Integrand size = 86, antiderivative size = 25 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=\frac {5 \left (4+\left (5-e^{x^3} (-6+x)\right )^2\right )}{2+x^2} \] Output:
5*(4+(5-(-6+x)*exp(x^3))^2)/(x^2+2)
Time = 3.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=\frac {5 \left (29-10 e^{x^3} (-6+x)+e^{2 x^3} (-6+x)^2\right )}{2+x^2} \] Input:
Integrate[(-290*x + E^x^3*(-100 - 600*x + 1850*x^2 - 300*x^3 + 900*x^4 - 1 50*x^5) + E^(2*x^3)*(-120 - 340*x + 2220*x^2 - 720*x^3 + 1140*x^4 - 360*x^ 5 + 30*x^6))/(4 + 4*x^2 + x^4),x]
Output:
(5*(29 - 10*E^x^3*(-6 + x) + E^(2*x^3)*(-6 + x)^2))/(2 + x^2)
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.
\(\displaystyle \int \frac {e^{x^3} \left (-150 x^5+900 x^4-300 x^3+1850 x^2-600 x-100\right )+e^{2 x^3} \left (30 x^6-360 x^5+1140 x^4-720 x^3+2220 x^2-340 x-120\right )-290 x}{x^4+4 x^2+4} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int -\frac {10 \left (5 e^{x^3} \left (3 x^5-18 x^4+6 x^3-37 x^2+12 x+2\right )+e^{2 x^3} \left (-3 x^6+36 x^5-114 x^4+72 x^3-222 x^2+34 x+12\right )+29 x\right )}{\left (x^2+2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -10 \int \frac {29 x+5 e^{x^3} \left (3 x^5-18 x^4+6 x^3-37 x^2+12 x+2\right )+e^{2 x^3} \left (-3 x^6+36 x^5-114 x^4+72 x^3-222 x^2+34 x+12\right )}{\left (x^2+2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -10 \int \left (\frac {29 x}{\left (x^2+2\right )^2}+\frac {5 e^{x^3} \left (3 x^5-18 x^4+6 x^3-37 x^2+12 x+2\right )}{\left (x^2+2\right )^2}-\frac {e^{2 x^3} (x-6) \left (3 x^5-18 x^4+6 x^3-36 x^2+6 x+2\right )}{\left (x^2+2\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -10 \left (\frac {5}{4} \left (12+35 i \sqrt {2}\right ) \int \frac {e^{x^3}}{i \sqrt {2}-x}dx+\frac {5 i \int \frac {e^{x^3}}{i \sqrt {2}-x}dx}{2 \sqrt {2}}-15 \int \frac {e^{x^3}}{i \sqrt {2}-x}dx-\frac {5}{4} \left (12-35 i \sqrt {2}\right ) \int \frac {e^{x^3}}{x+i \sqrt {2}}dx+\frac {5 i \int \frac {e^{x^3}}{x+i \sqrt {2}}dx}{2 \sqrt {2}}+15 \int \frac {e^{x^3}}{x+i \sqrt {2}}dx+60 \int \frac {e^{x^3} x}{\left (x^2+2\right )^2}dx-\frac {5 e^{x^3}}{2 \left (-x+i \sqrt {2}\right )}+\frac {5 e^{x^3}}{2 \left (x+i \sqrt {2}\right )}+\frac {30 x \Gamma \left (\frac {1}{3},-x^3\right )}{\sqrt [3]{-x^3}}-\frac {29}{2 \left (x^2+2\right )}-\frac {e^{2 x^3} (6-x) \left (-x^5+6 x^4-2 x^3+12 x^2\right )}{2 x^2 \left (x^2+2\right )^2}\right )\) |
Input:
Int[(-290*x + E^x^3*(-100 - 600*x + 1850*x^2 - 300*x^3 + 900*x^4 - 150*x^5 ) + E^(2*x^3)*(-120 - 340*x + 2220*x^2 - 720*x^3 + 1140*x^4 - 360*x^5 + 30 *x^6))/(4 + 4*x^2 + x^4),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
Time = 0.63 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\frac {145}{x^{2}+2}+\frac {5 \left (x^{2}-12 x +36\right ) {\mathrm e}^{2 x^{3}}}{x^{2}+2}-\frac {50 \left (-6+x \right ) {\mathrm e}^{x^{3}}}{x^{2}+2}\) | \(50\) |
norman | \(\frac {180 \,{\mathrm e}^{2 x^{3}}-50 \,{\mathrm e}^{x^{3}} x -60 \,{\mathrm e}^{2 x^{3}} x +5 \,{\mathrm e}^{2 x^{3}} x^{2}+300 \,{\mathrm e}^{x^{3}}+145}{x^{2}+2}\) | \(52\) |
parallelrisch | \(\frac {180 \,{\mathrm e}^{2 x^{3}}-50 \,{\mathrm e}^{x^{3}} x -60 \,{\mathrm e}^{2 x^{3}} x +5 \,{\mathrm e}^{2 x^{3}} x^{2}+300 \,{\mathrm e}^{x^{3}}+145}{x^{2}+2}\) | \(52\) |
parts | \(\frac {145}{x^{2}+2}+\frac {-50 \,{\mathrm e}^{x^{3}} x +300 \,{\mathrm e}^{x^{3}}}{x^{2}+2}+\frac {180 \,{\mathrm e}^{2 x^{3}}-60 \,{\mathrm e}^{2 x^{3}} x +5 \,{\mathrm e}^{2 x^{3}} x^{2}}{x^{2}+2}\) | \(70\) |
orering | \(\text {Expression too large to display}\) | \(1250\) |
Input:
