Integrand size = 176, antiderivative size = 22 \[ \int \frac {-36-6 x-3 x^2+e^x \left (-9+3 x-3 x^2\right )+\left (12+e^x (3-3 x)\right ) \log \left (x^2\right )}{64-96 x+4 x^2+24 x^3+4 x^4+e^{2 x} \left (4-8 x+4 x^2\right )+e^x \left (32-56 x+16 x^2+8 x^3\right )+\left (-128+64 x+56 x^2+8 x^3+e^{2 x} (-8+8 x)+e^x \left (-64+48 x+16 x^2\right )\right ) \log \left (x^2\right )+\left (64+4 e^{2 x}+32 x+4 x^2+e^x (32+8 x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {3 x}{4 \left (4+e^x+x\right ) \left (-1+x+\log \left (x^2\right )\right )} \]
Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-36-6 x-3 x^2+e^x \left (-9+3 x-3 x^2\right )+\left (12+e^x (3-3 x)\right ) \log \left (x^2\right )}{64-96 x+4 x^2+24 x^3+4 x^4+e^{2 x} \left (4-8 x+4 x^2\right )+e^x \left (32-56 x+16 x^2+8 x^3\right )+\left (-128+64 x+56 x^2+8 x^3+e^{2 x} (-8+8 x)+e^x \left (-64+48 x+16 x^2\right )\right ) \log \left (x^2\right )+\left (64+4 e^{2 x}+32 x+4 x^2+e^x (32+8 x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {3 x}{4 \left (4+e^x+x\right ) \left (-1+x+\log \left (x^2\right )\right )} \]
Integrate[(-36 - 6*x - 3*x^2 + E^x*(-9 + 3*x - 3*x^2) + (12 + E^x*(3 - 3*x ))*Log[x^2])/(64 - 96*x + 4*x^2 + 24*x^3 + 4*x^4 + E^(2*x)*(4 - 8*x + 4*x^ 2) + E^x*(32 - 56*x + 16*x^2 + 8*x^3) + (-128 + 64*x + 56*x^2 + 8*x^3 + E^ (2*x)*(-8 + 8*x) + E^x*(-64 + 48*x + 16*x^2))*Log[x^2] + (64 + 4*E^(2*x) + 32*x + 4*x^2 + E^x*(32 + 8*x))*Log[x^2]^2),x]
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.
\(\displaystyle \int \frac {-3 x^2+e^x \left (-3 x^2+3 x-9\right )+\left (e^x (3-3 x)+12\right ) \log \left (x^2\right )-6 x-36}{4 x^4+24 x^3+4 x^2+e^{2 x} \left (4 x^2-8 x+4\right )+\left (4 x^2+32 x+4 e^{2 x}+e^x (8 x+32)+64\right ) \log ^2\left (x^2\right )+e^x \left (8 x^3+16 x^2-56 x+32\right )+\left (8 x^3+56 x^2+e^x \left (16 x^2+48 x-64\right )+64 x+e^{2 x} (8 x-8)-128\right ) \log \left (x^2\right )-96 x+64} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 \left (-x^2-e^x \left (x^2-x+3\right )-\left (e^x (x-1)-4\right ) \log \left (x^2\right )-2 x-12\right )}{4 \left (x+e^x+4\right )^2 \left (-\log \left (x^2\right )-x+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{4} \int -\frac {x^2+2 x+e^x \left (x^2-x+3\right )-\left (e^x (1-x)+4\right ) \log \left (x^2\right )+12}{\left (x+e^x+4\right )^2 \left (-x-\log \left (x^2\right )+1\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {3}{4} \int \frac {x^2+2 x+e^x \left (x^2-x+3\right )-\left (e^x (1-x)+4\right ) \log \left (x^2\right )+12}{\left (x+e^x+4\right )^2 \left (-x-\log \left (x^2\right )+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {3}{4} \int \left (\frac {x^2+\log \left (x^2\right ) x-x-\log \left (x^2\right )+3}{\left (x+e^x+4\right ) \left (x+\log \left (x^2\right )-1\right )^2}-\frac {x (x+3)}{\left (x+e^x+4\right )^2 \left (x+\log \left (x^2\right )-1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3}{4} \left (3 \int \frac {1}{\left (x+e^x+4\right ) \left (x+\log \left (x^2\right )-1\right )^2}dx-\int \frac {x}{\left (x+e^x+4\right ) \left (x+\log \left (x^2\right )-1\right )^2}dx+\int \frac {x^2}{\left (x+e^x+4\right ) \left (x+\log \left (x^2\right )-1\right )^2}dx-\int \frac {\log \left (x^2\right )}{\left (x+e^x+4\right ) \left (x+\log \left (x^2\right )-1\right )^2}dx+\int \frac {x \log \left (x^2\right )}{\left (x+e^x+4\right ) \left (x+\log \left (x^2\right )-1\right )^2}dx-3 \int \frac {x}{\left (x+e^x+4\right )^2 \left (x+\log \left (x^2\right )-1\right )}dx-\int \frac {x^2}{\left (x+e^x+4\right )^2 \left (x+\log \left (x^2\right )-1\right )}dx\right )\) |
Int[(-36 - 6*x - 3*x^2 + E^x*(-9 + 3*x - 3*x^2) + (12 + E^x*(3 - 3*x))*Log [x^2])/(64 - 96*x + 4*x^2 + 24*x^3 + 4*x^4 + E^(2*x)*(4 - 8*x + 4*x^2) + E ^x*(32 - 56*x + 16*x^2 + 8*x^3) + (-128 + 64*x + 56*x^2 + 8*x^3 + E^(2*x)* (-8 + 8*x) + E^x*(-64 + 48*x + 16*x^2))*Log[x^2] + (64 + 4*E^(2*x) + 32*x + 4*x^2 + E^x*(32 + 8*x))*Log[x^2]^2),x]
3.14.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs. \(2(19)=38\).
Time = 1.62 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86
method | result | size |
parallelrisch | \(\frac {3 x}{4 \left ({\mathrm e}^{x} \ln \left (x^{2}\right )+{\mathrm e}^{x} x +x \ln \left (x^{2}\right )+x^{2}-{\mathrm e}^{x}+4 \ln \left (x^{2}\right )+3 x -4\right )}\) | \(41\) |
risch | \(\frac {3 x}{2 \left (4+x +{\mathrm e}^{x}\right ) \left (-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 x +4 \ln \left (x \right )-2\right )}\) | \(71\) |
int((((-3*x+3)*exp(x)+12)*ln(x^2)+(-3*x^2+3*x-9)*exp(x)-3*x^2-6*x-36)/((4* exp(x)^2+(8*x+32)*exp(x)+4*x^2+32*x+64)*ln(x^2)^2+((8*x-8)*exp(x)^2+(16*x^ 2+48*x-64)*exp(x)+8*x^3+56*x^2+64*x-128)*ln(x^2)+(4*x^2-8*x+4)*exp(x)^2+(8 *x^3+16*x^2-56*x+32)*exp(x)+4*x^4+24*x^3+4*x^2-96*x+64),x,method=_RETURNVE RBOSE)
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-36-6 x-3 x^2+e^x \left (-9+3 x-3 x^2\right )+\left (12+e^x (3-3 x)\right ) \log \left (x^2\right )}{64-96 x+4 x^2+24 x^3+4 x^4+e^{2 x} \left (4-8 x+4 x^2\right )+e^x \left (32-56 x+16 x^2+8 x^3\right )+\left (-128+64 x+56 x^2+8 x^3+e^{2 x} (-8+8 x)+e^x \left (-64+48 x+16 x^2\right )\right ) \log \left (x^2\right )+\left (64+4 e^{2 x}+32 x+4 x^2+e^x (32+8 x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {3 \, x}{4 \, {\left (x^{2} + {\left (x - 1\right )} e^{x} + {\left (x + e^{x} + 4\right )} \log \left (x^{2}\right ) + 3 \, x - 4\right )}} \]
