Integrand size = 99, antiderivative size = 30 \begin {dmath*} \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-x+\frac {9+\log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{2-x} \end {dmath*}
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \begin {dmath*} \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-x+\frac {-9-\log \left (3-2 e^4 (-1+x) x\right )}{-2+x} \end {dmath*}
Integrate[(-15 - 12*x + 3*x^2 + E^4*(-4 - 2*x^2 + 10*x^3 - 2*x^4) + (-3 + E^4*(-2*x + 2*x^2))*Log[3 + E^4*(2*x - 2*x^2)])/(-12 + 12*x - 3*x^2 + E^4* (-8*x + 16*x^2 - 10*x^3 + 2*x^4)),x]
Leaf count is larger than twice the leaf count of optimal. \(715\) vs. \(2(30)=60\).
Time = 2.36 (sec) , antiderivative size = 715, normalized size of antiderivative = 23.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2463, 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.
\(\displaystyle \int \frac {3 x^2+\left (e^4 \left (2 x^2-2 x\right )-3\right ) \log \left (e^4 \left (2 x-2 x^2\right )+3\right )+e^4 \left (-2 x^4+10 x^3-2 x^2-4\right )-12 x-15}{-3 x^2+e^4 \left (2 x^4-10 x^3+16 x^2-8 x\right )+12 x-12} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {6 e^4 \left (3 x^2+\left (e^4 \left (2 x^2-2 x\right )-3\right ) \log \left (e^4 \left (2 x-2 x^2\right )+3\right )+e^4 \left (-2 x^4+10 x^3-2 x^2-4\right )-12 x-15\right )}{\left (4 e^4-3\right )^2 (x-2)}+\frac {2 e^4 \left (6 e^4 x+2 e^4+3\right ) \left (3 x^2+\left (e^4 \left (2 x^2-2 x\right )-3\right ) \log \left (e^4 \left (2 x-2 x^2\right )+3\right )+e^4 \left (-2 x^4+10 x^3-2 x^2-4\right )-12 x-15\right )}{\left (4 e^4-3\right )^2 \left (2 e^4 x^2-2 e^4 x-3\right )}+\frac {3 x^2+\left (e^4 \left (2 x^2-2 x\right )-3\right ) \log \left (e^4 \left (2 x-2 x^2\right )+3\right )+e^4 \left (-2 x^4+10 x^3-2 x^2-4\right )-12 x-15}{\left (4 e^4-3\right ) (x-2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 e^2 \sqrt {6+e^4} \left (3+5 e^4\right ) \text {arctanh}\left (\frac {e^2 (1-2 x)}{\sqrt {6+e^4}}\right )}{\left (3-4 e^4\right )^2}+\frac {2 e^2 \sqrt {6+e^4} \text {arctanh}\left (\frac {e^2 (1-2 x)}{\sqrt {6+e^4}}\right )}{3-4 e^4}+\frac {18 e^6 \sqrt {6+e^4} \text {arctanh}\left (\frac {e^2 (1-2 x)}{\sqrt {6+e^4}}\right )}{\left (3-4 