Integrand size = 162, antiderivative size = 30 \[ \int \frac {e^{2 \log ^2(x)} \left (64 x+32 x^2-96 x^3+\left (64 x+32 x^2-96 x^3\right ) \log (15)+\left (16 x+8 x^2-24 x^3\right ) \log ^2(15)+\left (128 x-64 x^2+64 x^3+\left (128 x-64 x^2+64 x^3\right ) \log (15)+\left (32 x-16 x^2+16 x^3\right ) \log ^2(15)\right ) \log (x)\right )}{32-80 x+160 x^2-200 x^3+210 x^4-161 x^5+105 x^6-50 x^7+20 x^8-5 x^9+x^{10}} \, dx=\frac {e^{2 \log ^2(x)} x^2 (4+2 \log (15))^2}{\left (2-x+x^2\right )^4} \]
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{2 \log ^2(x)} \left (64 x+32 x^2-96 x^3+\left (64 x+32 x^2-96 x^3\right ) \log (15)+\left (16 x+8 x^2-24 x^3\right ) \log ^2(15)+\left (128 x-64 x^2+64 x^3+\left (128 x-64 x^2+64 x^3\right ) \log (15)+\left (32 x-16 x^2+16 x^3\right ) \log ^2(15)\right ) \log (x)\right )}{32-80 x+160 x^2-200 x^3+210 x^4-161 x^5+105 x^6-50 x^7+20 x^8-5 x^9+x^{10}} \, dx=\frac {4 e^{2 \log ^2(x)} x^2 (2+\log (15))^2}{\left (2-x+x^2\right )^4} \]
Integrate[(E^(2*Log[x]^2)*(64*x + 32*x^2 - 96*x^3 + (64*x + 32*x^2 - 96*x^ 3)*Log[15] + (16*x + 8*x^2 - 24*x^3)*Log[15]^2 + (128*x - 64*x^2 + 64*x^3 + (128*x - 64*x^2 + 64*x^3)*Log[15] + (32*x - 16*x^2 + 16*x^3)*Log[15]^2)* Log[x]))/(32 - 80*x + 160*x^2 - 200*x^3 + 210*x^4 - 161*x^5 + 105*x^6 - 50 *x^7 + 20*x^8 - 5*x^9 + x^10),x]
Time = 11.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2463, 7239, 27, 2726}
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.
| \(\displaystyle \int \frac {e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+\left (64 x^3-64 x^2+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)+128 x\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)+64 x\right )}{x^{10}-5 x^9+20 x^8-50 x^7+105 x^6-161 x^5+210 x^4-200 x^3+160 x^2-80 x+32} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {20 i e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \sqrt {7} \left (-2 x+i \sqrt {7}+1\right )}+\frac {20 i e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \sqrt {7} \left (2 x+i \sqrt {7}-1\right )}-\frac {20 e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \left (-2 x+i \sqrt {7}+1\right )^2}-\frac {20 e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \left (2 x+i \sqrt {7}-1\right )^2}-\frac {120 i e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \sqrt {7} \left (-2 x+i \sqrt {7}+1\right )^3}-\frac {120 i e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \sqrt {7} \left (2 x+i \sqrt {7}-1\right )^3}+\frac {80 e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \left (-2 x+i \sqrt {7}+1\right )^4}+\frac {80 e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{343 \left (2 x+i \sqrt {7}-1\right )^4}+\frac {32 i e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{49 \sqrt {7} \left (-2 x+i \sqrt {7}+1\right )^5}+\frac {32 i e^{2 \log ^2(x)} \left (-96 x^3+32 x^2+64 x+\left (64 x^3-64 x^2+128 x+\left (16 x^3-16 x^2+32 x\right ) \log ^2(15)+\left (64 x^3-64 x^2+128 x\right ) \log (15)\right ) \log (x)+\left (-24 x^3+8 x^2+16 x\right ) \log ^2(15)+\left (-96 x^3+32 x^2+64 x\right ) \log (15)\right )}{49 \sqrt {7} \left (2 x+i \sqrt {7}-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8 x (2+\log (15))^2 e^{2 \log ^2(x)} \left (-3 x^2+2 \left (x^2-x+2\right ) \log (x)+x+2\right )}{\left (x^2-x+2\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 (2+\log (15))^2 \int \frac {e^{2 \log ^2(x)} x \left (-3 x^2+x+2 \left (x^2-x+2\right ) \log (x)+2\right )}{\left (x^2-x+2\right )^5}dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {4 x^2 (2+\log (15))^2 e^{2 \log ^2(x)}}{\left (x^2-x+2\right )^4}\) |
Int[(E^(2*Log[x]^2)*(64*x + 32*x^2 - 96*x^3 + (64*x + 32*x^2 - 96*x^3)*Log [15] + (16*x + 8*x^2 - 24*x^3)*Log[15]^2 + (128*x - 64*x^2 + 64*x^3 + (128 *x - 64*x^2 + 64*x^3)*Log[15] + (32*x - 16*x^2 + 16*x^3)*Log[15]^2)*Log[x] ))/(32 - 80*x + 160*x^2 - 200*x^3 + 210*x^4 - 161*x^5 + 105*x^6 - 50*x^7 + 20*x^8 - 5*x^9 + x^10),x]
3.4.33.3.1 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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, 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. \(76\) vs. \(2(29)=58\).
