3.46.94 \(\int \frac {e^{\frac {1}{9} (2 x^2+4 x^3+2 x^4+(4 x+4 x^2) \log (\frac {1}{8} (7 x+8 \log (x)))+2 \log ^2(\frac {1}{8} (7 x+8 \log (x))))} (32 x+60 x^2+56 x^3+84 x^4+56 x^5+(32 x^2+96 x^3+64 x^4) \log (x)+(32+28 x+28 x^2+56 x^3+(32 x+64 x^2) \log (x)) \log (\frac {1}{8} (7 x+8 \log (x))))}{63 x^2+72 x \log (x)} \, dx\)

Optimal. Leaf size=22 \[ e^{\frac {2}{9} \left (x+x^2+\log \left (\frac {7 x}{8}+\log (x)\right )\right )^2} \]

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Rubi [F]  time = 97.75, 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 {\exp \left (\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )\right ) \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x^2 + 4*x^3 + 2*x^4 + (4*x + 4*x^2)*Log[(7*x + 8*Log[x])/8] + 2*Log[(7*x + 8*Log[x])/8]^2)/9)*(32*x
+ 60*x^2 + 56*x^3 + 84*x^4 + 56*x^5 + (32*x^2 + 96*x^3 + 64*x^4)*Log[x] + (32 + 28*x + 28*x^2 + 56*x^3 + (32*
x + 64*x^2)*Log[x])*Log[(7*x + 8*Log[x])/8]))/(63*x^2 + 72*x*Log[x]),x]

[Out]

