3.98.48 \(\int \frac {234 x+156 e^4 x+26 e^8 x+(1725+234 x+156 e^4 x+26 e^8 x) \log (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8})}{1725+234 x+156 e^4 x+26 e^8 x} \, dx\)

Optimal. Leaf size=18 \[ x \log \left (-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right ) \]

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Rubi [B]  time = 0.12, antiderivative size = 58, normalized size of antiderivative = 3.22, number of steps used = 10, number of rules used = 5, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6, 6688, 43, 2389, 2295} \begin {gather*} \frac {1}{26} \left (26 x+\frac {1725}{\left (3+e^4\right )^2}\right ) \log \left (-\frac {26 x}{69}-\frac {25}{\left (3+e^4\right )^2}\right )-\frac {1725 \log \left (26 \left (3+e^4\right )^2 x+1725\right )}{26 \left (3+e^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(234*x + 156*E^4*x + 26*E^8*x + (1725 + 234*x + 156*E^4*x + 26*E^8*x)*Log[(-1725 - 234*x - 156*E^4*x - 26*
E^8*x)/(621 + 414*E^4 + 69*E^8)])/(1725 + 234*x + 156*E^4*x + 26*E^8*x),x]

[Out]

((1725/(3 + E^4)^2 + 26*x)*Log[-25/(3 + E^4)^2 - (26*x)/69])/26 - (1725*Log[1725 + 26*(3 + E^4)^2*x])/(26*(3 +
E^4)^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, 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} &=\int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+26 e^8 x+\left (234+156 e^4\right ) x} \, dx\\ &=\int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+\left (234+156 e^4+26 e^8\right ) x} \, dx\\ &=\int \frac {26 e^8 x+\left (234+156 e^4\right ) x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+\left (234+156 e^4+26 e^8\right ) x} \, dx\\ &=\int \frac {\left (234+156 e^4+26 e^8\right ) x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+\left (234+156 e^4+26 e^8\right ) x} \, dx\\ &=\int \left (\frac {26 \left (3+e^4\right )^2 x}{1725+26 \left (3+e^4\right )^2 x}+\log \left (-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right )\right ) \, dx\\ &=\left (26 \left (3+e^4\right )^2\right ) \int \frac {x}{1725+26 \left (3+e^4\right )^2 x} \, dx+\int \log \left (-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right ) \, dx\\ &=-\left (\frac {69}{26} \operatorname {Subst}\left (\int \log (x) \, dx,x,-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right )\right )+\left (26 \left (3+e^4\right )^2\right ) \int \left (\frac {1}{26 \left (3+e^4\right )^2}+\frac {1725}{26 \left (3+e^4\right )^2 \left (-1725-26 \left (3+e^4\right )^2 x\right )}\right ) \, dx\\ &=\frac {1}{26} \left (\frac {1725}{\left (3+e^4\right )^2}+26 x\right ) \log \left (-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right )-\frac {1725 \log \left (1725+26 \left (3+e^4\right )^2 x\right )}{26 \left (3+e^4\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 56, normalized size = 3.11 \begin {gather*} \frac {\left (1725+26 \left (3+e^4\right )^2 x\right ) \log \left (-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right )-1725 \log \left (1725+26 \left (3+e^4\right )^2 x\right )}{26 \left (3+e^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(234*x + 156*E^4*x + 26*E^8*x + (1725 + 234*x + 156*E^4*x + 26*E^8*x)*Log[(-1725 - 234*x - 156*E^4*x
- 26*E^8*x)/(621 + 414*E^4 + 69*E^8)])/(1725 + 234*x + 156*E^4*x + 26*E^8*x),x]

[Out]

((1725 + 26*(3 + E^4)^2*x)*Log[-25/(3 + E^4)^2 - (26*x)/69] - 1725*Log[1725 + 26*(3 + E^4)^2*x])/(26*(3 + E^4)
^2)

