Integrand size = 152, antiderivative size = 25 \[ \int \frac {e^{\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )} \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx=e^{\left (1+x^2\right ) \left (4-\frac {4 x}{3}+\log \left (7-x^2\right )\right )^2} \]
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {e^{\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )} \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx=e^{\frac {1}{9} \left (1+x^2\right ) \left (16 (-3+x)^2+9 \log ^2\left (7-x^2\right )\right )} \left (7-x^2\right )^{8-\frac {8}{3} x \left (1-3 x+x^2\right )} \]
Integrate[(E^((144 - 96*x + 160*x^2 - 96*x^3 + 16*x^4 + (72 - 24*x + 72*x^ 2 - 24*x^3)*Log[7 - x^2] + (9 + 9*x^2)*Log[7 - x^2]^2)/9)*(672 - 2096*x + 1872*x^2 + 16*x^3 - 336*x^4 + 64*x^5 + (168 - 972*x + 480*x^2 + 180*x^3 - 72*x^4)*Log[7 - x^2] + (-126*x + 18*x^3)*Log[7 - x^2]^2))/(-63 + 9*x^2),x]
E^(((1 + x^2)*(16*(-3 + x)^2 + 9*Log[7 - x^2]^2))/9)*(7 - x^2)^(8 - (8*x*( 1 - 3*x + x^2))/3)
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.
\(\displaystyle \int \frac {\left (64 x^5-336 x^4+16 x^3+1872 x^2+\left (18 x^3-126 x\right ) \log ^2\left (7-x^2\right )+\left (-72 x^4+180 x^3+480 x^2-972 x+168\right ) \log \left (7-x^2\right )-2096 x+672\right ) \exp \left (\frac {1}{9} \left (16 x^4-96 x^3+160 x^2+\left (9 x^2+9\right ) \log ^2\left (7-x^2\right )+\left (-24 x^3+72 x^2-24 x+72\right ) \log \left (7-x^2\right )-96 x+144\right )\right )}{9 x^2-63} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (2 x \log ^2\left (7-x^2\right ) \exp \left (\frac {1}{9} \left (x^2+1\right ) \left (-3 \log \left (7-x^2\right )+4 x-12\right )^2\right )-\frac {4 \left (6 x^4-15 x^3-40 x^2+81 x-14\right ) \log \left (7-x^2\right ) \exp \left (\frac {1}{9} \left (x^2+1\right ) \left (-3 \log \left (7-x^2\right )+4 x-12\right )^2\right )}{3 \left (x^2-7\right )}+\frac {16 \left (4 x^5-21 x^4+x^3+117 x^2-131 x+42\right ) \exp \left (\frac {1}{9} \left (x^2+1\right ) \left (-3 \log \left (7-x^2\right )+4 x-12\right )^2\right )}{9 \left (x^2-7\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} x \log ^2\left (7-x^2\right )dx-\frac {160}{3} \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2}dx-\frac {64}{3} \left (3-\sqrt {7}\right ) \int \frac {e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2}}{\sqrt {7}-x}dx+\frac {464}{9} \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} xdx-\frac {112}{3} \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} x^2dx+\frac {64}{3} \left (3+\sqrt {7}\right ) \int \frac {e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2}}{x+\sqrt {7}}dx-\frac {8}{3} \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} \log \left (7-x^2\right )dx-16 \int \frac {e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} \log \left (7-x^2\right )}{\sqrt {7}-x}dx+20 \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} x \log \left (7-x^2\right )dx-8 \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} x^2 \log \left (7-x^2\right )dx+16 \int \frac {e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} \log \left (7-x^2\right )}{x+\sqrt {7}}dx+\frac {64}{9} \int e^{\frac {1}{9} \left (x^2+1\right ) \left (4 x-3 \log \left (7-x^2\right )-12\right )^2} x^3dx\) |
Int[(E^((144 - 96*x + 160*x^2 - 96*x^3 + 16*x^4 + (72 - 24*x + 72*x^2 - 24 *x^3)*Log[7 - x^2] + (9 + 9*x^2)*Log[7 - x^2]^2)/9)*(672 - 2096*x + 1872*x ^2 + 16*x^3 - 336*x^4 + 64*x^5 + (168 - 972*x + 480*x^2 + 180*x^3 - 72*x^4 )*Log[7 - x^2] + (-126*x + 18*x^3)*Log[7 - x^2]^2))/(-63 + 9*x^2),x]
3.2.40.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(22)=44\).
