\(\int \frac {e^x (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4))}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+(8 x^2-8 x^7+2 x^{12}) \log (4)+\log ^2(4)} \, dx\) [6401]

Optimal result

Integrand size = 82, antiderivative size = 20 \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {e^x}{x^2 \left (-2+x^5\right )^2+\log (4)} \]

[Out]

exp(x)/(x^2*(x^5-2)^2+2*ln(2))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(20)=40\).

Time = 0.63 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.40, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2326} \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {e^x \left (x^{12}-4 x^7+4 x^2+\log (4)\right )}{x^{24}-8 x^{19}+24 x^{14}-32 x^9+16 x^4+2 \left (x^{12}-4 x^7+4 x^2\right ) \log (4)+\log ^2(4)} \]

[In]

Int[(E^x*(-8*x + 4*x^2 + 28*x^6 - 4*x^7 - 12*x^11 + x^12 + Log[4]))/(16*x^4 - 32*x^9 + 24*x^14 - 8*x^19 + x^24
+ (8*x^2 - 8*x^7 + 2*x^12)*Log[4] + Log[4]^2),x]

[Out]

(E^x*(4*x^2 - 4*x^7 + x^12 + Log[4]))/(16*x^4 - 32*x^9 + 24*x^14 - 8*x^19 + x^24 + 2*(4*x^2 - 4*x^7 + x^12)*Lo
g[4] + Log[4]^2)

Rule 2326

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]

Rubi steps \begin{align*} \text {integral}& = \frac {e^x \left (4 x^2-4 x^7+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+2 \left (4 x^2-4 x^7+x^{12}\right ) \log (4)+\log ^2(4)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {e^x}{4 x^2-4 x^7+x^{12}+\log (4)} \]

[In]

Integrate[(E^x*(-8*x + 4*x^2 + 28*x^6 - 4*x^7 - 12*x^11 + x^12 + Log[4]))/(16*x^4 - 32*x^9 + 24*x^14 - 8*x^19
+ x^24 + (8*x^2 - 8*x^7 + 2*x^12)*Log[4] + Log[4]^2),x]

[Out]

E^x/(4*x^2 - 4*x^7 + x^12 + Log[4])

Maple [A] (verified)

Time = 2.92 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20

method	result	size
gosper	\(\frac {{\mathrm e}^{x}}{x^{12}-4 x^{7}+4 x^{2}+2 \ln \left (2\right )}\)	\(24\)
risch	\(\frac {{\mathrm e}^{x}}{x^{12}-4 x^{7}+4 x^{2}+2 \ln \left (2\right )}\)	\(24\)
parallelrisch	\(\frac {{\mathrm e}^{x}}{x^{12}-4 x^{7}+4 x^{2}+2 \ln \left (2\right )}\)	\(24\)
default	\(\text {Expression too large to display}\)	\(2854\)

[In]

int((2*ln(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*ln(2)^2+2*(2*x^12-8*x^7+8*x^2)*ln(2)+x^24-8*x^19+2
4*x^14-32*x^9+16*x^4),x,method=_RETURNVERBOSE)

[Out]

exp(x)/(x^12-4*x^7+4*x^2+2*ln(2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {e^{x}}{x^{12} - 4 \, x^{7} + 4 \, x^{2} + 2 \, \log \left (2\right )} \]

[In]

integrate((2*log(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*log(2)^2+2*(2*x^12-8*x^7+8*x^2)*log(2)+x^24
-8*x^19+24*x^14-32*x^9+16*x^4),x, algorithm="fricas")

[Out]

e^x/(x^12 - 4*x^7 + 4*x^2 + 2*log(2))

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {e^{x}}{x^{12} - 4 x^{7} + 4 x^{2} + 2 \log {\left (2 \right )}} \]

[In]

integrate((2*ln(2)+x**12-12*x**11-4*x**7+28*x**6+4*x**2-8*x)*exp(x)/(4*ln(2)**2+2*(2*x**12-8*x**7+8*x**2)*ln(2
)+x**24-8*x**19+24*x**14-32*x**9+16*x**4),x)

[Out]

exp(x)/(x**12 - 4*x**7 + 4*x**2 + 2*log(2))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {e^{x}}{x^{12} - 4 \, x^{7} + 4 \, x^{2} + 2 \, \log \left (2\right )} \]

[In]

integrate((2*log(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*log(2)^2+2*(2*x^12-8*x^7+8*x^2)*log(2)+x^24
-8*x^19+24*x^14-32*x^9+16*x^4),x, algorithm="maxima")

[Out]

e^x/(x^12 - 4*x^7 + 4*x^2 + 2*log(2))

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {e^{x}}{x^{12} - 4 \, x^{7} + 4 \, x^{2} + 2 \, \log \left (2\right )} \]

[In]

integrate((2*log(2)+x^12-12*x^11-4*x^7+28*x^6+4*x^2-8*x)*exp(x)/(4*log(2)^2+2*(2*x^12-8*x^7+8*x^2)*log(2)+x^24
-8*x^19+24*x^14-32*x^9+16*x^4),x, algorithm="giac")

[Out]

e^x/(x^12 - 4*x^7 + 4*x^2 + 2*log(2))

Mupad [B] (verification not implemented)

Time = 12.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {e^x \left (-8 x+4 x^2+28 x^6-4 x^7-12 x^{11}+x^{12}+\log (4)\right )}{16 x^4-32 x^9+24 x^{14}-8 x^{19}+x^{24}+\left (8 x^2-8 x^7+2 x^{12}\right ) \log (4)+\log ^2(4)} \, dx=\frac {{\mathrm {e}}^x}{x^{12}-4\,x^7+4\,x^2+\ln \left (4\right )} \]

[In]

int((exp(x)*(2*log(2) - 8*x + 4*x^2 + 28*x^6 - 4*x^7 - 12*x^11 + x^12))/(4*log(2)^2 + 2*log(2)*(8*x^2 - 8*x^7
+ 2*x^12) + 16*x^4 - 32*x^9 + 24*x^14 - 8*x^19 + x^24),x)

[Out]

exp(x)/(log(4) + 4*x^2 - 4*x^7 + x^12)