3.51.88 \(\int -\frac {10 e^5}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx\)

Optimal. Leaf size=15 \[ \log \left (21-10 e^5 (2-x+\log (4))\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 33, 31} \begin {gather*} \log \left (21-10 e^5 (-x+2+\log (4))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10*E^5)/(-21 + E^5*(20 - 10*x) + 10*E^5*Log[4]),x]

[Out]

Log[21 - 10*E^5*(2 - x + Log[4])]

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 (\left (10 e^5\right ) \int \frac {1}{-21+e^5 (20-10 x)+10 e^5 \log (4)} \, dx\right )\\ &=e^5 \operatorname {Subst}\left (\int \frac {1}{-21+e^5 x+10 e^5 \log (4)} \, dx,x,20-10 x\right )\\ &=\log \left (21-10 e^5 (2-x+\log (4))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \log \left (21+10 e^5 (-2+x-\log (4))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*E^5)/(-21 + E^5*(20 - 10*x) + 10*E^5*Log[4]),x]

[Out]

Log[21 + 10*E^5*(-2 + x - Log[4])]

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fricas [A]  time = 1.09, size = 16, normalized size = 1.07 \begin {gather*} \log \left (10 \, {\left (x - 2\right )} e^{5} - 20 \, e^{5} \log \relax (2) + 21\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-10*exp(5)/(20*exp(5)*log(2)+(20-10*x)*exp(5)-21),x, algorithm="fricas")

[Out]

log(10*(x - 2)*e^5 - 20*e^5*log(2) + 21)

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giac [A]  time = 0.12, size = 17, normalized size = 1.13 \begin {gather*} \log \left ({\left | 10 \, {\left (x - 2\right )} e^{5} - 20 \, e^{5} \log \relax (2) + 21 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-10*exp(5)/(20*exp(5)*log(2)+(20-10*x)*exp(5)-21),x, algorithm="giac")

[Out]

log(abs(10*(x - 2)*e^5 - 20*e^5*log(2) + 21))

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maple [A]  time = 0.20, size = 19, normalized size = 1.27

method	result	size
default	\(\ln \left (20 \,{\mathrm e}^{5} \ln \relax (2)-10 x \,{\mathrm e}^{5}+20 \,{\mathrm e}^{5}-21\right )\)	\(19\)
norman	\(\ln \left (20 \,{\mathrm e}^{5} \ln \relax (2)-10 x \,{\mathrm e}^{5}+20 \,{\mathrm e}^{5}-21\right )\)	\(19\)
risch	\(\ln \left (20 \,{\mathrm e}^{5} \ln \relax (2)-10 x \,{\mathrm e}^{5}+20 \,{\mathrm e}^{5}-21\right )\)	\(19\)
meijerg	\(\ln \left (1-\frac {10 x \,{\mathrm e}^{5}}{20 \,{\mathrm e}^{5} \ln \relax (2)+20 \,{\mathrm e}^{5}-21}\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-10*exp(5)/(20*exp(5)*ln(2)+(20-10*x)*exp(5)-21),x,method=_RETURNVERBOSE)

[Out]

ln(20*exp(5)*ln(2)-10*x*exp(5)+20*exp(5)-21)

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maxima [A]  time = 0.34, size = 16, normalized size = 1.07 \begin {gather*} \log \left (10 \, {\left (x - 2\right )} e^{5} - 20 \, e^{5} \log \relax (2) + 21\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-10*exp(5)/(20*exp(5)*log(2)+(20-10*x)*exp(5)-21),x, algorithm="maxima")

[Out]

log(10*(x - 2)*e^5 - 20*e^5*log(2) + 21)

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mupad [B]  time = 0.31, size = 12, normalized size = 0.80 \begin {gather*} \ln \left (x+\frac {21\,{\mathrm {e}}^{-5}}{10}-\ln \relax (4)-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*exp(5))/(exp(5)*(10*x - 20) - 20*exp(5)*log(2) + 21),x)

[Out]

log(x + (21*exp(-5))/10 - log(4) - 2)

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sympy [A]  time = 0.07, size = 22, normalized size = 1.47 \begin {gather*} \log {\left (10 x e^{5} - 20 e^{5} - 20 e^{5} \log {\relax (2 )} + 21 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-10*exp(5)/(20*exp(5)*ln(2)+(20-10*x)*exp(5)-21),x)

[Out]

log(10*x*exp(5) - 20*exp(5) - 20*exp(5)*log(2) + 21)

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