3.26.88 \(\int \frac {(-7-x-e x) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2))+(4 x^4+4 e x^4) \log (4 \log (5) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2)))+(28 x^3+4 x^4+4 e x^4) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2)) \log ^2(4 \log (5) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2)))}{(7+x+e x) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2))} \, dx\)

Optimal. Leaf size=24 \[ -x+x^4 \log ^2\left (4 \log (5) \log \left ((7+x+e x)^2\right )\right ) \]

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Rubi [F]  time = 1.90, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-7 - x - E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)] + (4*x^4 + 4*E*x^4)*Log[4*Log[5]*Log[49
+ 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]] + (28*x^3 + 4*x^4 + 4*E*x^4)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14
*x + 2*x^2)]*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]]^2)/((7 + x + E*x)*Log[49 + 14*x +
x^2 + E^2*x^2 + E*(14*x + 2*x^2)]),x]

[Out]

-x + (2401*Log[4*Log[5]*Log[(7 + (1 + E)*x)^2]]^2)/(1 + E)^4 + (196*Defer[Int][(x*Log[4*Log[5]*Log[(7 + (1 + E
)*x)^2]])/Log[(7 + (1 + E)*x)^2], x])/(1 + E)^2 - (28*Defer[Int][(x^2*Log[4*Log[5]*Log[(7 + (1 + E)*x)^2]])/Lo
g[(7 + (1 + E)*x)^2], x])/(1 + E) + 4*Defer[Int][(x^3*Log[4*Log[5]*Log[(7 + (1 + E)*x)^2]])/Log[(7 + (1 + E)*x
)^2], x] + 4*Defer[Int][x^3*Log[4*Log[5]*Log[(7 + (1 + E)*x)^2]]^2, x] - (1372*Defer[Subst][Defer[Int][Log[4*L
og[5]*Log[x^2]]/Log[x^2], x], x, 7 + (1 + E)*x])/(1 + E)^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+(1+e) x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx\\ &=\int \left (-1+\frac {4 (1+e) x^4 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(7+(1+e) x) \log \left ((7+(1+e) x)^2\right )}+4 x^3 \log ^2\left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )\right ) \, dx\\ &=-x+4 \int x^3 \log ^2\left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right ) \, dx+(4 (1+e)) \int \frac {x^4 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(7+(1+e) x) \log \left ((7+(1+e) x)^2\right )} \, dx\\ &=-x+4 \int x^3 \log ^2\left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right ) \, dx+(4 (1+e)) \int \left (-\frac {343 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(1+e)^4 \log \left ((7+(1+e) x)^2\right )}+\frac {49 x \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(1+e)^3 \log \left ((7+(1+e) x)^2\right )}-\frac {7 x^2 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(1+e)^2 \log \left ((7+(1+e) x)^2\right )}+\frac {x^3 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(1+e) \log \left ((7+(1+e) x)^2\right )}+\frac {2401 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(1+e)^4 (7+(1+e) x) \log \left ((7+(1+e) x)^2\right )}\right ) \, dx\\ &=-x+4 \int \frac {x^3 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{\log \left ((7+(1+e) x)^2\right )} \, dx+4 \int x^3 \log ^2\left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right ) \, dx-\frac {1372 \int \frac {\log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{\log \left ((7+(1+e) x)^2\right )} \, dx}{(1+e)^3}+\frac {9604 \int \frac {\log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(7+(1+e) x) \log \left ((7+(1+e) x)^2\right )} \, dx}{(1+e)^3}+\frac {196 \int \frac {x \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{\log \left ((7+(1+e) x)^2\right )} \, dx}{(1+e)^2}-\frac {28 \int \frac {x^2 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{\log \left ((7+(1+e) x)^2\right )} \, dx}{1+e}\\ &=-x+\frac {2401 \log ^2\left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{(1+e)^4}+4 \int \frac {x^3 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{\log \left ((7+(1+e) x)^2\right )} \, dx+4 \int x^3 \log ^2\left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right ) \, dx-\frac {1372 \operatorname {Subst}\left (\int \frac {\log \left (4 \log (5) \log \left (x^2\right )\right )}{\log \left (x^2\right )} \, dx,x,7+(1+e) x\right )}{(1+e)^4}+\frac {196 \int \frac {x \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{\log \left ((7+(1+e) x)^2\right )} \, dx}{(1+e)^2}-\frac {28 \int \frac {x^2 \log \left (4 \log (5) \log \left ((7+(1+e) x)^2\right )\right )}{\log \left ((7+(1+e) x)^2\right )} \, dx}{1+e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 24, normalized size = 1.00 \begin {gather*} -x+x^4 \log ^2\left (4 \log (5) \log \left ((7+x+e x)^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-7 - x - E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)] + (4*x^4 + 4*E*x^4)*Log[4*Log[5]*L
og[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]] + (28*x^3 + 4*x^4 + 4*E*x^4)*Log[49 + 14*x + x^2 + E^2*x^2 +
E*(14*x + 2*x^2)]*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]]^2)/((7 + x + E*x)*Log[49 +
14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]),x]