Time = 6.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57
| method | result | size |
| risch | \(\frac {4 \left (\ln \left (5\right )^{2}+2 \ln \left (3\right ) \ln \left (5\right )+\ln \left (3\right )^{2}+4 \ln \left (5\right )+4 \ln \left (3\right )+4\right ) x^{2} {\mathrm e}^{2 \ln \left (x \right )^{2}}}{x^{8}-4 x^{7}+14 x^{6}-28 x^{5}+49 x^{4}-56 x^{3}+56 x^{2}-32 x +16}\) | \(77\) |
| parallelrisch | \(\frac {4 \ln \left (15\right )^{2} x^{2} {\mathrm e}^{2 \ln \left (x \right )^{2}}+16 \ln \left (15\right ) x^{2} {\mathrm e}^{2 \ln \left (x \right )^{2}}+16 \,{\mathrm e}^{2 \ln \left (x \right )^{2}} x^{2}}{x^{8}-4 x^{7}+14 x^{6}-28 x^{5}+49 x^{4}-56 x^{3}+56 x^{2}-32 x +16}\) | \(85\) |
int((((16*x^3-16*x^2+32*x)*ln(15)^2+(64*x^3-64*x^2+128*x)*ln(15)+64*x^3-64 *x^2+128*x)*ln(x)+(-24*x^3+8*x^2+16*x)*ln(15)^2+(-96*x^3+32*x^2+64*x)*ln(1 5)-96*x^3+32*x^2+64*x)*exp(ln(x)^2)^2/(x^10-5*x^9+20*x^8-50*x^7+105*x^6-16 1*x^5+210*x^4-200*x^3+160*x^2-80*x+32),x,method=_RETURNVERBOSE)
4*(ln(5)^2+2*ln(3)*ln(5)+ln(3)^2+4*ln(5)+4*ln(3)+4)*x^2/(x^8-4*x^7+14*x^6- 28*x^5+49*x^4-56*x^3+56*x^2-32*x+16)*exp(2*ln(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {e^{2 \log ^2(x)} \left (64 x+32 x^2-96 x^3+\left (64 x+32 x^2-96 x^3\right ) \log (15)+\left (16 x+8 x^2-24 x^3\right ) \log ^2(15)+\left (128 x-64 x^2+64 x^3+\left (128 x-64 x^2+64 x^3\right ) \log (15)+\left (32 x-16 x^2+16 x^3\right ) \log ^2(15)\right ) \log (x)\right )}{32-80 x+160 x^2-200 x^3+210 x^4-161 x^5+105 x^6-50 x^7+20 x^8-5 x^9+x^{10}} \, dx=\frac {4 \, {\left (x^{2} \log \left (15\right )^{2} + 4 \, x^{2} \log \left (15\right ) + 4 \, x^{2}\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{8} - 4 \, x^{7} + 14 \, x^{6} - 28 \, x^{5} + 49 \, x^{4} - 56 \, x^{3} + 56 \, x^{2} - 32 \, x + 16} \]
integrate((((16*x^3-16*x^2+32*x)*log(15)^2+(64*x^3-64*x^2+128*x)*log(15)+6 4*x^3-64*x^2+128*x)*log(x)+(-24*x^3+8*x^2+16*x)*log(15)^2+(-96*x^3+32*x^2+ 64*x)*log(15)-96*x^3+32*x^2+64*x)*exp(log(x)^2)^2/(x^10-5*x^9+20*x^8-50*x^ 7+105*x^6-161*x^5+210*x^4-200*x^3+160*x^2-80*x+32),x, algorithm=\