e^4\right )^2}+\frac {2 e^4 x^3}{3 \left (3-4 e^4\right )}-\frac {2 e^4 \left (3-22 e^4\right ) x^3}{3 \left (3-4 e^4\right )^2}-\frac {12 e^8 x^3}{\left (3-4 e^4\right )^2}+\frac {2 e^4 \left (6+13 e^4\right ) x^2}{\left (3-4 e^4\right )^2}-\frac {3 e^4 \left (3+10 e^4\right ) x^2}{\left (3-4 e^4\right )^2}-\frac {e^4 x^2}{3-4 e^4}-\frac {6 e^8 x^2 \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{\left (3-4 e^4\right )^2}-\frac {2 e^4 x \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{3-4 e^4}-\frac {12 e^8 x \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{\left (3-4 e^4\right )^2}+\frac {\left (6 e^4 x+2 e^4+3\right )^2 \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{6 \left (3-4 e^4\right )^2}+\frac {\log \left (-2 e^4 x^2+2 e^4 x+3\right )}{2-x}-\frac {\left (9+84 e^4+34 e^8\right ) \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{6 \left (3-4 e^4\right )^2}+\frac {3 e^4 \left (3+e^4\right ) \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{\left (3-4 e^4\right )^2}+\frac {e^4 \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{3-4 e^4}+\frac {6 e^8 \log \left (-2 e^4 x^2+2 e^4 x+3\right )}{\left (3-4 e^4\right )^2}+\frac {6 e^4 \left (3+8 e^4\right ) x}{\left (3-4 e^4\right )^2}-\frac {2 e^4 \left (6+7 e^4\right ) x}{\left (3-4 e^4\right )^2}-\frac {10 e^4 x}{3-4 e^4}-\frac {3 x}{3-4 e^4}+\frac {12 e^4 \left (3-10 e^4\right ) x}{\left (3-4 e^4\right )^2}+\frac {30 e^8 x}{\left (3-4 e^4\right )^2}-\frac {36 e^4}{\left (3-4 e^4\right ) (2-x)}+\frac {27}{\left (3-4 e^4\right ) (2-x)}\) |
Int[(-15 - 12*x + 3*x^2 + E^4*(-4 - 2*x^2 + 10*x^3 - 2*x^4) + (-3 + E^4*(- 2*x + 2*x^2))*Log[3 + E^4*(2*x - 2*x^2)])/(-12 + 12*x - 3*x^2 + E^4*(-8*x + 16*x^2 - 10*x^3 + 2*x^4)),x]
27/((3 - 4*E^4)*(2 - x)) - (36*E^4)/((3 - 4*E^4)*(2 - x)) + (30*E^8*x)/(3 - 4*E^4)^2 + (12*E^4*(3 - 10*E^4)*x)/(3 - 4*E^4)^2 - (3*x)/(3 - 4*E^4) - ( 10*E^4*x)/(3 - 4*E^4) - (2*E^4*(6 + 7*E^4)*x)/(3 - 4*E^4)^2 + (6*E^4*(3 + 8*E^4)*x)/(3 - 4*E^4)^2 - (E^4*x^2)/(3 - 4*E^4) - (3*E^4*(3 + 10*E^4)*x^2) /(3 - 4*E^4)^2 + (2*E^4*(6 + 13*E^4)*x^2)/(3 - 4*E^4)^2 - (12*E^8*x^3)/(3 - 4*E^4)^2 - (2*E^4*(3 - 22*E^4)*x^3)/(3*(3 - 4*E^4)^2) + (2*E^4*x^3)/(3*( 3 - 4*E^4)) + (18*E^6*Sqrt[6 + E^4]*ArcTanh[(E^2*(1 - 2*x))/Sqrt[6 + E^4]] )/(3 - 4*E^4)^2 + (2*E^2*Sqrt[6 + E^4]*ArcTanh[(E^2*(1 - 2*x))/Sqrt[6 + E^ 4]])/(3 - 4*E^4) - (2*E^2*Sqrt[6 + E^4]*(3 + 5*E^4)*ArcTanh[(E^2*(1 - 2*x) )/Sqrt[6 + E^4]])/(3 - 4*E^4)^2 + (6*E^8*Log[3 + 2*E^4*x - 2*E^4*x^2])/(3 - 4*E^4)^2 + (E^4*Log[3 + 2*E^4*x - 2*E^4*x^2])/(3 - 4*E^4) + (3*E^4*(3 + E^4)*Log[3 + 2*E^4*x - 2*E^4*x^2])/(3 - 4*E^4)^2 - ((9 + 84*E^4 + 34*E^8)* Log[3 + 2*E^4*x - 2*E^4*x^2])/(6*(3 - 4*E^4)^2) + Log[3 + 2*E^4*x - 2*E^4* x^2]/(2 - x) - (12*E^8*x*Log[3 + 2*E^4*x - 2*E^4*x^2])/(3 - 4*E^4)^2 - (2* E^4*x*Log[3 + 2*E^4*x - 2*E^4*x^2])/(3 - 4*E^4) - (6*E^8*x^2*Log[3 + 2*E^4 *x - 2*E^4*x^2])/(3 - 4*E^4)^2 + ((3 + 2*E^4 + 6*E^4*x)^2*Log[3 + 2*E^4*x - 2*E^4*x^2])/(6*(3 - 4*E^4)^2)
3.2.60.3.1 Defintions of rubi rules used
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]
Time = 1.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {-\ln \left (\left (-2 x^{2}+2 x \right ) {\mathrm e}^{4}+3\right )-x^{2}-5}{-2+x}\) | \(31\) |
parallelrisch | \(\frac {-18 x^{2}-45 x -18 \ln \left (\left (-2 x^{2}+2 x \right ) {\mathrm e}^{4}+3\right )}{-36+18 x}\) | \(34\) |
risch | \(-\frac {\ln \left (\left (-2 x^{2}+2 x \right ) {\mathrm e}^{4}+3\right )}{-2+x}-\frac {x^{2}-2 x +9}{-2+x}\) | \(39\) |
int((((2*x^2-2*x)*exp(4)-3)*ln((-2*x^2+2*x)*exp(4)+3)+(-2*x^4+10*x^3-2*x^2 -4)*exp(4)+3*x^2-12*x-15)/((2*x^4-10*x^3+16*x^2-8*x)*exp(4)-3*x^2+12*x-12) ,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-\frac {x^{2} - 2 \, x + \log \left (-2 \, {\left (x^{2} - x\right )} e^{4} + 3\right ) + 9}{x - 2} \end {dmath*}
integrate((((2*x^2-2*x)*exp(4)-3)*log((-2*x^2+2*x)*exp(4)+3)+(-2*x^4+10*x^ 3-2*x^2-4)*exp(4)+3*x^2-12*x-15)/((2*x^4-10*x^3+16*x^2-8*x)*exp(4)-3*x^2+1 2*x-12),x, algorithm=\
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \begin {dmath*} \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=- x - \frac {\log {\left (\left (- 2 x^{2} + 2 x\right ) e^{4} + 3 \right )}}{x - 2} - \frac {9}{x - 2} \end {dmath*}
integrate((((2*x**2-2*x)*exp(4)-3)*ln((-2*x**2+2*x)*exp(4)+3)+(-2*x**4+10* x**3-2*x**2-4)*exp(4)+3*x**2-12*x-15)/((2*x**4-10*x**3+16*x**2-8*x)*exp(4) -3*x**2+12*x-12),x)
Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (27) = 54\).
