Integrand size = 53, antiderivative size = 28 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=e-e^{2+x-(4+x)^2 \log (7)}+\frac {1}{2 x}-2 x \] Output:
exp(1)-2*x+1/2/x-exp(x+2-ln(7)*(4+x)^2)
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=-7^{-(4+x)^2} e^{2+x}+\frac {1}{2 x}-2 x \] Input:
Integrate[(-1 - 4*x^2 + E^(2 + x + (-16 - 8*x - x^2)*Log[7])*(-2*x^2 + (16 *x^2 + 4*x^3)*Log[7]))/(2*x^2),x]
Output:
-(E^(2 + x)/7^(4 + x)^2) + 1/(2*x) - 2*x
Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {27, 25, 2010, 2009}
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.
\(\displaystyle \int \frac {-4 x^2+e^{\left (-x^2-8 x-16\right ) \log (7)+x+2} \left (\left (4 x^3+16 x^2\right ) \log (7)-2 x^2\right )-1}{2 x^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {4 x^2+2\ 7^{-x^2-8 x-16} e^{x+2} \left (x^2-2 \left (x^3+4 x^2\right ) \log (7)\right )+1}{x^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {4 x^2+2\ 7^{-x^2-8 x-16} e^{x+2} \left (x^2-2 \left (x^3+4 x^2\right ) \log (7)\right )+1}{x^2}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {4 x^2+1}{x^2}-2\ 7^{-(x+4)^2} e^{x+2} (\log (49) x+8 \log (7)-1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {7^{-x^2} \log (49) e^{x (1-8 \log (7))+2 (1-8 \log (7))}}{\log (7)}-4 x+\frac {1}{x}\right )\) |
Input:
Int[(-1 - 4*x^2 + E^(2 + x + (-16 - 8*x - x^2)*Log[7])*(-2*x^2 + (16*x^2 + 4*x^3)*Log[7]))/(2*x^2),x]
Output:
(x^(-1) - 4*x - (E^(2*(1 - 8*Log[7]) + x*(1 - 8*Log[7]))*Log[49])/(7^x^2*L og[7]))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 0.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-2 x +\frac {1}{2 x}-\left (\frac {1}{7}\right )^{\left (4+x \right )^{2}} {\mathrm e}^{2+x}\) | \(23\) |
norman | \(\frac {\frac {1}{2}-2 x^{2}-x \,{\mathrm e}^{\left (-x^{2}-8 x -16\right ) \ln \left (7\right )+2+x}}{x}\) | \(32\) |
parallelrisch | \(-\frac {4 x^{2}+2 x \,{\mathrm e}^{\left (-x^{2}-8 x -16\right ) \ln \left (7\right )+2+x}-1}{2 x}\) | \(33\) |
parts | \(-{\mathrm e}^{-\ln \left (7\right ) x^{2}+\left (-8 \ln \left (7\right )+1\right ) x -16 \ln \left (7\right )+2}-2 x +\frac {1}{2 x}\) | \(34\) |
default | \(-{\mathrm e}^{-\ln \left (7\right ) x^{2}+\left (-8 \ln \left (7\right )+1\right ) x -16 \ln \left (7\right )+2}+\frac {\left (-8 \ln \left (7\right )+1\right ) \sqrt {\pi }\, {\mathrm e}^{-16 \ln \left (7\right )+2+\frac {\left (-8 \ln \left (7\right )+1\right )^{2}}{4 \ln \left (7\right )}} \operatorname {erf}\left (\sqrt {\ln \left (7\right )}\, x -\frac {-8 \ln \left (7\right )+1}{2 \sqrt {\ln \left (7\right )}}\right )}{2 \sqrt {\ln \left (7\right )}}+4 \sqrt {\ln \left (7\right )}\, \sqrt {\pi }\, {\mathrm e}^{-16 \ln \left (7\right )+2+\frac {\left (-8 \ln \left (7\right )+1\right )^{2}}{4 \ln \left (7\right )}} \operatorname {erf}\left (\sqrt {\ln \left (7\right )}\, x -\frac {-8 \ln \left (7\right )+1}{2 \sqrt {\ln \left (7\right )}}\right )-\frac {\sqrt {\pi }\, {\mathrm e}^{-16 \ln \left (7\right )+2+\frac {\left (-8 \ln \left (7\right )+1\right )^{2}}{4 \ln \left (7\right )}} \operatorname {erf}\left (\sqrt {\ln \left (7\right )}\, x -\frac {-8 \ln \left (7\right )+1}{2 \sqrt {\ln \left (7\right )}}\right )}{2 \sqrt {\ln \left (7\right )}}-2 x +\frac {1}{2 x}\) | \(190\) |
Input:
int(1/2*(((4*x^3+16*x^2)*ln(7)-2*x^2)*exp((-x^2-8*x-16)*ln(7)+2+x)-4*x^2-1 )/x^2,x,method=_RETURNVERBOSE)
Output:
-2*x+1/2/x-(1/7)^((4+x)^2)*exp(2+x)
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=-\frac {4 \, x^{2} + 2 \, x e^{\left (-{\left (x^{2} + 8 \, x + 16\right )} \log \left (7\right ) + x + 2\right )} - 1}{2 \, x} \] Input:
integrate(1/2*(((4*x^3+16*x^2)*log(7)-2*x^2)*exp((-x^2-8*x-16)*log(7)+2+x) -4*x^2-1)/x^2,x, algorithm="fricas")
Output:
-1/2*(4*x^2 + 2*x*e^(-(x^2 + 8*x + 16)*log(7) + x + 2) - 1)/x