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\left (-x^{2}+7\right )^{-\frac {8 \left (-3+x \right ) \left (x^{2}+1\right )}{3}} {\mathrm e}^{\frac {\left (x^{2}+1\right ) \left (9 \ln \left (-x^{2}+7\right )^{2}+16 x^{2}-96 x +144\right )}{9}}\) | \(50\) |
parallelrisch | \({\mathrm e}^{\frac {\left (9 x^{2}+9\right ) \ln \left (-x^{2}+7\right )^{2}}{9}+\frac {\left (-24 x^{3}+72 x^{2}-24 x +72\right ) \ln \left (-x^{2}+7\right )}{9}+\frac {16 x^{4}}{9}-\frac {32 x^{3}}{3}+\frac {160 x^{2}}{9}-\frac {32 x}{3}+16}\) | \(66\) |
int(((18*x^3-126*x)*ln(-x^2+7)^2+(-72*x^4+180*x^3+480*x^2-972*x+168)*ln(-x ^2+7)+64*x^5-336*x^4+16*x^3+1872*x^2-2096*x+672)*exp(1/9*(9*x^2+9)*ln(-x^2 +7)^2+1/9*(-24*x^3+72*x^2-24*x+72)*ln(-x^2+7)+16/9*x^4-32/3*x^3+160/9*x^2- 32/3*x+16)/(9*x^2-63),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {e^{\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )} \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx=e^{\left (\frac {16}{9} \, x^{4} - \frac {32}{3} \, x^{3} + {\left (x^{2} + 1\right )} \log \left (-x^{2} + 7\right )^{2} + \frac {160}{9} \, x^{2} - \frac {8}{3} \, {\left (x^{3} - 3 \, x^{2} + x - 3\right )} \log \left (-x^{2} + 7\right ) - \frac {32}{3} \, x + 16\right )} \]
integrate(((18*x^3-126*x)*log(-x^2+7)^2+(-72*x^4+180*x^3+480*x^2-972*x+168 )*log(-x^2+7)+64*x^5-336*x^4+16*x^3+1872*x^2-2096*x+672)*exp(1/9*(9*x^2+9) *log(-x^2+7)^2+1/9*(-24*x^3+72*x^2-24*x+72)*log(-x^2+7)+16/9*x^4-32/3*x^3+ 160/9*x^2-32/3*x+16)/(9*x^2-63),x, algorithm=\
e^(16/9*x^4 - 32/3*x^3 + (x^2 + 1)*log(-x^2 + 7)^2 + 160/9*x^2 - 8/3*(x^3 - 3*x^2 + x - 3)*log(-x^2 + 7) - 32/3*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (20) = 40\).
Time = 0.50 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {e^{\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )} \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx=e^{\frac {16 x^{4}}{9} - \frac {32 x^{3}}{3} + \frac {160 x^{2}}{9} - \frac {32 x}{3} + \left (x^{2} + 1\right ) \log {\left (7 - x^{2} \right )}^{2} + \left (- \frac {8 x^{3}}{3} + 8 x^{2} - \frac {8 x}{3} + 8\right ) \log {\left (7 - x^{2} \right )} + 16} \]
integrate(((18*x**3-126*x)*ln(-x**2+7)**2+(-72*x**4+180*x**3+480*x**2-972* x+168)*ln(-x**2+7)+64*x**5-336*x**4+16*x**3+1872*x**2-2096*x+672)*exp(1/9* (9*x**2+9)*ln(-x**2+7)**2+1/9*(-24*x**3+72*x**2-24*x+72)*ln(-x**2+7)+16/9* x**4-32/3*x**3+160/9*x**2-32/3*x+16)/(9*x**2-63),x)
exp(16*x**4/9 - 32*x**3/3 + 160*x**2/9 - 32*x/3 + (x**2 + 1)*log(7 - x**2) **2 + (-8*x**3/3 + 8*x**2 - 8*x/3 + 8)*log(7 - x**2) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (25) = 50\).
Time = 0.47 (sec) , antiderivative size = 142, normalized size of antiderivative = 5.68 \[ \int \frac {e^{\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )} \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx={\left (x^{16} e^{16} - 56 \, x^{14} e^{16} + 1372 \, x^{12} e^{16} - 19208 \, x^{10} e^{16} + 168070 \, x^{8} e^{16} - 941192 \, x^{6} e^{16} + 3294172 \, x^{4} e^{16} - 6588344 \, x^{2} e^{16} + 5764801 \, e^{16}\right )} e^{\left (\frac {16}{9} \, x^{4} - \frac {8}{3} \, x^{3} \log \left (-x^{2} + 7\right ) + x^{2} \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x^{3} + 8 \, x^{2} \log \left (-x^{2} + 7\right ) + \frac {160}{9} \, x^{2} - \frac {8}{3} \, x \log \left (-x^{2} + 7\right ) + \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x\right )} \]
integrate(((18*x^3-126*x)*log(-x^2+7)^2+(-72*x^4+180*x^3+480*x^2-972*x+168 )*log(-x^2+7)+64*x^5-336*x^4+16*x^3+1872*x^2-2096*x+672)*exp(1/9*(9*x^2+9) *log(-x^2+7)^2+1/9*(-24*x^3+72*x^2-24*x+72)*log(-x^2+7)+16/9*x^4-32/3*x^3+ 160/9*x^2-32/3*x+16)/(9*x^2-63),x, algorithm=\
(x^16*e^16 - 56*x^14*e^16 + 1372*x^12*e^16 - 19208*x^10*e^16 + 168070*x^8* e^16 - 941192*x^6*e^16 + 3294172*x^4*e^16 - 6588344*x^2*e^16 + 5764801*e^1 6)*e^(16/9*x^4 - 8/3*x^3*log(-x^2 + 7) + x^2*log(-x^2 + 7)^2 - 32/3*x^3 + 8*x^2*log(-x^2 + 7) + 160/9*x^2 - 8/3*x*log(-x^2 + 7) + log(-x^2 + 7)^2 - 32/3*x)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (25) = 50\).
