Integrand size = 68, antiderivative size = 21 \[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx=\log \left (25+\frac {4 x}{(-4+x) \left (1+e^4+2 x\right )}\right ) \] Output:
ln(25+4*x/(-4+x)/(1+2*x+exp(4)))
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(21)=42\).
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.52 \[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx=-8 \left (\frac {1}{8} \log \left (9+e^4+2 (-4+x)\right )-\frac {1}{8} \log \left (16+229 (-4+x)+25 e^4 (-4+x)+50 (-4+x)^2\right )+\frac {1}{8} \log (-4+x)\right ) \] Input:
Integrate[(-16 - 16*E^4 - 8*x^2)/(400 + 1384*x + 797*x^2 - 692*x^3 + 100*x ^4 + E^8*(400 - 200*x + 25*x^2) + E^4*(800 + 1184*x - 746*x^2 + 100*x^3)), x]
Output:
-8*(Log[9 + E^4 + 2*(-4 + x)]/8 - Log[16 + 229*(-4 + x) + 25*E^4*(-4 + x) + 50*(-4 + x)^2]/8 + Log[-4 + x]/8)
Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(21)=42\).
Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2462, 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.
\(\displaystyle \int \frac {-8 x^2-16 e^4-16}{100 x^4-692 x^3+797 x^2+e^8 \left (25 x^2-200 x+400\right )+e^4 \left (100 x^3-746 x^2+1184 x+800\right )+1384 x+400} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {-100 x-25 e^4+171}{-50 x^2+\left (171-25 e^4\right ) x+100 \left (1+e^4\right )}+\frac {1}{4-x}-\frac {2}{2 x+e^4+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (-50 x^2+\left (171-25 e^4\right ) x+100 \left (1+e^4\right )\right )-\log (4-x)-\log \left (2 x+e^4+1\right )\) |
Input:
Int[(-16 - 16*E^4 - 8*x^2)/(400 + 1384*x + 797*x^2 - 692*x^3 + 100*x^4 + E ^8*(400 - 200*x + 25*x^2) + E^4*(800 + 1184*x - 746*x^2 + 100*x^3)),x]
Output:
-Log[4 - x] - Log[1 + E^4 + 2*x] + Log[100*(1 + E^4) + (171 - 25*E^4)*x - 50*x^2]
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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71
method | result | size |
parallelrisch | \(-\ln \left (x -4\right )-\ln \left (x +\frac {{\mathrm e}^{4}}{2}+\frac {1}{2}\right )+\ln \left (\frac {x \,{\mathrm e}^{4}}{2}+x^{2}-2 \,{\mathrm e}^{4}-\frac {171 x}{50}-2\right )\) | \(36\) |
norman | \(-\ln \left (x -4\right )-\ln \left (1+2 x +{\mathrm e}^{4}\right )+\ln \left (25 x \,{\mathrm e}^{4}+50 x^{2}-100 \,{\mathrm e}^{4}-171 x -100\right )\) | \(38\) |
risch | \(\ln \left (-50 x^{2}+\left (-25 \,{\mathrm e}^{4}+171\right ) x +100 \,{\mathrm e}^{4}+100\right )-\ln \left (-2 x^{2}+\left (-{\mathrm e}^{4}+7\right ) x +4 \,{\mathrm e}^{4}+4\right )\) | \(44\) |
default | \(-\ln \left (x -4\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (100 \textit {\_Z}^{3}+\left (100 \,{\mathrm e}^{4}-292\right ) \textit {\_Z}^{2}+\left (-346 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}-371\right ) \textit {\_Z} -200 \,{\mathrm e}^{4}-100 \,{\mathrm e}^{8}-100\right )}{\sum }\frac {\left (100 \textit {\_R} \,{\mathrm e}^{4}+100 \textit {\_R}^{2}+54 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}+100 \textit {\_R} +29\right ) \ln \left (x -\textit {\_R} \right )}{200 \textit {\_R} \,{\mathrm e}^{4}+300 \textit {\_R}^{2}-346 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}-584 \textit {\_R} -371}\right )\) | \(104\) |
Input:
int((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746*x^2+11 84*x+800)*exp(4)+100*x^4-692*x^3+797*x^2+1384*x+400),x,method=_RETURNVERBO SE)
Output:
-ln(x-4)-ln(x+1/2*exp(4)+1/2)+ln(1/2*x*exp(4)+x^2-2*exp(4)-171/50*x-2)
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81 \[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx=\log \left (50 \, x^{2} + 25 \, {\left (x - 4\right )} e^{4} - 171 \, x - 100\right ) - \log \left (2 \, x^{2} + {\left (x - 4\right )} e^{4} - 7 \, x - 4\right ) \] Input:
integrate((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746* x^2+1184*x+800)*exp(4)+100*x^4-692*x^3+797*x^2+1384*x+400),x, algorithm="f ricas")
Output:
log(50*x^2 + 25*(x - 4)*e^4 - 171*x - 100) - log(2*x^2 + (x - 4)*e^4 - 7*x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (17) = 34\).
