Integrand size = 160, antiderivative size = 22 \[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx=e^{\frac {2}{9} \left (x+x^2+\log \left (\frac {7 x}{8}+\log (x)\right )\right )^2} \] Output:
exp(2/9*(x^2+x+ln(ln(x)+7/8*x))^2)
\[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx=\int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx \] Input:
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 ]
Output:
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]
Time = 5.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {3041, 7292, 27, 7257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (56 x^5+84 x^4+56 x^3+60 x^2+\left (56 x^3+28 x^2+\left (64 x^2+32 x\right ) \log (x)+28 x+32\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+\left (64 x^4+96 x^3+32 x^2\right ) \log (x)+32 x\right ) \exp \left (\frac {1}{9} \left (2 x^4+4 x^3+2 x^2+\left (4 x^2+4 x\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )\right )}{63 x^2+72 x \log (x)} \, dx\) |
| \(\Big \downarrow \) 3041 |
| \(\displaystyle \int \frac {\left (56 x^5+84 x^4+56 x^3+60 x^2+\left (56 x^3+28 x^2+\left (64 x^2+32 x\right ) \log (x)+28 x+32\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+\left (64 x^4+96 x^3+32 x^2\right ) \log (x)+32 x\right ) \exp \left (\frac {1}{9} \left (2 x^4+4 x^3+2 x^2+\left (4 x^2+4 x\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )\right )}{x (63 x+72 \log (x))}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 e^{\frac {2}{9} \left (x^2+x+\log \left (\frac {7 x}{8}+\log (x)\right )\right )^2} \left (14 x^3+7 x^2+16 x^2 \log (x)+7 x+8 x \log (x)+8\right ) \left (x^2+x+\log \left (\frac {7 x}{8}+\log (x)\right )\right )}{9 x (7 x+8 \log (x))}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{9} \int \frac {e^{\frac {2}{9} \left (x^2+x+\log \left (\frac {7 x}{8}+\log (x)\right )\right )^2} \left (14 x^3+16 \log (x) x^2+7 x^2+8 \log (x) x+7 x+8\right ) \left (x^2+x+\log \left (\frac {7 x}{8}+\log (x)\right )\right )}{x (7 x+8 \log (x))}dx\) |
| \(\Big \downarrow \) 7257 |
| \(\displaystyle e^{\frac {2}{9} \left (x^2+x+\log \left (\frac {7 x}{8}+\log (x)\right )\right )^2}\) |
Input:
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]
Output:
E^((2*(x + x^2 + Log[(7*x)/8 + Log[x]])^2)/9)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.)) ^(p_.), x_Symbol] :> Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Leaf count of result is larger than twice the leaf count of optimal. \(42\) vs. \(2(17)=34\).
Time = 1.60 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95
| method | result | size |
| risch | \(\left (\ln \left (x \right )+\frac {7 x}{8}\right )^{\frac {4 \left (1+x \right ) x}{9}} {\mathrm e}^{\frac {2 \ln \left (\ln \left (x \right )+\frac {7 x}{8}\right )^{2}}{9}+\frac {2 x^{4}}{9}+\frac {4 x^{3}}{9}+\frac {2 x^{2}}{9}}\) | \(43\) |
| parallelrisch | \({\mathrm e}^{\frac {2 \ln \left (\ln \left (x \right )+\frac {7 x}{8}\right )^{2}}{9}+\frac {\left (4 x^{2}+4 x \right ) \ln \left (\ln \left (x \right )+\frac {7 x}{8}\right )}{9}+\frac {2 x^{4}}{9}+\frac {4 x^{3}}{9}+\frac {2 x^{2}}{9}}\) | \(47\) |
Input:
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+9 6*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)
Output:
(ln(x)+7/8*x)^(4/9*(1+x)*x)*exp(2/9*ln(ln(x)+7/8*x)^2+2/9*x^4+4/9*x^3+2/9* x^2)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (17) = 34\).