Defer[Int][2^(5 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*(7*x + 8*Log[x])^(-
1 + (4*x)/9 + (4*x^2)/9), x]/9 + (5*Defer[Int][2^(2 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8
+ Log[x]]^2))/9)*x*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9), x])/3 + (7*Defer[Int][2^(3 - (4*x)/3 - (4*x^2)
/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x^2*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9), x])/9
+ (7*Defer[Int][2^(2 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x^3*(7*x + 8*
Log[x])^(-1 + (4*x)/9 + (4*x^2)/9), x])/3 + (7*Defer[Int][2^(3 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + L
og[(7*x)/8 + Log[x]]^2))/9)*x^4*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9), x])/9 + Defer[Int][2^(5 - (4*x)/3
- (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x*Log[x]*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4
*x^2)/9), x]/9 + Defer[Int][2^(5 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x^
2*Log[x]*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9), x]/3 + Defer[Int][2^(6 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2
*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x^3*Log[x]*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9), x]/9 + (7*De
fer[Int][2^(2 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*(7*x + 8*Log[x])^(-1
+ (4*x)/9 + (4*x^2)/9)*Log[(7*x)/8 + Log[x]], x])/9 + Defer[Int][(2^(5 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 +
x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9)*Log[(7*x)/8 + Log[x]])/x, x]/9
+ (7*Defer[Int][2^(2 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x*(7*x + 8*Lo
g[x])^(-1 + (4*x)/9 + (4*x^2)/9)*Log[(7*x)/8 + Log[x]], x])/9 + (7*Defer[Int][2^(3 - (4*x)/3 - (4*x^2)/3)*E^((
2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x^2*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9)*Log[(7*x)/8 +
Log[x]], x])/9 + Defer[Int][2^(5 - (4*x)/3 - (4*x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*Lo
g[x]*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/9)*Log[(7*x)/8 + Log[x]], x]/9 + Defer[Int][2^(6 - (4*x)/3 - (4*
x^2)/3)*E^((2*(x^2*(1 + x)^2 + Log[(7*x)/8 + Log[x]]^2))/9)*x*Log[x]*(7*x + 8*Log[x])^(-1 + (4*x)/9 + (4*x^2)/
9)*Log[(7*x)/8 + Log[x]], x]/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )\right ) \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{x (63 x+72 \log (x))} \, dx\\ &=\int \frac {2^{2-\frac {4 x}{3}-\frac {4 x^2}{3}} \exp \left (\frac {2}{9} \left (x^2 (1+x)^2+\log ^2\left (\frac {7 x}{8}+\log (x)\right )\right )\right ) (7 x+8 \log (x))^{-1+\frac {4 x}{9}+\frac {4 x^2}{9}} \left (8+7 x+7 x^2+14 x^3+8 x (1+2 x) \log (x)\right ) \left (x+x^2+\log \left (\frac {7 x}{8}+\log (x)\right )\right )}{9 x} \, dx\\ &=\frac {1}{9} \int \frac {2^{2-\frac {4 x}{3}-\frac {4 x^2}{3}} \exp \left (\frac {2}{9} \left (x^2 (1+x)^2+\log ^2\left (\frac {7 x}{8}+\log (x)\right )\right )\right ) (7 x+8 \log (x))^{-1+\frac {4 x}{9}+\frac {4 x^2}{9}} \left (8+7 x+7 x^2+14 x^3+8 x (1+2 x) \log (x)\right ) \left (x+x^2+\log \left (\frac {7 x}{8}+\log (x)\right )\right )}{x} \, dx\\ &=\frac {1}{9} \int \left (2^{2-\frac {4 x}{3}-\frac {4 x^2}{3}} \exp \left (\frac {2}{9} \left (x^2 (1+x)^2+\log ^2\left (\frac {7 x}{8}+\log (x)\right )\right )\right ) (1+x) (7 x+8 \log (x))^{-1+\frac {4 x}{9}+\frac {4 x^2}{9}} \left (8+7 x+7 x^2+14 x^3+8 x \log (x)+16 x^2 \log (x)\right )+\frac {2^{2-\frac {4 x}{3}-\frac {4 x^2}{3}} \exp \left (\frac {2}{9} \left (x^2 (1+x)^2+\log ^2\left (\frac {7 x}{8}+\log (x)\right )\right )\right ) (7 x+8 \log (x))^{-1+\frac {4 x}{9}+\frac {4 x^2}{9}} \left (8+7 x+7 x^2+14 x^3+8 x \log (x)+16 x^2 \log (x)\right ) \log \left (\frac {7 x}{8}+\log (x)\right )}{x}\right ) \, dx\\ &=\frac {1}{9} \int 2^{2-\frac {4 x}{3}-\frac {4 x^2}{3}} \exp \left (\frac {2}{9} \left (x^2 (1+x)^2+\log ^2\left (\frac {7 x}{8}+\log (x)\right )\right )\right ) (1+x) (7 x+8 \log (x))^{-1+\frac {4 x}{9}+\frac {4 x^2}{9}} \left (8+7 x+7 x^2+14 x^3+8 x \log (x)+16 x^2 \log (x)\right ) \, dx+\frac {1}{9} \int \frac {2^{2-\frac {4 x}{3}-\frac {4 x^2}{3}} \exp \left (\frac {2}{9} \left (x^2 (1+x)^2+\log ^2\left (\frac {7 x}{8}+\log (x)\right )\right )\right ) (7 x+8 \log (x))^{-1+\frac {4 x}{9}+\frac {4 x^2}{9}} \left (8+7 x+7 x^2+14 x^3+8 x \log (x)+16 x^2 \log (x)\right ) \log \left (\frac {7 x}{8}+\log (x)\right )}{x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 7.78, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((2*x^2 + 4*x^3 + 2*x^4 + (4*x + 4*x^2)*Log[(7*x + 8*Log[x])/8] + 2*Log[(7*x + 8*Log[x])/8]^2)/9)
*(32*x + 60*x^2 + 56*x^3 + 84*x^4 + 56*x^5 + (32*x^2 + 96*x^3 + 64*x^4)*Log[x] + (32 + 28*x + 28*x^2 + 56*x^3
+ (32*x + 64*x^2)*Log[x])*Log[(7*x + 8*Log[x])/8]))/(63*x^2 + 72*x*Log[x]),x]

[Out]

Integrate[(E^((2*x^2 + 4*x^3 + 2*x^4 + (4*x + 4*x^2)*Log[(7*x + 8*Log[x])/8] + 2*Log[(7*x + 8*Log[x])/8]^2)/9)
*(32*x + 60*x^2 + 56*x^3 + 84*x^4 + 56*x^5 + (32*x^2 + 96*x^3 + 64*x^4)*Log[x] + (32 + 28*x + 28*x^2 + 56*x^3
+ (32*x + 64*x^2)*Log[x])*Log[(7*x + 8*Log[x])/8]))/(63*x^2 + 72*x*Log[x]), x]