[Out]

-x + x^4*Log[4*Log[5]*Log[(7 + x + E*x)^2]]^2

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fricas [A]  time = 0.84, size = 41, normalized size = 1.71 \begin {gather*} x^{4} \log \left (4 \, \log \relax (5) \log \left (x^{2} e^{2} + x^{2} + 2 \, {\left (x^{2} + 7 \, x\right )} e + 14 \, x + 49\right )\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49)*log(4*log(5)*log(x^2*
exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*e
xp(1)+x^2+14*x+49))+(-x*exp(1)-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))/(x*exp(1)+x+7)/log(x^2*
exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49),x, algorithm="fricas")

[Out]

x^4*log(4*log(5)*log(x^2*e^2 + x^2 + 2*(x^2 + 7*x)*e + 14*x + 49))^2 - x

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giac [A]  time = 4.54, size = 42, normalized size = 1.75 \begin {gather*} x^{4} \log \left (4 \, \log \relax (5) \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right )\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49)*log(4*log(5)*log(x^2*
exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*e
xp(1)+x^2+14*x+49))+(-x*exp(1)-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))/(x*exp(1)+x+7)/log(x^2*
exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49),x, algorithm="giac")

[Out]

x^4*log(4*log(5)*log(x^2*e^2 + 2*x^2*e + x^2 + 14*x*e + 14*x + 49))^2 - x

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (4 x^{4} {\mathrm e}+4 x^{4}+28 x^{3}\right ) \ln \left (x^{2} {\mathrm e}^{2}+\left (2 x^{2}+14 x \right ) {\mathrm e}+x^{2}+14 x +49\right ) \ln \left (4 \ln \relax (5) \ln \left (x^{2} {\mathrm e}^{2}+\left (2 x^{2}+14 x \right ) {\mathrm e}+x^{2}+14 x +49\right )\right )^{2}+\left (4 x^{4} {\mathrm e}+4 x^{4}\right ) \ln \left (4 \ln \relax (5) \ln \left (x^{2} {\mathrm e}^{2}+\left (2 x^{2}+14 x \right ) {\mathrm e}+x^{2}+14 x +49\right )\right )+\left (-x \,{\mathrm e}-x -7\right ) \ln \left (x^{2} {\mathrm e}^{2}+\left (2 x^{2}+14 x \right ) {\mathrm e}+x^{2}+14 x +49\right )}{\left (x \,{\mathrm e}+x +7\right ) \ln \left (x^{2} {\mathrm e}^{2}+\left (2 x^{2}+14 x \right ) {\mathrm e}+x^{2}+14 x +49\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4*exp(1)+4*x^4+28*x^3)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49)*ln(4*ln(5)*ln(x^2*exp(1)^2+(
2*x^2+14*x)*exp(1)+x^2+14*x+49))^2+(4*x^4*exp(1)+4*x^4)*ln(4*ln(5)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*
x+49))+(-x*exp(1)-x-7)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))/(x*exp(1)+x+7)/ln(x^2*exp(1)^2+(2*x^2
+14*x)*exp(1)+x^2+14*x+49),x)

[Out]

int(((4*x^4*exp(1)+4*x^4+28*x^3)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49)*ln(4*ln(5)*ln(x^2*exp(1)^2+(
2*x^2+14*x)*exp(1)+x^2+14*x+49))^2+(4*x^4*exp(1)+4*x^4)*ln(4*ln(5)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*
x+49))+(-x*exp(1)-x-7)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))/(x*exp(1)+x+7)/ln(x^2*exp(1)^2+(2*x^2
+14*x)*exp(1)+x^2+14*x+49),x)