integrate((((-3*x+3)*exp(x)+12)*log(x^2)+(-3*x^2+3*x-9)*exp(x)-3*x^2-6*x-3 6)/((4*exp(x)^2+(8*x+32)*exp(x)+4*x^2+32*x+64)*log(x^2)^2+((8*x-8)*exp(x)^ 2+(16*x^2+48*x-64)*exp(x)+8*x^3+56*x^2+64*x-128)*log(x^2)+(4*x^2-8*x+4)*ex p(x)^2+(8*x^3+16*x^2-56*x+32)*exp(x)+4*x^4+24*x^3+4*x^2-96*x+64),x, algori thm=\
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {-36-6 x-3 x^2+e^x \left (-9+3 x-3 x^2\right )+\left (12+e^x (3-3 x)\right ) \log \left (x^2\right )}{64-96 x+4 x^2+24 x^3+4 x^4+e^{2 x} \left (4-8 x+4 x^2\right )+e^x \left (32-56 x+16 x^2+8 x^3\right )+\left (-128+64 x+56 x^2+8 x^3+e^{2 x} (-8+8 x)+e^x \left (-64+48 x+16 x^2\right )\right ) \log \left (x^2\right )+\left (64+4 e^{2 x}+32 x+4 x^2+e^x (32+8 x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {3 x}{4 x^{2} + 4 x \log {\left (x^{2} \right )} + 12 x + \left (4 x + 4 \log {\left (x^{2} \right )} - 4\right ) e^{x} + 16 \log {\left (x^{2} \right )} - 16} \]
integrate((((-3*x+3)*exp(x)+12)*ln(x**2)+(-3*x**2+3*x-9)*exp(x)-3*x**2-6*x -36)/((4*exp(x)**2+(8*x+32)*exp(x)+4*x**2+32*x+64)*ln(x**2)**2+((8*x-8)*ex p(x)**2+(16*x**2+48*x-64)*exp(x)+8*x**3+56*x**2+64*x-128)*ln(x**2)+(4*x**2 -8*x+4)*exp(x)**2+(8*x**3+16*x**2-56*x+32)*exp(x)+4*x**4+24*x**3+4*x**2-96 *x+64),x)
Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {-36-6 x-3 x^2+e^x \left (-9+3 x-3 x^2\right )+\left (12+e^x (3-3 x)\right ) \log \left (x^2\right )}{64-96 x+4 x^2+24 x^3+4 x^4+e^{2 x} \left (4-8 x+4 x^2\right )+e^x \left (32-56 x+16 x^2+8 x^3\right )+\left (-128+64 x+56 x^2+8 x^3+e^{2 x} (-8+8 x)+e^x \left (-64+48 x+16 x^2\right )\right ) \log \left (x^2\right )+\left (64+4 e^{2 x}+32 x+4 x^2+e^x (32+8 x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {3 \, x}{4 \, {\left (x^{2} + {\left (x + 2 \, \log \left (x\right ) - 1\right )} e^{x} + 2 \, {\left (x + 4\right )} \log \left (x\right ) + 3 \, x - 4\right )}} \]
integrate((((-3*x+3)*exp(x)+12)*log(x^2)+(-3*x^2+3*x-9)*exp(x)-3*x^2-6*x-3 6)/((4*exp(x)^2+(8*x+32)*exp(x)+4*x^2+32*x+64)*log(x^2)^2+((8*x-8)*exp(x)^ 2+(16*x^2+48*x-64)*exp(x)+8*x^3+56*x^2+64*x-128)*log(x^2)+(4*x^2-8*x+4)*ex p(x)^2+(8*x^3+16*x^2-56*x+32)*exp(x)+4*x^4+24*x^3+4*x^2-96*x+64),x, algori thm=\
Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {-36-6 x-3 x^2+e^x \left (-9+3 x-3 x^2\right )+\left (12+e^x (3-3 x)\right ) \log \left (x^2\right )}{64-96 x+4 x^2+24 x^3+4 x^4+e^{2 x} \left (4-8 x+4 x^2\right )+e^x \left (32-56 x+16 x^2+8 x^3\right )+\left (-128+64 x+56 x^2+8 x^3+e^{2 x} (-8+8 x)+e^x \left (-64+48 x+16 x^2\right )\right ) \log \left (x^2\right )+\left (64+4 e^{2 x}+32 x+4 x^2+e^x (32+8 x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {3 \, x}{4 \, {\left (x^{2} + x e^{x} + x \log \left (x^{2}\right ) + e^{x} \log \left (x^{2}\right ) + 3 \, x - e^{x} + 4 \, \log \left (x^{2}\right ) - 4\right )}} \]
integrate((((-3*x+3)*exp(x)+12)*log(x^2)+(-3*x^2+3*x-9)*exp(x)-3*x^2-6*x-3 6)/((4*exp(x)^2+(8*x+32)*exp(x)+4*x^2+32*x+64)*log(x^2)^2+((8*x-8)*exp(x)^ 2+(16*x^2+48*x-64)*exp(x)+8*x^3+56*x^2+64*x-128)*log(x^2)+(4*x^2-8*x+4)*ex p(x)^2+(8*x^3+16*x^2-56*x+32)*exp(x)+4*x^4+24*x^3+4*x^2-96*x+64),x, algori thm=\
Timed out. \[ \int \frac {-36-6 x-3 x^2+e^x \left (-9+3 x-3 x^2\right )+\left (12+e^x (3-3 x)\right ) \log \left (x^2\right )}{64-96 x+4 x^2+24 x^3+4 x^4+e^{2 x} \left (4-8 x+4 x^2\right )+e^x \left (32-56 x+16 x^2+8 x^3\right )+\left (-128+64 x+56 x^2+8 x^3+e^{2 x} (-8+8 x)+e^x \left (-64+48 x+16 x^2\right )\right ) \log \left (x^2\right )+\left (64+4 e^{2 x}+32 x+4 x^2+e^x (32+8 x)\right ) \log ^2\left (x^2\right )} \, dx=\int -\frac {6\,x+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (3\,x-3\right )-12\right )+{\mathrm {e}}^x\,\left (3\,x^2-3\,x+9\right )+3\,x^2+36}{{\mathrm {e}}^{2\,x}\,\left (4\,x^2-8\,x+4\right )-96\,x+{\ln \left (x^2\right )}^2\,\left (32\,x+4\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (8\,x+32\right )+4\,x^2+64\right )+\ln \left (x^2\right )\,\left (64\,x+{\mathrm {e}}^x\,\left (16\,x^2+48\,x-64\right )+{\mathrm {e}}^{2\,x}\,\left (8\,x-8\right )+56\,x^2+8\,x^3-128\right )+4\,x^2+24\,x^3+4\,x^4+{\mathrm {e}}^x\,\left (8\,x^3+16\,x^2-56\,x+32\right )+64} \,d x \]
int(-(6*x + log(x^2)*(exp(x)*(3*x - 3) - 12) + exp(x)*(3*x^2 - 3*x + 9) + 3*x^2 + 36)/(exp(2*x)*(4*x^2 - 8*x + 4) - 96*x + log(x^2)^2*(32*x + 4*exp( 2*x) + exp(x)*(8*x + 32) + 4*x^2 + 64) + log(x^2)*(64*x + exp(x)*(48*x + 1 6*x^2 - 64) + exp(2*x)*(8*x - 8) + 56*x^2 + 8*x^3 - 128) + 4*x^2 + 24*x^3 + 4*x^4 + exp(x)*(16*x^2 - 56*x + 8*x^3 + 32) + 64),x)
int(-(6*x + log(x^2)*(exp(x)*(3*x - 3) - 12) + exp(x)*(3*x^2 - 3*x + 9) + 3*x^2 + 36)/(exp(2*x)*(4*x^2 - 8*x + 4) - 96*x + log(x^2)^2*(32*x + 4*exp( 2*x) + exp(x)*(8*x + 32) + 4*x^2 + 64) + log(x^2)*(64*x + exp(x)*(48*x + 1 6*x^2 - 64) + exp(2*x)*(8*x - 8) + 56*x^2 + 8*x^3 - 128) + 4*x^2 + 24*x^3 + 4*x^4 + exp(x)*(16*x^2 - 56*x + 8*x^3 + 32) + 64), x)