int(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2+(-150 *x^5+900*x^4-300*x^3+1850*x^2-600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+4),x,m ethod=_RETURNVERBOSE)
Output:
145/(x^2+2)+5*(x^2-12*x+36)/(x^2+2)*exp(x^3)^2-50*(-6+x)/(x^2+2)*exp(x^3)
Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=\frac {5 \, {\left ({\left (x^{2} - 12 \, x + 36\right )} e^{\left (2 \, x^{3}\right )} - 10 \, {\left (x - 6\right )} e^{\left (x^{3}\right )} + 29\right )}}{x^{2} + 2} \] Input:
integrate(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2 +(-150*x^5+900*x^4-300*x^3+1850*x^2-600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+ 4),x, algorithm="fricas")
Output:
5*((x^2 - 12*x + 36)*e^(2*x^3) - 10*(x - 6)*e^(x^3) + 29)/(x^2 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=\frac {\left (- 50 x^{3} + 300 x^{2} - 100 x + 600\right ) e^{x^{3}} + \left (5 x^{4} - 60 x^{3} + 190 x^{2} - 120 x + 360\right ) e^{2 x^{3}}}{x^{4} + 4 x^{2} + 4} + \frac {290}{2 x^{2} + 4} \] Input:
integrate(((30*x**6-360*x**5+1140*x**4-720*x**3+2220*x**2-340*x-120)*exp(x **3)**2+(-150*x**5+900*x**4-300*x**3+1850*x**2-600*x-100)*exp(x**3)-290*x) /(x**4+4*x**2+4),x)
Output:
((-50*x**3 + 300*x**2 - 100*x + 600)*exp(x**3) + (5*x**4 - 60*x**3 + 190*x **2 - 120*x + 360)*exp(2*x**3))/(x**4 + 4*x**2 + 4) + 290/(2*x**2 + 4)
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=\frac {5 \, {\left ({\left (x^{2} - 12 \, x + 36\right )} e^{\left (2 \, x^{3}\right )} - 10 \, {\left (x - 6\right )} e^{\left (x^{3}\right )}\right )}}{x^{2} + 2} + \frac {145}{x^{2} + 2} \] Input:
integrate(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2 +(-150*x^5+900*x^4-300*x^3+1850*x^2-600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+ 4),x, algorithm="maxima")
Output:
5*((x^2 - 12*x + 36)*e^(2*x^3) - 10*(x - 6)*e^(x^3))/(x^2 + 2) + 145/(x^2 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=\frac {5 \, {\left (x^{2} e^{\left (2 \, x^{3}\right )} - 12 \, x e^{\left (2 \, x^{3}\right )} - 10 \, x e^{\left (x^{3}\right )} + 36 \, e^{\left (2 \, x^{3}\right )} + 60 \, e^{\left (x^{3}\right )} + 29\right )}}{x^{2} + 2} \] Input:
integrate(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2 +(-150*x^5+900*x^4-300*x^3+1850*x^2-600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+ 4),x, algorithm="giac")
Output:
5*(x^2*e^(2*x^3) - 12*x*e^(2*x^3) - 10*x*e^(x^3) + 36*e^(2*x^3) + 60*e^(x^ 3) + 29)/(x^2 + 2)
Time = 8.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=5\,{\mathrm {e}}^{2\,x^3}+\frac {300\,{\mathrm {e}}^{x^3}+170\,{\mathrm {e}}^{2\,x^3}-x\,\left (50\,{\mathrm {e}}^{x^3}+60\,{\mathrm {e}}^{2\,x^3}\right )+145}{x^2+2} \] Input:
int(-(290*x + exp(2*x^3)*(340*x - 2220*x^2 + 720*x^3 - 1140*x^4 + 360*x^5 - 30*x^6 + 120) + exp(x^3)*(600*x - 1850*x^2 + 300*x^3 - 900*x^4 + 150*x^5 + 100))/(4*x^2 + x^4 + 4),x)
Output:
5*exp(2*x^3) + (300*exp(x^3) + 170*exp(2*x^3) - x*(50*exp(x^3) + 60*exp(2* x^3)) + 145)/(x^2 + 2)
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {-290 x+e^{x^3} \left (-100-600 x+1850 x^2-300 x^3+900 x^4-150 x^5\right )+e^{2 x^3} \left (-120-340 x+2220 x^2-720 x^3+1140 x^4-360 x^5+30 x^6\right )}{4+4 x^2+x^4} \, dx=\frac {10 e^{2 x^{3}} x^{2}-120 e^{2 x^{3}} x +360 e^{2 x^{3}}-100 e^{x^{3}} x +600 e^{x^{3}}-145 x^{2}}{2 x^{2}+4} \] Input:
int(((30*x^6-360*x^5+1140*x^4-720*x^3+2220*x^2-340*x-120)*exp(x^3)^2+(-150 *x^5+900*x^4-300*x^3+1850*x^2-600*x-100)*exp(x^3)-290*x)/(x^4+4*x^2+4),x)
Output:
(5*(2*e**(2*x**3)*x**2 - 24*e**(2*x**3)*x + 72*e**(2*x**3) - 20*e**(x**3)* x + 120*e**(x**3) - 29*x**2))/(2*(x**2 + 2))