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maxima [B]  time = 0.98, size = 267, normalized size = 11.12 \begin {gather*} 2 \, x^{4} {\left (3 \, \log \relax (2) + \log \left (\log \relax (5)\right )\right )} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right ) + x^{4} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )^{2} + {\left (9 \, \log \relax (2)^{2} + 6 \, \log \relax (2) \log \left (\log \relax (5)\right ) + \log \left (\log \relax (5)\right )^{2}\right )} x^{4} - {\left (\frac {x}{e + 1} - \frac {7 \, \log \left (x {\left (e + 1\right )} + 7\right )}{e^{2} + 2 \, e + 1}\right )} e - \frac {7 \, \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right ) \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )}{2 \, {\left (e + 1\right )}} - \frac {x}{e + 1} + \frac {7 \, {\left (\frac {{\left (e \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right ) + \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right )\right )} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )}{e + 1} - 2 \, \log \left (x {\left (e + 1\right )} + 7\right )\right )}}{2 \, {\left (e + 1\right )}} + \frac {7 \, \log \left (x {\left (e + 1\right )} + 7\right )}{e^{2} + 2 \, e + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49)*log(4*log(5)*log(x^2*
exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*e
xp(1)+x^2+14*x+49))+(-x*exp(1)-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))/(x*exp(1)+x+7)/log(x^2*
exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49),x, algorithm="maxima")

[Out]

2*x^4*(3*log(2) + log(log(5)))*log(log(x*(e + 1) + 7)) + x^4*log(log(x*(e + 1) + 7))^2 + (9*log(2)^2 + 6*log(2
)*log(log(5)) + log(log(5))^2)*x^4 - (x/(e + 1) - 7*log(x*(e + 1) + 7)/(e^2 + 2*e + 1))*e - 7/2*log(x^2*e^2 +
2*x^2*e + x^2 + 14*x*e + 14*x + 49)*log(log(x*(e + 1) + 7))/(e + 1) - x/(e + 1) + 7/2*((e*log(x^2*e^2 + 2*x^2*
e + x^2 + 14*x*e + 14*x + 49) + log(x^2*e^2 + 2*x^2*e + x^2 + 14*x*e + 14*x + 49))*log(log(x*(e + 1) + 7))/(e
+ 1) - 2*log(x*(e + 1) + 7))/(e + 1) + 7*log(x*(e + 1) + 7)/(e^2 + 2*e + 1)

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mupad [B]  time = 2.26, size = 42, normalized size = 1.75 \begin {gather*} x\,\left (x^3\,{\ln \left (4\,\ln \left (14\,x+\mathrm {e}\,\left (2\,x^2+14\,x\right )+x^2\,{\mathrm {e}}^2+x^2+49\right )\,\ln \relax (5)\right )}^2-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*log(5))*(4*x^4*exp(1) + 4*x^4) - log(14*x
+ exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*(x + x*exp(1) + 7) + log(14*x + exp(1)*(14*x + 2*x^2) + x^2*
exp(2) + x^2 + 49)*log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*log(5))^2*(4*x^4*exp(1) + 2
8*x^3 + 4*x^4))/(log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*(x + x*exp(1) + 7)),x)

[Out]

x*(x^3*log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*log(5))^2 - 1)

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sympy [A]  time = 1.68, size = 41, normalized size = 1.71 \begin {gather*} x^{4} \log {\left (4 \log {\relax (5 )} \log {\left (x^{2} + x^{2} e^{2} + 14 x + e \left (2 x^{2} + 14 x\right ) + 49 \right )} \right )}^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4*exp(1)+4*x**4+28*x**3)*ln(x**2*exp(1)**2+(2*x**2+14*x)*exp(1)+x**2+14*x+49)*ln(4*ln(5)*ln(x
**2*exp(1)**2+(2*x**2+14*x)*exp(1)+x**2+14*x+49))**2+(4*x**4*exp(1)+4*x**4)*ln(4*ln(5)*ln(x**2*exp(1)**2+(2*x*
*2+14*x)*exp(1)+x**2+14*x+49))+(-x*exp(1)-x-7)*ln(x**2*exp(1)**2+(2*x**2+14*x)*exp(1)+x**2+14*x+49))/(x*exp(1)
+x+7)/ln(x**2*exp(1)**2+(2*x**2+14*x)*exp(1)+x**2+14*x+49),x)

[Out]

x**4*log(4*log(5)*log(x**2 + x**2*exp(2) + 14*x + E*(2*x**2 + 14*x) + 49))**2 - x

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