4*(x^2*log(15)^2 + 4*x^2*log(15) + 4*x^2)*e^(2*log(x)^2)/(x^8 - 4*x^7 + 14 *x^6 - 28*x^5 + 49*x^4 - 56*x^3 + 56*x^2 - 32*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {e^{2 \log ^2(x)} \left (64 x+32 x^2-96 x^3+\left (64 x+32 x^2-96 x^3\right ) \log (15)+\left (16 x+8 x^2-24 x^3\right ) \log ^2(15)+\left (128 x-64 x^2+64 x^3+\left (128 x-64 x^2+64 x^3\right ) \log (15)+\left (32 x-16 x^2+16 x^3\right ) \log ^2(15)\right ) \log (x)\right )}{32-80 x+160 x^2-200 x^3+210 x^4-161 x^5+105 x^6-50 x^7+20 x^8-5 x^9+x^{10}} \, dx=\frac {\left (16 x^{2} + 4 x^{2} \log {\left (15 \right )}^{2} + 16 x^{2} \log {\left (15 \right )}\right ) e^{2 \log {\left (x \right )}^{2}}}{x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 49 x^{4} - 56 x^{3} + 56 x^{2} - 32 x + 16} \]
integrate((((16*x**3-16*x**2+32*x)*ln(15)**2+(64*x**3-64*x**2+128*x)*ln(15 )+64*x**3-64*x**2+128*x)*ln(x)+(-24*x**3+8*x**2+16*x)*ln(15)**2+(-96*x**3+ 32*x**2+64*x)*ln(15)-96*x**3+32*x**2+64*x)*exp(ln(x)**2)**2/(x**10-5*x**9+ 20*x**8-50*x**7+105*x**6-161*x**5+210*x**4-200*x**3+160*x**2-80*x+32),x)
(16*x**2 + 4*x**2*log(15)**2 + 16*x**2*log(15))*exp(2*log(x)**2)/(x**8 - 4 *x**7 + 14*x**6 - 28*x**5 + 49*x**4 - 56*x**3 + 56*x**2 - 32*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (28) = 56\).
Time = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.47 \[ \int \frac {e^{2 \log ^2(x)} \left (64 x+32 x^2-96 x^3+\left (64 x+32 x^2-96 x^3\right ) \log (15)+\left (16 x+8 x^2-24 x^3\right ) \log ^2(15)+\left (128 x-64 x^2+64 x^3+\left (128 x-64 x^2+64 x^3\right ) \log (15)+\left (32 x-16 x^2+16 x^3\right ) \log ^2(15)\right ) \log (x)\right )}{32-80 x+160 x^2-200 x^3+210 x^4-161 x^5+105 x^6-50 x^7+20 x^8-5 x^9+x^{10}} \, dx=\frac {4 \, {\left (\log \left (5\right )^{2} + 2 \, {\left (\log \left (5\right ) + 2\right )} \log \left (3\right ) + \log \left (3\right )^{2} + 4 \, \log \left (5\right ) + 4\right )} x^{2} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{8} - 4 \, x^{7} + 14 \, x^{6} - 28 \, x^{5} + 49 \, x^{4} - 56 \, x^{3} + 56 \, x^{2} - 32 \, x + 16} \]
integrate((((16*x^3-16*x^2+32*x)*log(15)^2+(64*x^3-64*x^2+128*x)*log(15)+6 4*x^3-64*x^2+128*x)*log(x)+(-24*x^3+8*x^2+16*x)*log(15)^2+(-96*x^3+32*x^2+ 64*x)*log(15)-96*x^3+32*x^2+64*x)*exp(log(x)^2)^2/(x^10-5*x^9+20*x^8-50*x^ 7+105*x^6-161*x^5+210*x^4-200*x^3+160*x^2-80*x+32),x, algorithm=\
4*(log(5)^2 + 2*(log(5) + 2)*log(3) + log(3)^2 + 4*log(5) + 4)*x^2*e^(2*lo g(x)^2)/(x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 49*x^4 - 56*x^3 + 56*x^2 - 32*x + 16)