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fricas [B]  time = 0.64, size = 42, normalized size = 1.91 \begin {gather*} e^{\left (\frac {2}{9} \, x^{4} + \frac {4}{9} \, x^{3} + \frac {2}{9} \, x^{2} + \frac {4}{9} \, {\left (x^{2} + x\right )} \log \left (\frac {7}{8} \, x + \log \relax (x)\right ) + \frac {2}{9} \, \log \left (\frac {7}{8} \, x + \log \relax (x)\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+32*x)*log(x)+56*x^3+28*x^2+28*x+32)*log(log(x)+7/8*x)+(64*x^4+96*x^3+32*x^2)*log(x)+56*x^5
+84*x^4+56*x^3+60*x^2+32*x)*exp(2/9*log(log(x)+7/8*x)^2+1/9*(4*x^2+4*x)*log(log(x)+7/8*x)+2/9*x^4+4/9*x^3+2/9*
x^2)/(72*x*log(x)+63*x^2),x, algorithm="fricas")

[Out]

e^(2/9*x^4 + 4/9*x^3 + 2/9*x^2 + 4/9*(x^2 + x)*log(7/8*x + log(x)) + 2/9*log(7/8*x + log(x))^2)

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giac [B]  time = 6.03, size = 50, normalized size = 2.27 \begin {gather*} e^{\left (\frac {2}{9} \, x^{4} + \frac {4}{9} \, x^{3} + \frac {4}{9} \, x^{2} \log \left (\frac {7}{8} \, x + \log \relax (x)\right ) + \frac {2}{9} \, x^{2} + \frac {4}{9} \, x \log \left (\frac {7}{8} \, x + \log \relax (x)\right ) + \frac {2}{9} \, \log \left (\frac {7}{8} \, x + \log \relax (x)\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+32*x)*log(x)+56*x^3+28*x^2+28*x+32)*log(log(x)+7/8*x)+(64*x^4+96*x^3+32*x^2)*log(x)+56*x^5
+84*x^4+56*x^3+60*x^2+32*x)*exp(2/9*log(log(x)+7/8*x)^2+1/9*(4*x^2+4*x)*log(log(x)+7/8*x)+2/9*x^4+4/9*x^3+2/9*
x^2)/(72*x*log(x)+63*x^2),x, algorithm="giac")

[Out]

e^(2/9*x^4 + 4/9*x^3 + 4/9*x^2*log(7/8*x + log(x)) + 2/9*x^2 + 4/9*x*log(7/8*x + log(x)) + 2/9*log(7/8*x + log
(x))^2)

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maple [B]  time = 0.05, size = 43, normalized size = 1.95

method	result	size
risch	\(\left (\ln \relax (x )+\frac {7 x}{8}\right )^{\frac {4 \left (x +1\right ) x}{9}} {\mathrm e}^{\frac {2 \ln \left (\ln \relax (x )+\frac {7 x}{8}\right )^{2}}{9}+\frac {2 x^{4}}{9}+\frac {4 x^{3}}{9}+\frac {2 x^{2}}{9}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((64*x^2+32*x)*ln(x)+56*x^3+28*x^2+28*x+32)*ln(ln(x)+7/8*x)+(64*x^4+96*x^3+32*x^2)*ln(x)+56*x^5+84*x^4+56
*x^3+60*x^2+32*x)*exp(2/9*ln(ln(x)+7/8*x)^2+1/9*(4*x^2+4*x)*ln(ln(x)+7/8*x)+2/9*x^4+4/9*x^3+2/9*x^2)/(72*x*ln(
x)+63*x^2),x,method=_RETURNVERBOSE)

[Out]

(ln(x)+7/8*x)^(4/9*(x+1)*x)*exp(2/9*ln(ln(x)+7/8*x)^2+2/9*x^4+4/9*x^3+2/9*x^2)