Time = 0.89 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx=- \log {\left (x^{2} + x \left (- \frac {7}{2} + \frac {e^{4}}{2}\right ) - 2 e^{4} - 2 \right )} + \log {\left (x^{2} + x \left (- \frac {171}{50} + \frac {e^{4}}{2}\right ) - 2 e^{4} - 2 \right )} \] Input:
integrate((-16*exp(4)-8*x**2-16)/((25*x**2-200*x+400)*exp(4)**2+(100*x**3- 746*x**2+1184*x+800)*exp(4)+100*x**4-692*x**3+797*x**2+1384*x+400),x)
Output:
-log(x**2 + x*(-7/2 + exp(4)/2) - 2*exp(4) - 2) + log(x**2 + x*(-171/50 + exp(4)/2) - 2*exp(4) - 2)
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx=\log \left (50 \, x^{2} + x {\left (25 \, e^{4} - 171\right )} - 100 \, e^{4} - 100\right ) - \log \left (2 \, x + e^{4} + 1\right ) - \log \left (x - 4\right ) \] Input:
integrate((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746* x^2+1184*x+800)*exp(4)+100*x^4-692*x^3+797*x^2+1384*x+400),x, algorithm="m axima")
Output:
log(50*x^2 + x*(25*e^4 - 171) - 100*e^4 - 100) - log(2*x + e^4 + 1) - log( x - 4)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx=\log \left ({\left | 50 \, x^{2} + 25 \, x e^{4} - 171 \, x - 100 \, e^{4} - 100 \right |}\right ) - \log \left ({\left | 2 \, x + e^{4} + 1 \right |}\right ) - \log \left ({\left | x - 4 \right |}\right ) \] Input:
integrate((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746* x^2+1184*x+800)*exp(4)+100*x^4-692*x^3+797*x^2+1384*x+400),x, algorithm="g iac")
Output:
log(abs(50*x^2 + 25*x*e^4 - 171*x - 100*e^4 - 100)) - log(abs(2*x + e^4 + 1)) - log(abs(x - 4))
Time = 0.66 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.62 \[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx=-\mathrm {atan}\left (\frac {{\mathrm {e}}^4\,3788800{}\mathrm {i}-{\mathrm {e}}^8\,640000{}\mathrm {i}+x^2\,\left (160000\,{\mathrm {e}}^4-1107200\right )\,2{}\mathrm {i}+x\,\left (40000\,{\mathrm {e}}^8-1833600\,{\mathrm {e}}^4+635200\right )\,2{}\mathrm {i}+4428800{}\mathrm {i}}{392211200\,{\mathrm {e}}^4+76640000\,{\mathrm {e}}^8+4000000\,{\mathrm {e}}^{12}-2\,x^2\,\left (18160000\,{\mathrm {e}}^4+1000000\,{\mathrm {e}}^8+79899200\right )+2\,x\,\left (22893600\,{\mathrm {e}}^4-5620000\,{\mathrm {e}}^8-500000\,{\mathrm {e}}^{12}+276384800\right )+12800\,x^2+319571200}\right )\,2{}\mathrm {i} \] Input:
