3.16.1 \(\int \frac {-4+e^x (631-x-50 \log (4)+\log ^2(4))}{-2520+4 x+200 \log (4)-4 \log ^2(4)+e^x (630-x-50 \log (4)+\log ^2(4))} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\frac {-4+e^x}{5-x+(25-\log (4))^2}\right ) \]

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Rubi [A]  time = 0.66, antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 7, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6741, 6742, 2282, 36, 31, 29, 43} \begin {gather*} \log \left (4-e^x\right )-\log \left (-x+630+\log ^2(4)-50 \log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + E^x*(631 - x - 50*Log[4] + Log[4]^2))/(-2520 + 4*x + 200*Log[4] - 4*Log[4]^2 + E^x*(630 - x - 50*Log
[4] + Log[4]^2)),x]

[Out]

Log[4 - E^x] - Log[630 - x - 50*Log[4] + Log[4]^2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 6741

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

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 \frac {4-e^x \left (631-x-50 \log (4)+\log ^2(4)\right )}{\left (4-e^x\right ) \left (630-x-50 \log (4)+\log ^2(4)\right )} \, dx\\ &=\int \left (\frac {4}{-4+e^x}+\frac {-631+x+50 \log (4)-\log ^2(4)}{-630+x+50 \log (4)-\log ^2(4)}\right ) \, dx\\ &=4 \int \frac {1}{-4+e^x} \, dx+\int \frac {-631+x+50 \log (4)-\log ^2(4)}{-630+x+50 \log (4)-\log ^2(4)} \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {1}{(-4+x) x} \, dx,x,e^x\right )+\int \left (1+\frac {1}{630-x-50 \log (4)+\log ^2(4)}\right ) \, dx\\ &=x-\log \left (630-x-50 \log (4)+\log ^2(4)\right )+\operatorname {Subst}\left (\int \frac {1}{-4+x} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )\\ &=\log \left (4-e^x\right )-\log \left (630-x-50 \log (4)+\log ^2(4)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 1.14 \begin {gather*} \log \left (4-e^x\right )-\log \left (630-x-50 \log (4)+\log ^2(4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + E^x*(631 - x - 50*Log[4] + Log[4]^2))/(-2520 + 4*x + 200*Log[4] - 4*Log[4]^2 + E^x*(630 - x -
50*Log[4] + Log[4]^2)),x]

[Out]

Log[4 - E^x] - Log[630 - x - 50*Log[4] + Log[4]^2]

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fricas [A]  time = 0.70, size = 22, normalized size = 1.00 \begin {gather*} -\log \left (-4 \, \log \relax (2)^{2} + x + 100 \, \log \relax (2) - 630\right ) + \log \left (e^{x} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(2)^2-100*log(2)-x+631)*exp(x)-4)/((4*log(2)^2-100*log(2)-x+630)*exp(x)-16*log(2)^2+400*log(2
)+4*x-2520),x, algorithm="fricas")

[Out]

-log(-4*log(2)^2 + x + 100*log(2) - 630) + log(e^x - 4)

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giac [A]  time = 0.29, size = 22, normalized size = 1.00 \begin {gather*} -\log \left (-4 \, \log \relax (2)^{2} + x + 100 \, \log \relax (2) - 630\right ) + \log \left (e^{x} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(2)^2-100*log(2)-x+631)*exp(x)-4)/((4*log(2)^2-100*log(2)-x+630)*exp(x)-16*log(2)^2+400*log(2
)+4*x-2520),x, algorithm="giac")

[Out]

-log(-4*log(2)^2 + x + 100*log(2) - 630) + log(e^x - 4)

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maple [A]  time = 0.07, size = 23, normalized size = 1.05

method	result	size
risch	\(-\ln \left (-4 \ln \relax (2)^{2}+100 \ln \relax (2)+x -630\right )+\ln \left ({\mathrm e}^{x}-4\right )\)	\(23\)
norman	\(-\ln \left (4 \ln \relax (2)^{2}-100 \ln \relax (2)-x +630\right )+\ln \left ({\mathrm e}^{x}-4\right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*ln(2)^2-100*ln(2)-x+631)*exp(x)-4)/((4*ln(2)^2-100*ln(2)-x+630)*exp(x)-16*ln(2)^2+400*ln(2)+4*x-2520),
x,method=_RETURNVERBOSE)

[Out]

-ln(-4*ln(2)^2+100*ln(2)+x-630)+ln(exp(x)-4)

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maxima [A]  time = 0.88, size = 22, normalized size = 1.00 \begin {gather*} -\log \left (-4 \, \log \relax (2)^{2} + x + 100 \, \log \relax (2) - 630\right ) + \log \left (e^{x} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(2)^2-100*log(2)-x+631)*exp(x)-4)/((4*log(2)^2-100*log(2)-x+630)*exp(x)-16*log(2)^2+400*log(2
)+4*x-2520),x, algorithm="maxima")

[Out]

-log(-4*log(2)^2 + x + 100*log(2) - 630) + log(e^x - 4)

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mupad [B]  time = 1.10, size = 22, normalized size = 1.00 \begin {gather*} \ln \left ({\mathrm {e}}^x-4\right )-\ln \left (x+100\,\ln \relax (2)-4\,{\ln \relax (2)}^2-630\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(x + 100*log(2) - 4*log(2)^2 - 631) + 4)/(exp(x)*(x + 100*log(2) - 4*log(2)^2 - 630) - 400*log(2)
- 4*x + 16*log(2)^2 + 2520),x)

[Out]

log(exp(x) - 4) - log(x + 100*log(2) - 4*log(2)^2 - 630)

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sympy [A]  time = 0.14, size = 22, normalized size = 1.00 \begin {gather*} \log {\left (e^{x} - 4 \right )} - \log {\left (x - 630 - 4 \log {\relax (2 )}^{2} + 100 \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*ln(2)**2-100*ln(2)-x+631)*exp(x)-4)/((4*ln(2)**2-100*ln(2)-x+630)*exp(x)-16*ln(2)**2+400*ln(2)+4
*x-2520),x)

[Out]

log(exp(x) - 4) - log(x - 630 - 4*log(2)**2 + 100*log(2))

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