\[ \int \frac {e^{2 \log ^2(x)} \left (64 x+32 x^2-96 x^3+\left (64 x+32 x^2-96 x^3\right ) \log (15)+\left (16 x+8 x^2-24 x^3\right ) \log ^2(15)+\left (128 x-64 x^2+64 x^3+\left (128 x-64 x^2+64 x^3\right ) \log (15)+\left (32 x-16 x^2+16 x^3\right ) \log ^2(15)\right ) \log (x)\right )}{32-80 x+160 x^2-200 x^3+210 x^4-161 x^5+105 x^6-50 x^7+20 x^8-5 x^9+x^{10}} \, dx=\int { -\frac {8 \, {\left (12 \, x^{3} + {\left (3 \, x^{3} - x^{2} - 2 \, x\right )} \log \left (15\right )^{2} - 4 \, x^{2} + 4 \, {\left (3 \, x^{3} - x^{2} - 2 \, x\right )} \log \left (15\right ) - 2 \, {\left (4 \, x^{3} + {\left (x^{3} - x^{2} + 2 \, x\right )} \log \left (15\right )^{2} - 4 \, x^{2} + 4 \, {\left (x^{3} - x^{2} + 2 \, x\right )} \log \left (15\right ) + 8 \, x\right )} \log \left (x\right ) - 8 \, x\right )} e^{\left (2 \, \log \left (x\right )^{2}\right )}}{x^{10} - 5 \, x^{9} + 20 \, x^{8} - 50 \, x^{7} + 105 \, x^{6} - 161 \, x^{5} + 210 \, x^{4} - 200 \, x^{3} + 160 \, x^{2} - 80 \, x + 32} \,d x } \]
integrate((((16*x^3-16*x^2+32*x)*log(15)^2+(64*x^3-64*x^2+128*x)*log(15)+6 4*x^3-64*x^2+128*x)*log(x)+(-24*x^3+8*x^2+16*x)*log(15)^2+(-96*x^3+32*x^2+ 64*x)*log(15)-96*x^3+32*x^2+64*x)*exp(log(x)^2)^2/(x^10-5*x^9+20*x^8-50*x^ 7+105*x^6-161*x^5+210*x^4-200*x^3+160*x^2-80*x+32),x, algorithm=\
integrate(-8*(12*x^3 + (3*x^3 - x^2 - 2*x)*log(15)^2 - 4*x^2 + 4*(3*x^3 - x^2 - 2*x)*log(15) - 2*(4*x^3 + (x^3 - x^2 + 2*x)*log(15)^2 - 4*x^2 + 4*(x ^3 - x^2 + 2*x)*log(15) + 8*x)*log(x) - 8*x)*e^(2*log(x)^2)/(x^10 - 5*x^9 + 20*x^8 - 50*x^7 + 105*x^6 - 161*x^5 + 210*x^4 - 200*x^3 + 160*x^2 - 80*x + 32), x)
Time = 8.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {e^{2 \log ^2(x)} \left (64 x+32 x^2-96 x^3+\left (64 x+32 x^2-96 x^3\right ) \log (15)+\left (16 x+8 x^2-24 x^3\right ) \log ^2(15)+\left (128 x-64 x^2+64 x^3+\left (128 x-64 x^2+64 x^3\right ) \log (15)+\left (32 x-16 x^2+16 x^3\right ) \log ^2(15)\right ) \log (x)\right )}{32-80 x+160 x^2-200 x^3+210 x^4-161 x^5+105 x^6-50 x^7+20 x^8-5 x^9+x^{10}} \, dx=\frac {4\,{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}\,{\left (\ln \left (15\right )+2\right )}^2}{{\left (x^2-x+2\right )}^3}+\frac {4\,{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}\,{\left (\ln \left (15\right )+2\right )}^2\,\left (x-2\right )}{{\left (x^2-x+2\right )}^4} \]
int((exp(2*log(x)^2)*(64*x + log(x)*(128*x + log(15)*(128*x - 64*x^2 + 64* x^3) + log(15)^2*(32*x - 16*x^2 + 16*x^3) - 64*x^2 + 64*x^3) + log(15)*(64 *x + 32*x^2 - 96*x^3) + log(15)^2*(16*x + 8*x^2 - 24*x^3) + 32*x^2 - 96*x^ 3))/(160*x^2 - 80*x - 200*x^3 + 210*x^4 - 161*x^5 + 105*x^6 - 50*x^7 + 20* x^8 - 5*x^9 + x^10 + 32),x)