ^2\left (261+4 e^x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2282, 2390, 2301} \begin {gather*} 4 \log ^2\left (4 e^x+261\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32*E^x*Log[261 + 4*E^x])/(261 + 4*E^x),x]

[Out]

4*Log[261 + 4*E^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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=32 \int \frac {e^x \log \left (261+4 e^x\right )}{261+4 e^x} \, dx\\ &=32 \operatorname {Subst}\left (\int \frac {\log (261+4 x)}{261+4 x} \, dx,x,e^x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,261+4 e^x\right )\\ &=4 \log ^2\left (261+4 e^x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} 4 \log ^2\left (261+4 e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32*E^x*Log[261 + 4*E^x])/(261 + 4*E^x),x]

[Out]

4*Log[261 + 4*E^x]^2

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fricas [A]  time = 0.76, size = 11, normalized size = 0.92 \begin {gather*} 4 \, \log \left (4 \, e^{x} + 261\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(32*exp(x)*log(4*exp(x)+261)/(4*exp(x)+261),x, algorithm="fricas")

[Out]

4*log(4*e^x + 261)^2

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giac [A]  time = 0.20, size = 11, normalized size = 0.92 \begin {gather*} 4 \, \log \left (4 \, e^{x} + 261\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(32*exp(x)*log(4*exp(x)+261)/(4*exp(x)+261),x, algorithm="giac")

[Out]

4*log(4*e^x + 261)^2

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maple [A]  time = 0.04, size = 12, normalized size = 1.00

method	result	size
derivativedivides	\(4 \ln \left (4 \,{\mathrm e}^{x}+261\right )^{2}\)	\(12\)
default	\(4 \ln \left (4 \,{\mathrm e}^{x}+261\right )^{2}\)	\(12\)
norman	\(4 \ln \left (4 \,{\mathrm e}^{x}+261\right )^{2}\)	\(12\)
risch	\(4 \ln \left (4 \,{\mathrm e}^{x}+261\right )^{2}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(32*exp(x)*ln(4*exp(x)+261)/(4*exp(x)+261),x,method=_RETURNVERBOSE)

[Out]

4*ln(4*exp(x)+261)^2

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maxima [A]  time = 0.38, size = 11, normalized size = 0.92 \begin {gather*} 4 \, \log \left (4 \, e^{x} + 261\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(32*exp(x)*log(4*exp(x)+261)/(4*exp(x)+261),x, algorithm="maxima")

[Out]

4*log(4*e^x + 261)^2

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mupad [B]  time = 3.52, size = 11, normalized size = 0.92 \begin {gather*} 4\,{\ln \left (4\,{\mathrm {e}}^x+261\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*log(4*exp(x) + 261)*exp(x))/(4*exp(x) + 261),x)

[Out]

4*log(4*exp(x) + 261)^2

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sympy [A]  time = 0.11, size = 10, normalized size = 0.83 \begin {gather*} 4 \log {\left (4 e^{x} + 261 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(32*exp(x)*ln(4*exp(x)+261)/(4*exp(x)+261),x)

[Out]

4*log(4*exp(x) + 261)**2

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