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fricas [A]  time = 0.52, size = 30, normalized size = 1.67 \begin {gather*} x \log \left (-\frac {26 \, x e^{8} + 156 \, x e^{4} + 234 \, x + 1725}{69 \, {\left (e^{8} + 6 \, e^{4} + 9\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*log((-26*x*exp(4)^2-156*x*exp(4)-234*x-1725)/(69*exp(4)^2+4
14*exp(4)+621))+26*x*exp(4)^2+156*x*exp(4)+234*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x, algorithm="fricas
")

[Out]

x*log(-1/69*(26*x*e^8 + 156*x*e^4 + 234*x + 1725)/(e^8 + 6*e^4 + 9))

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giac [A]  time = 0.20, size = 30, normalized size = 1.67 \begin {gather*} x \log \left (-\frac {26 \, x e^{8} + 156 \, x e^{4} + 234 \, x + 1725}{69 \, {\left (e^{8} + 6 \, e^{4} + 9\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*log((-26*x*exp(4)^2-156*x*exp(4)-234*x-1725)/(69*exp(4)^2+4
14*exp(4)+621))+26*x*exp(4)^2+156*x*exp(4)+234*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x, algorithm="giac")

[Out]

x*log(-1/69*(26*x*e^8 + 156*x*e^4 + 234*x + 1725)/(e^8 + 6*e^4 + 9))

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maple [B]  time = 0.10, size = 32, normalized size = 1.78

method	result	size
risch	\(x \ln \left (\frac {-26 x \,{\mathrm e}^{8}-156 x \,{\mathrm e}^{4}-234 x -1725}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}\right )\)	\(32\)
norman	\(x \ln \left (\frac {-26 x \,{\mathrm e}^{8}-156 x \,{\mathrm e}^{4}-234 x -1725}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}\right )\)	\(36\)
derivativedivides	\(-\frac {69 \,{\mathrm e}^{8} \left (\left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right ) \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )+\frac {26 x}{69}+\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {69 \,{\mathrm e}^{8} \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {207 \,{\mathrm e}^{4} \left (\left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right ) \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )+\frac {26 x}{69}+\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{13 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {207 \,{\mathrm e}^{4} \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{13 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {621 \left (\left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right ) \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )+\frac {26 x}{69}+\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {621 \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {1725 \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}\)	\(353\)
default	\(-\frac {69 \,{\mathrm e}^{8} \left (\left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right ) \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )+\frac {26 x}{69}+\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {69 \,{\mathrm e}^{8} \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {207 \,{\mathrm e}^{4} \left (\left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right ) \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )+\frac {26 x}{69}+\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{13 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {207 \,{\mathrm e}^{4} \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{13 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {621 \left (\left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right ) \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )+\frac {26 x}{69}+\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {621 \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}-\frac {1725 \ln \left (-\frac {26 x}{69}-\frac {25}{{\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9}\right )}{26 \left ({\mathrm e}^{8}+6 \,{\mathrm e}^{4}+9\right )}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*ln((-26*x*exp(4)^2-156*x*exp(4)-234*x-1725)/(69*exp(4)^2+414*exp(
4)+621))+26*x*exp(4)^2+156*x*exp(4)+234*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x,method=_RETURNVERBOSE)

[Out]

x*ln((-26*x*exp(8)-156*x*exp(4)-234*x-1725)/(69*exp(8)+414*exp(4)+621))

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maxima [B]  time = 0.46, size = 858, normalized size = 47.67 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*log((-26*x*exp(4)^2-156*x*exp(4)-234*x-1725)/(69*exp(4)^2+4
14*exp(4)+621))+26*x*exp(4)^2+156*x*exp(4)+234*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x, algorithm="maxima
")

[Out]