int(-(16*exp(4) + 8*x^2 + 16)/(1384*x + exp(8)*(25*x^2 - 200*x + 400) + ex p(4)*(1184*x - 746*x^2 + 100*x^3 + 800) + 797*x^2 - 692*x^3 + 100*x^4 + 40 0),x)
Output:
-atan((exp(4)*3788800i - exp(8)*640000i + x^2*(160000*exp(4) - 1107200)*2i + x*(40000*exp(8) - 1833600*exp(4) + 635200)*2i + 4428800i)/(392211200*ex p(4) + 76640000*exp(8) + 4000000*exp(12) - 2*x^2*(18160000*exp(4) + 100000 0*exp(8) + 79899200) + 2*x*(22893600*exp(4) - 5620000*exp(8) - 500000*exp( 12) + 276384800) + 12800*x^2 + 319571200))*2i
\[ \int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 \left (400-200 x+25 x^2\right )+e^4 \left (800+1184 x-746 x^2+100 x^3\right )} \, dx =\text {Too large to display} \] Input:
int((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746*x^2+11 84*x+800)*exp(4)+100*x^4-692*x^3+797*x^2+1384*x+400),x)
Output:
(2*(1250*int(x/(625*e**12*x**2 - 5000*e**12*x + 10000*e**12 + 2500*e**8*x* *3 - 12975*e**8*x**2 - 15800*e**8*x + 110800*e**8 + 2500*e**4*x**4 + 5400* e**4*x**3 - 149417*e**4*x**2 + 303368*e**4*x + 191600*e**4 + 22700*x**4 - 157084*x**3 + 180919*x**2 + 314168*x + 90800),x)*e**12 + 14050*int(x/(625* e**12*x**2 - 5000*e**12*x + 10000*e**12 + 2500*e**8*x**3 - 12975*e**8*x**2 - 15800*e**8*x + 110800*e**8 + 2500*e**4*x**4 + 5400*e**4*x**3 - 149417*e **4*x**2 + 303368*e**4*x + 191600*e**4 + 22700*x**4 - 157084*x**3 + 180919 *x**2 + 314168*x + 90800),x)*e**8 - 52434*int(x/(625*e**12*x**2 - 5000*e** 12*x + 10000*e**12 + 2500*e**8*x**3 - 12975*e**8*x**2 - 15800*e**8*x + 110 800*e**8 + 2500*e**4*x**4 + 5400*e**4*x**3 - 149417*e**4*x**2 + 303368*e** 4*x + 191600*e**4 + 22700*x**4 - 157084*x**3 + 180919*x**2 + 314168*x + 90 800),x)*e**4 - 698706*int(x/(625*e**12*x**2 - 5000*e**12*x + 10000*e**12 + 2500*e**8*x**3 - 12975*e**8*x**2 - 15800*e**8*x + 110800*e**8 + 2500*e**4 *x**4 + 5400*e**4*x**3 - 149417*e**4*x**2 + 303368*e**4*x + 191600*e**4 + 22700*x**4 - 157084*x**3 + 180919*x**2 + 314168*x + 90800),x) - 10000*int( 1/(625*e**12*x**2 - 5000*e**12*x + 10000*e**12 + 2500*e**8*x**3 - 12975*e* *8*x**2 - 15800*e**8*x + 110800*e**8 + 2500*e**4*x**4 + 5400*e**4*x**3 - 1 49417*e**4*x**2 + 303368*e**4*x + 191600*e**4 + 22700*x**4 - 157084*x**3 + 180919*x**2 + 314168*x + 90800),x)*e**12 - 190800*int(1/(625*e**12*x**2 - 5000*e**12*x + 10000*e**12 + 2500*e**8*x**3 - 12975*e**8*x**2 - 15800*...