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=- 2 x - e^{x + \left (- x^{2} - 8 x - 16\right ) \log {\left (7 \right )} + 2} + \frac {1}{2 x} \] Input:
integrate(1/2*(((4*x**3+16*x**2)*ln(7)-2*x**2)*exp((-x**2-8*x-16)*ln(7)+2+ x)-4*x**2-1)/x**2,x)
Output:
-2*x - exp(x + (-x**2 - 8*x - 16)*log(7) + 2) + 1/(2*x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 231, normalized size of antiderivative = 8.25 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=\frac {4}{33232930569601} \, \sqrt {\pi } \operatorname {erf}\left (x \sqrt {\log \left (7\right )} + \frac {8 \, \log \left (7\right ) - 1}{2 \, \sqrt {\log \left (7\right )}}\right ) e^{\left (\frac {{\left (8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )} + 2\right )} \sqrt {\log \left (7\right )} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )}^{2}}{\log \left (7\right )}}\right ) - 1\right )} {\left (8 \, \log \left (7\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )}^{2}}{\log \left (7\right )}} \left (-\log \left (7\right )\right )^{\frac {3}{2}}} + \frac {2 \, e^{\left (-\frac {{\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )}\right )} \log \left (7\right )}{\left (-\log \left (7\right )\right )^{\frac {3}{2}}}\right )} e^{\left (\frac {{\left (8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )} + 2\right )} \log \left (7\right )}{66465861139202 \, \sqrt {-\log \left (7\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {\log \left (7\right )} + \frac {8 \, \log \left (7\right ) - 1}{2 \, \sqrt {\log \left (7\right )}}\right ) e^{\left (\frac {{\left (8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )} + 2\right )}}{66465861139202 \, \sqrt {\log \left (7\right )}} - 2 \, x + \frac {1}{2 \, x} \] Input:
integrate(1/2*(((4*x^3+16*x^2)*log(7)-2*x^2)*exp((-x^2-8*x-16)*log(7)+2+x) -4*x^2-1)/x^2,x, algorithm="maxima")
Output:
4/33232930569601*sqrt(pi)*erf(x*sqrt(log(7)) + 1/2*(8*log(7) - 1)/sqrt(log (7)))*e^(1/4*(8*log(7) - 1)^2/log(7) + 2)*sqrt(log(7)) - 1/66465861139202* (sqrt(pi)*(2*x*log(7) + 8*log(7) - 1)*(erf(1/2*sqrt((2*x*log(7) + 8*log(7) - 1)^2/log(7))) - 1)*(8*log(7) - 1)/(sqrt((2*x*log(7) + 8*log(7) - 1)^2/l og(7))*(-log(7))^(3/2)) + 2*e^(-1/4*(2*x*log(7) + 8*log(7) - 1)^2/log(7))* log(7)/(-log(7))^(3/2))*e^(1/4*(8*log(7) - 1)^2/log(7) + 2)*log(7)/sqrt(-l og(7)) - 1/66465861139202*sqrt(pi)*erf(x*sqrt(log(7)) + 1/2*(8*log(7) - 1) /sqrt(log(7)))*e^(1/4*(8*log(7) - 1)^2/log(7) + 2)/sqrt(log(7)) - 2*x + 1/ 2/x
Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=-\frac {132931722278404 \, x^{2} + 2 \, x e^{\left (-x^{2} \log \left (7\right ) - 8 \, x \log \left (7\right ) + x + 2\right )} - 33232930569601}{66465861139202 \, x} \] Input:
integrate(1/2*(((4*x^3+16*x^2)*log(7)-2*x^2)*exp((-x^2-8*x-16)*log(7)+2+x) -4*x^2-1)/x^2,x, algorithm="giac")
Output:
-1/66465861139202*(132931722278404*x^2 + 2*x*e^(-x^2*log(7) - 8*x*log(7) + x + 2) - 33232930569601)/x
Time = 4.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=\frac {1}{2\,x}-2\,x-\frac {{\mathrm {e}}^2\,{\mathrm {e}}^x}{33232930569601\,7^{8\,x}\,7^{x^2}} \] Input:
int(-(2*x^2 - (exp(x - log(7)*(8*x + x^2 + 16) + 2)*(log(7)*(16*x^2 + 4*x^ 3) - 2*x^2))/2 + 1/2)/x^2,x)
Output:
1/(2*x) - 2*x - (exp(2)*exp(x))/(33232930569601*7^(8*x)*7^(x^2))
Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{2 x^2} \, dx=\frac {-2 e^{x} e^{2} x -132931722278404 \,7^{8 x} 7^{x^{2}} x^{2}+33232930569601 \,7^{8 x} 7^{x^{2}}}{66465861139202 \,7^{8 x} 7^{x^{2}} x} \] Input:
int(1/2*(((4*x^3+16*x^2)*log(7)-2*x^2)*exp((-x^2-8*x-16)*log(7)+2+x)-4*x^2 -1)/x^2,x)
Output:
( - 2*e**x*e**2*x - 132931722278404*7**(8*x)*7**(x**2)*x**2 + 332329305696 01*7**(8*x)*7**(x**2))/(66465861139202*7**(8*x)*7**(x**2)*x)