Time = 0.35 (sec) , antiderivative size = 1175, normalized size of antiderivative = 39.17 \begin {dmath*} \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=\text {Too large to display} \end {dmath*}
integrate((((2*x^2-2*x)*exp(4)-3)*log((-2*x^2+2*x)*exp(4)+3)+(-2*x^4+10*x^ 3-2*x^2-4)*exp(4)+3*x^2-12*x-15)/((2*x^4-10*x^3+16*x^2-8*x)*exp(4)-3*x^2+1 2*x-12),x, algorithm=\
-1/2*(2*x*e^(-4) + (16*e^12 + 120*e^8 + 189*e^4 + 27)*e^(-2)*log((2*x*e^4 - sqrt(e^4 + 6)*e^2 - e^4)/(2*x*e^4 + sqrt(e^4 + 6)*e^2 - e^4))/((16*e^12 - 24*e^8 + 9*e^4)*sqrt(e^4 + 6)) + (16*e^8 + 72*e^4 + 45)*log(2*x^2*e^4 - 2*x*e^4 - 3)/(16*e^12 - 24*e^8 + 9*e^4) + 128*(e^4 - 3)*log(x - 2)/(16*e^8 - 24*e^4 + 9) - 64/(x*(4*e^4 - 3) - 8*e^4 + 6))*e^4 + 5/2*((16*e^8 + 96*e ^4 + 81)*e^(-2)*log((2*x*e^4 - sqrt(e^4 + 6)*e^2 - e^4)/(2*x*e^4 + sqrt(e^ 4 + 6)*e^2 - e^4))/((16*e^8 - 24*e^4 + 9)*sqrt(e^4 + 6)) + (16*e^8 + 48*e^ 4 + 9)*log(2*x^2*e^4 - 2*x*e^4 - 3)/(16*e^12 - 24*e^8 + 9*e^4) - 144*log(x - 2)/(16*e^8 - 24*e^4 + 9) - 32/(x*(4*e^4 - 3) - 8*e^4 + 6))*e^4 - ((8*e^ 8 + 36*e^4 + 9)*e^(-2)*log((2*x*e^4 - sqrt(e^4 + 6)*e^2 - e^4)/(2*x*e^4 + sqrt(e^4 + 6)*e^2 - e^4))/((16*e^8 - 24*e^4 + 9)*sqrt(e^4 + 6)) + 4*(2*e^4 + 3)*log(2*x^2*e^4 - 2*x*e^4 - 3)/(16*e^8 - 24*e^4 + 9) - 8*(2*e^4 + 3)*l og(x - 2)/(16*e^8 - 24*e^4 + 9) - 8/(x*(4*e^4 - 3) - 8*e^4 + 6))*e^4 - 4*( (5*e^8 + 3*e^4)*e^(-2)*log((2*x*e^4 - sqrt(e^4 + 6)*e^2 - e^4)/(2*x*e^4 + sqrt(e^4 + 6)*e^2 - e^4))/((16*e^8 - 24*e^4 + 9)*sqrt(e^4 + 6)) + 3*e^4*lo g(2*x^2*e^4 - 2*x*e^4 - 3)/(16*e^8 - 24*e^4 + 9) - 6*e^4*log(x - 2)/(16*e^ 8 - 24*e^4 + 9) - 1/(x*(4*e^4 - 3) - 8*e^4 + 6))*e^4 + 3/2*(8*e^8 + 36*e^4 + 9)*e^(-2)*log((2*x*e^4 - sqrt(e^4 + 6)*e^2 - e^4)/(2*x*e^4 + sqrt(e^4 + 6)*e^2 - e^4))/((16*e^8 - 24*e^4 + 9)*sqrt(e^4 + 6)) - 6*(8*e^8 + 21*e^4) *e^(-2)*log((2*x*e^4 - sqrt(e^4 + 6)*e^2 - e^4)/(2*x*e^4 + sqrt(e^4 + 6...
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-\frac {x^{2} - 2 \, x + \log \left (-2 \, x^{2} e^{4} + 2 \, x e^{4} + 3\right ) + 9}{x - 2} \end {dmath*}
integrate((((2*x^2-2*x)*exp(4)-3)*log((-2*x^2+2*x)*exp(4)+3)+(-2*x^4+10*x^ 3-2*x^2-4)*exp(4)+3*x^2-12*x-15)/((2*x^4-10*x^3+16*x^2-8*x)*exp(4)-3*x^2+1 2*x-12),x, algorithm=\
Time = 14.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-15-12 x+3 x^2+e^4 \left (-4-2 x^2+10 x^3-2 x^4\right )+\left (-3+e^4 \left (-2 x+2 x^2\right )\right ) \log \left (3+e^4 \left (2 x-2 x^2\right )\right )}{-12+12 x-3 x^2+e^4 \left (-8 x+16 x^2-10 x^3+2 x^4\right )} \, dx=-\frac {\ln \left ({\mathrm {e}}^4\,\left (2\,x-2\,x^2\right )+3\right )-2\,x+x^2+9}{x-2} \end {dmath*}