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maxima [B]  time = 0.70, size = 87, normalized size = 3.95 \begin {gather*} e^{\left (\frac {2}{9} \, x^{4} + \frac {4}{9} \, x^{3} - \frac {4}{3} \, x^{2} \log \relax (2) + \frac {4}{9} \, x^{2} \log \left (7 \, x + 8 \, \log \relax (x)\right ) + \frac {2}{9} \, x^{2} - \frac {4}{3} \, x \log \relax (2) + 2 \, \log \relax (2)^{2} + \frac {4}{9} \, x \log \left (7 \, x + 8 \, \log \relax (x)\right ) - \frac {4}{3} \, \log \relax (2) \log \left (7 \, x + 8 \, \log \relax (x)\right ) + \frac {2}{9} \, \log \left (7 \, x + 8 \, \log \relax (x)\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^2+32*x)*log(x)+56*x^3+28*x^2+28*x+32)*log(log(x)+7/8*x)+(64*x^4+96*x^3+32*x^2)*log(x)+56*x^5
+84*x^4+56*x^3+60*x^2+32*x)*exp(2/9*log(log(x)+7/8*x)^2+1/9*(4*x^2+4*x)*log(log(x)+7/8*x)+2/9*x^4+4/9*x^3+2/9*
x^2)/(72*x*log(x)+63*x^2),x, algorithm="maxima")

[Out]

e^(2/9*x^4 + 4/9*x^3 - 4/3*x^2*log(2) + 4/9*x^2*log(7*x + 8*log(x)) + 2/9*x^2 - 4/3*x*log(2) + 2*log(2)^2 + 4/
9*x*log(7*x + 8*log(x)) - 4/3*log(2)*log(7*x + 8*log(x)) + 2/9*log(7*x + 8*log(x))^2)

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mupad [B]  time = 3.88, size = 47, normalized size = 2.14 \begin {gather*} {\mathrm {e}}^{\frac {2\,x^2}{9}}\,{\mathrm {e}}^{\frac {2\,x^4}{9}}\,{\mathrm {e}}^{\frac {4\,x^3}{9}}\,{\mathrm {e}}^{\frac {2\,{\ln \left (\frac {7\,x}{8}+\ln \relax (x)\right )}^2}{9}}\,{\left (\frac {7\,x}{8}+\ln \relax (x)\right )}^{\frac {4\,x^2}{9}+\frac {4\,x}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log((7*x)/8 + log(x))*(4*x + 4*x^2))/9 + (2*x^2)/9 + (4*x^3)/9 + (2*x^4)/9 + (2*log((7*x)/8 + log(x)
)^2)/9)*(32*x + log((7*x)/8 + log(x))*(28*x + log(x)*(32*x + 64*x^2) + 28*x^2 + 56*x^3 + 32) + log(x)*(32*x^2
+ 96*x^3 + 64*x^4) + 60*x^2 + 56*x^3 + 84*x^4 + 56*x^5))/(72*x*log(x) + 63*x^2),x)

[Out]

exp((2*x^2)/9)*exp((2*x^4)/9)*exp((4*x^3)/9)*exp((2*log((7*x)/8 + log(x))^2)/9)*((7*x)/8 + log(x))^((4*x)/9 +
(4*x^2)/9)

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sympy [B]  time = 1.71, size = 58, normalized size = 2.64 \begin {gather*} e^{\frac {2 x^{4}}{9} + \frac {4 x^{3}}{9} + \frac {2 x^{2}}{9} + \left (\frac {4 x^{2}}{9} + \frac {4 x}{9}\right ) \log {\left (\frac {7 x}{8} + \log {\relax (x )} \right )} + \frac {2 \log {\left (\frac {7 x}{8} + \log {\relax (x )} \right )}^{2}}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x**2+32*x)*ln(x)+56*x**3+28*x**2+28*x+32)*ln(ln(x)+7/8*x)+(64*x**4+96*x**3+32*x**2)*ln(x)+56*x
**5+84*x**4+56*x**3+60*x**2+32*x)*exp(2/9*ln(ln(x)+7/8*x)**2+1/9*(4*x**2+4*x)*ln(ln(x)+7/8*x)+2/9*x**4+4/9*x**
3+2/9*x**2)/(72*x*ln(x)+63*x**2),x)

[Out]

exp(2*x**4/9 + 4*x**3/9 + 2*x**2/9 + (4*x**2/9 + 4*x/9)*log(7*x/8 + log(x)) + 2*log(7*x/8 + log(x))**2/9)

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