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx=e^{\left (\frac {2}{9} \, x^{4} + \frac {4}{9} \, x^{3} + \frac {2}{9} \, x^{2} + \frac {4}{9} \, {\left (x^{2} + x\right )} \log \left (\frac {7}{8} \, x + \log \left (x\right )\right ) + \frac {2}{9} \, \log \left (\frac {7}{8} \, x + \log \left (x\right )\right )^{2}\right )} \] Input:
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*lo g(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")
Output:
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*l og(7/8*x + log(x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 1.00 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64 \[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx=e^{\frac {2 x^{4}}{9} + \frac {4 x^{3}}{9} + \frac {2 x^{2}}{9} + \left (\frac {4 x^{2}}{9} + \frac {4 x}{9}\right ) \log {\left (\frac {7 x}{8} + \log {\left (x \right )} \right )} + \frac {2 \log {\left (\frac {7 x}{8} + \log {\left (x \right )} \right )}^{2}}{9}} \] Input:
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)
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (17) = 34\).
Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95 \[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx=e^{\left (\frac {2}{9} \, x^{4} + \frac {4}{9} \, x^{3} - \frac {4}{3} \, x^{2} \log \left (2\right ) + \frac {4}{9} \, x^{2} \log \left (7 \, x + 8 \, \log \left (x\right )\right ) + \frac {2}{9} \, x^{2} - \frac {4}{3} \, x \log \left (2\right ) + 2 \, \log \left (2\right )^{2} + \frac {4}{9} \, x \log \left (7 \, x + 8 \, \log \left (x\right )\right ) - \frac {4}{3} \, \log \left (2\right ) \log \left (7 \, x + 8 \, \log \left (x\right )\right ) + \frac {2}{9} \, \log \left (7 \, x + 8 \, \log \left (x\right )\right )^{2}\right )} \] Input:
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*lo g(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")
Output:
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)*l og(7*x + 8*log(x)) + 2/9*log(7*x + 8*log(x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (17) = 34\).
Time = 0.48 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.27 \[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx=e^{\left (\frac {2}{9} \, x^{4} + \frac {4}{9} \, x^{3} + \frac {4}{9} \, x^{2} \log \left (\frac {7}{8} \, x + \log \left (x\right )\right ) + \frac {2}{9} \, x^{2} + \frac {4}{9} \, x \log \left (\frac {7}{8} \, x + \log \left (x\right )\right ) + \frac {2}{9} \, \log \left (\frac {7}{8} \, x + \log \left (x\right )\right )^{2}\right )} \] Input:
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*lo g(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")
Output:
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)
Time = 4.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx={\mathrm {e}}^{\frac {2\,x^2}{9}}\,{\mathrm {e}}^{\frac {2\,x^4}{9}}\,{\mathrm {e}}^{\frac {4\,x^3}{9}}\,{\mathrm {e}}^{\frac {2\,{\ln \left (\frac {7\,x}{8}+\ln \left (x\right )\right )}^2}{9}}\,{\left (\frac {7\,x}{8}+\ln \left (x\right )\right )}^{\frac {4\,x^2}{9}+\frac {4\,x}{9}} \] Input:
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)
Output:
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)
\[ \int \frac {e^{\frac {1}{9} \left (2 x^2+4 x^3+2 x^4+\left (4 x+4 x^2\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )+2 \log ^2\left (\frac {1}{8} (7 x+8 \log (x))\right )\right )} \left (32 x+60 x^2+56 x^3+84 x^4+56 x^5+\left (32 x^2+96 x^3+64 x^4\right ) \log (x)+\left (32+28 x+28 x^2+56 x^3+\left (32 x+64 x^2\right ) \log (x)\right ) \log \left (\frac {1}{8} (7 x+8 \log (x))\right )\right )}{63 x^2+72 x \log (x)} \, dx=\int \frac {\left (\left (\left (64 x^{2}+32 x \right ) \mathrm {log}\left (x \right )+56 x^{3}+28 x^{2}+28 x +32\right ) \mathrm {log}\left (\mathrm {log}\left (x \right )+\frac {7 x}{8}\right )+\left (64 x^{4}+96 x^{3}+32 x^{2}\right ) \mathrm {log}\left (x \right )+56 x^{5}+84 x^{4}+56 x^{3}+60 x^{2}+32 x \right ) {\mathrm e}^{\frac {2 \mathrm {log}\left (\mathrm {log}\left (x \right )+\frac {7 x}{8}\right )^{2}}{9}+\frac {\left (4 x^{2}+4 x \right ) \mathrm {log}\left (\mathrm {log}\left (x \right )+\frac {7 x}{8}\right )}{9}+\frac {2 x^{4}}{9}+\frac {4 x^{3}}{9}+\frac {2 x^{2}}{9}}}{72 \,\mathrm {log}\left (x \right ) x +63 x^{2}}d x \] Input:
int((((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)
Output:
int((((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)