3.62.81 \(\int \frac {51 e^4}{400+e^4 (800+51 x)} \, dx\)

Optimal. Leaf size=16 \[ 2+\log \left (16 \left (2+\frac {1}{e^4}\right )+\frac {51 x}{25}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 33, 31} \begin {gather*} \log \left (e^4 (51 x+800)+400\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(51*E^4)/(400 + E^4*(800 + 51*x)),x]

[Out]

Log[400 + E^4*(800 + 51*x)]

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (51 e^4\right ) \int \frac {1}{400+e^4 (800+51 x)} \, dx\\ &=e^4 \operatorname {Subst}\left (\int \frac {1}{400+e^4 x} \, dx,x,800+51 x\right )\\ &=\log \left (400+e^4 (800+51 x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.75 \begin {gather*} \log \left (400+e^4 (800+51 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(51*E^4)/(400 + E^4*(800 + 51*x)),x]

[Out]

Log[400 + E^4*(800 + 51*x)]

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fricas [A]  time = 0.61, size = 11, normalized size = 0.69 \begin {gather*} \log \left ({\left (51 \, x + 800\right )} e^{4} + 400\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(51*exp(4)/((51*x+800)*exp(4)+400),x, algorithm="fricas")

[Out]

log((51*x + 800)*e^4 + 400)

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giac [A]  time = 0.16, size = 12, normalized size = 0.75 \begin {gather*} \log \left ({\left | {\left (51 \, x + 800\right )} e^{4} + 400 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(51*exp(4)/((51*x+800)*exp(4)+400),x, algorithm="giac")

[Out]

log(abs((51*x + 800)*e^4 + 400))

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maple [A]  time = 0.36, size = 13, normalized size = 0.81

method	result	size
default	\(\ln \left (51 x \,{\mathrm e}^{4}+800 \,{\mathrm e}^{4}+400\right )\)	\(13\)
norman	\(\ln \left (51 x \,{\mathrm e}^{4}+800 \,{\mathrm e}^{4}+400\right )\)	\(13\)
risch	\(\ln \left (51 x \,{\mathrm e}^{4}+800 \,{\mathrm e}^{4}+400\right )\)	\(13\)
meijerg	\(\frac {400 \left (2 \,{\mathrm e}^{4}+1\right ) \ln \left (1+\frac {51 x \,{\mathrm e}^{4}}{400 \left (2 \,{\mathrm e}^{4}+1\right )}\right )}{800 \,{\mathrm e}^{4}+400}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(51*exp(4)/((51*x+800)*exp(4)+400),x,method=_RETURNVERBOSE)

[Out]

ln(51*x*exp(4)+800*exp(4)+400)

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maxima [A]  time = 0.34, size = 11, normalized size = 0.69 \begin {gather*} \log \left ({\left (51 \, x + 800\right )} e^{4} + 400\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(51*exp(4)/((51*x+800)*exp(4)+400),x, algorithm="maxima")

[Out]

log((51*x + 800)*e^4 + 400)

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mupad [B]  time = 4.48, size = 10, normalized size = 0.62 \begin {gather*} \ln \left (51\,x+400\,{\mathrm {e}}^{-4}+800\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((51*exp(4))/(exp(4)*(51*x + 800) + 400),x)

[Out]

log(51*x + 400*exp(-4) + 800)

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sympy [A]  time = 0.07, size = 14, normalized size = 0.88 \begin {gather*} \log {\left (51 x e^{4} + 400 + 800 e^{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(51*exp(4)/((51*x+800)*exp(4)+400),x)

[Out]

log(51*x*exp(4) + 400 + 800*exp(4))

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