Time = 1.66 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.68 \[ \int \frac {e^{\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )} \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx=e^{\left (\frac {16}{9} \, x^{4} - \frac {8}{3} \, x^{3} \log \left (-x^{2} + 7\right ) + x^{2} \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x^{3} + 8 \, x^{2} \log \left (-x^{2} + 7\right ) + \frac {160}{9} \, x^{2} - \frac {8}{3} \, x \log \left (-x^{2} + 7\right ) + \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x + 8 \, \log \left (-x^{2} + 7\right ) + 16\right )} \]
integrate(((18*x^3-126*x)*log(-x^2+7)^2+(-72*x^4+180*x^3+480*x^2-972*x+168 )*log(-x^2+7)+64*x^5-336*x^4+16*x^3+1872*x^2-2096*x+672)*exp(1/9*(9*x^2+9) *log(-x^2+7)^2+1/9*(-24*x^3+72*x^2-24*x+72)*log(-x^2+7)+16/9*x^4-32/3*x^3+ 160/9*x^2-32/3*x+16)/(9*x^2-63),x, algorithm=\
e^(16/9*x^4 - 8/3*x^3*log(-x^2 + 7) + x^2*log(-x^2 + 7)^2 - 32/3*x^3 + 8*x ^2*log(-x^2 + 7) + 160/9*x^2 - 8/3*x*log(-x^2 + 7) + log(-x^2 + 7)^2 - 32/ 3*x + 8*log(-x^2 + 7) + 16)
Time = 8.69 (sec) , antiderivative size = 780, normalized size of antiderivative = 31.20 \[ \int \frac {e^{\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )} \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx =\text {Too large to display} \]
int((exp((log(7 - x^2)^2*(9*x^2 + 9))/9 - (log(7 - x^2)*(24*x - 72*x^2 + 2 4*x^3 - 72))/9 - (32*x)/3 + (160*x^2)/9 - (32*x^3)/3 + (16*x^4)/9 + 16)*(l og(7 - x^2)*(480*x^2 - 972*x + 180*x^3 - 72*x^4 + 168) - log(7 - x^2)^2*(1 26*x - 18*x^3) - 2096*x + 1872*x^2 + 16*x^3 - 336*x^4 + 64*x^5 + 672))/(9* x^2 - 63),x)
5764801*exp(log(7 - x^2)^2 - (8*x^3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - x^2)^2 + (160*x^2)/9 - (32*x^3)/3 + (16*x^4)/9 - (8*x*log(7 - x^2))/3 + 16)*(7 - x^2)^(8*x^2) - 6588344*x^2*exp(log(7 - x^2)^2 - (8*x^3*log(7 - x ^2))/3 - (32*x)/3 + x^2*log(7 - x^2)^2 + (160*x^2)/9 - (32*x^3)/3 + (16*x^ 4)/9 - (8*x*log(7 - x^2))/3 + 16)*(7 - x^2)^(8*x^2) + 3294172*x^4*exp(log( 7 - x^2)^2 - (8*x^3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - x^2)^2 + (160 *x^2)/9 - (32*x^3)/3 + (16*x^4)/9 - (8*x*log(7 - x^2))/3 + 16)*(7 - x^2)^( 8*x^2) - 941192*x^6*exp(log(7 - x^2)^2 - (8*x^3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - x^2)^2 + (160*x^2)/9 - (32*x^3)/3 + (16*x^4)/9 - (8*x*log(7 - x^2))/3 + 16)*(7 - x^2)^(8*x^2) + 168070*x^8*exp(log(7 - x^2)^2 - (8*x^ 3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - x^2)^2 + (160*x^2)/9 - (32*x^3) /3 + (16*x^4)/9 - (8*x*log(7 - x^2))/3 + 16)*(7 - x^2)^(8*x^2) - 19208*x^1 0*exp(log(7 - x^2)^2 - (8*x^3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - x^2 )^2 + (160*x^2)/9 - (32*x^3)/3 + (16*x^4)/9 - (8*x*log(7 - x^2))/3 + 16)*( 7 - x^2)^(8*x^2) + 1372*x^12*exp(log(7 - x^2)^2 - (8*x^3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - x^2)^2 + (160*x^2)/9 - (32*x^3)/3 + (16*x^4)/9 - ( 8*x*log(7 - x^2))/3 + 16)*(7 - x^2)^(8*x^2) - 56*x^14*exp(log(7 - x^2)^2 - (8*x^3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - x^2)^2 + (160*x^2)/9 - (3 2*x^3)/3 + (16*x^4)/9 - (8*x*log(7 - x^2))/3 + 16)*(7 - x^2)^(8*x^2) + x^1 6*exp(log(7 - x^2)^2 - (8*x^3*log(7 - x^2))/3 - (32*x)/3 + x^2*log(7 - ...