1/26*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81
))*e^8*log(-26/69*x*e^8/(e^8 + 6*e^4 + 9) - 52/23*x*e^4/(e^8 + 6*e^4 + 9) - 78/23*x/(e^8 + 6*e^4 + 9) - 25/(e^
8 + 6*e^4 + 9)) + 3/13*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^16 + 12*e^12 + 54*
e^8 + 108*e^4 + 81))*e^4*log(-26/69*x*e^8/(e^8 + 6*e^4 + 9) - 52/23*x*e^4/(e^8 + 6*e^4 + 9) - 78/23*x/(e^8 + 6
*e^4 + 9) - 25/(e^8 + 6*e^4 + 9)) + 1/26*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^
16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*e^8 - 1/52*(52*x*(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) +
1725)^2 - 3450*log(26*x*(e^8 + 6*e^4 + 9) + 1725))*(e^8/(e^8 + 6*e^4 + 9) + 6*e^4/(e^8 + 6*e^4 + 9) + 9/(e^8 +
6*e^4 + 9))*e^8/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81) + 3/13*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8
+ 6*e^4 + 9) + 1725)/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*e^4 - 3/26*(52*x*(e^8 + 6*e^4 + 9) - 1725*log(2
6*x*(e^8 + 6*e^4 + 9) + 1725)^2 - 3450*log(26*x*(e^8 + 6*e^4 + 9) + 1725))*(e^8/(e^8 + 6*e^4 + 9) + 6*e^4/(e^8
+ 6*e^4 + 9) + 9/(e^8 + 6*e^4 + 9))*e^4/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81) + 9/26*(26*x/(e^8 + 6*e^4 +
9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*log(-26/69*x*e^8/(e^8 +
6*e^4 + 9) - 52/23*x*e^4/(e^8 + 6*e^4 + 9) - 78/23*x/(e^8 + 6*e^4 + 9) - 25/(e^8 + 6*e^4 + 9)) - 9/52*(52*x*(
e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)^2 - 3450*log(26*x*(e^8 + 6*e^4 + 9) + 1725))*(e^8/(
e^8 + 6*e^4 + 9) + 6*e^4/(e^8 + 6*e^4 + 9) + 9/(e^8 + 6*e^4 + 9))/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81) - 1
725/52*(2*(log(23) + log(3) + 2*log(e^4 + 3))*log(-26*x*(e^8 + 6*e^4 + 9) - 1725) - log(-26*x*(e^8 + 6*e^4 + 9
) - 1725)^2)/(e^8 + 6*e^4 + 9) + 9*x/(e^8 + 6*e^4 + 9) - 15525/26*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^16 + 1
2*e^12 + 54*e^8 + 108*e^4 + 81)

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mupad [B]  time = 1.11, size = 32, normalized size = 1.78 \begin {gather*} x\,\left (\ln \left (-234\,x-156\,x\,{\mathrm {e}}^4-26\,x\,{\mathrm {e}}^8-1725\right )-\ln \left (414\,{\mathrm {e}}^4+69\,{\mathrm {e}}^8+621\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((234*x + 156*x*exp(4) + 26*x*exp(8) + log(-(234*x + 156*x*exp(4) + 26*x*exp(8) + 1725)/(414*exp(4) + 69*ex
p(8) + 621))*(234*x + 156*x*exp(4) + 26*x*exp(8) + 1725))/(234*x + 156*x*exp(4) + 26*x*exp(8) + 1725),x)

[Out]

x*(log(- 234*x - 156*x*exp(4) - 26*x*exp(8) - 1725) - log(414*exp(4) + 69*exp(8) + 621))

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sympy [A]  time = 0.23, size = 34, normalized size = 1.89 \begin {gather*} x \log {\left (\frac {- 26 x e^{8} - 156 x e^{4} - 234 x - 1725}{621 + 414 e^{4} + 69 e^{8}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x*exp(4)**2+156*x*exp(4)+234*x+1725)*ln((-26*x*exp(4)**2-156*x*exp(4)-234*x-1725)/(69*exp(4)**2
+414*exp(4)+621))+26*x*exp(4)**2+156*x*exp(4)+234*x)/(26*x*exp(4)**2+156*x*exp(4)+234*x+1725),x)

[Out]

x*log((-26*x*exp(8) - 156*x*exp(4) - 234*x - 1725)/(621 + 414*exp(4) + 69*exp(8)))

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