3.43.8 \(\int \frac {-30 x^4+150 x^5+(-60 x^2+300 x^3) \log (4)+(-30+150 x) \log ^2(4)+e^x (-3 x^4+(-6 x-3 x^2) \log (4))}{8 x^4+8 e^{2 x} x^4-160 x^5+1200 x^6-4000 x^7+5000 x^8+(16 x^2-320 x^3+2400 x^4-8000 x^5+10000 x^6) \log (4)+(8-160 x+1200 x^2-4000 x^3+5000 x^4) \log ^2(4)+e^x (-16 x^4+160 x^5-400 x^6+(-16 x^2+160 x^3-400 x^4) \log (4))} \, dx\)

Optimal. Leaf size=30 \[ \frac {3}{8 \left (-(1-5 x)^2+\frac {e^x}{1+\frac {\log (4)}{x^2}}\right )} \]

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Rubi [F]  time = 5.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-30 x^4+150 x^5+\left (-60 x^2+300 x^3\right ) \log (4)+(-30+150 x) \log ^2(4)+e^x \left (-3 x^4+\left (-6 x-3 x^2\right ) \log (4)\right )}{8 x^4+8 e^{2 x} x^4-160 x^5+1200 x^6-4000 x^7+5000 x^8+\left (16 x^2-320 x^3+2400 x^4-8000 x^5+10000 x^6\right ) \log (4)+\left (8-160 x+1200 x^2-4000 x^3+5000 x^4\right ) \log ^2(4)+e^x \left (-16 x^4+160 x^5-400 x^6+\left (-16 x^2+160 x^3-400 x^4\right ) \log (4)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-30*x^4 + 150*x^5 + (-60*x^2 + 300*x^3)*Log[4] + (-30 + 150*x)*Log[4]^2 + E^x*(-3*x^4 + (-6*x - 3*x^2)*Lo
g[4]))/(8*x^4 + 8*E^(2*x)*x^4 - 160*x^5 + 1200*x^6 - 4000*x^7 + 5000*x^8 + (16*x^2 - 320*x^3 + 2400*x^4 - 8000
*x^5 + 10000*x^6)*Log[4] + (8 - 160*x + 1200*x^2 - 4000*x^3 + 5000*x^4)*Log[4]^2 + E^x*(-16*x^4 + 160*x^5 - 40
0*x^6 + (-16*x^2 + 160*x^3 - 400*x^4)*Log[4])),x]

[Out]

(27*Log[4]^2*Defer[Int][(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x*Log[4] - x^2*(1 + 25*Log[4]))^(-2), x])/8 -
(3*Log[4]^2*Defer[Int][1/(x*(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x*Log[4] - x^2*(1 + 25*Log[4]))^2), x])/
4 - (3*Log[4]*(1 - Log[1024])*Defer[Int][x/(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x*Log[4] - x^2*(1 + 25*Log
[4]))^2, x])/4 - (3*Log[4]*(1 + 5*Log[32])*Defer[Int][x^2/(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x*Log[4] -
x^2*(1 + 25*Log[4]))^2, x])/4 + (105*Log[4]*Defer[Int][x^3/(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x*Log[4] -
x^2*(1 + 25*Log[4]))^2, x])/4 - (3*(11 + 50*Log[4])*Defer[Int][x^4/(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x
*Log[4] - x^2*(1 + 25*Log[4]))^2, x])/8 + (45*Defer[Int][x^5/(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x*Log[4]
- x^2*(1 + 25*Log[4]))^2, x])/2 - (75*Defer[Int][x^6/(E^x*x^2 + 10*x^3 - 25*x^4 - Log[4] + 10*x*Log[4] - x^2*
(1 + 25*Log[4]))^2, x])/8 + (3*Log[4]*Defer[Int][(-(E^x*x^2) - 10*x^3 + 25*x^4 + Log[4] - 10*x*Log[4] + x^2*(1
+ 25*Log[4]))^(-1), x])/8 + (3*Log[16]*Defer[Int][1/(x*(-(E^x*x^2) - 10*x^3 + 25*x^4 + Log[4] - 10*x*Log[4] +
x^2*(1 + 25*Log[4]))), x])/8 + (3*Defer[Int][x^2/(-(E^x*x^2) - 10*x^3 + 25*x^4 + Log[4] - 10*x*Log[4] + x^2*(
1 + 25*Log[4])), x])/8

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-\left (\left (10+e^x\right ) x^4\right )+50 x^5-\left (20+e^x\right ) x^2 \log (4)+100 x^3 \log (4)-2 x \left (e^x-25 \log (4)\right ) \log (4)-10 \log ^2(4)\right )}{8 \left (10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 \left (1-e^x+25 \log (4)\right )\right )^2} \, dx\\ &=\frac {3}{8} \int \frac {-\left (\left (10+e^x\right ) x^4\right )+50 x^5-\left (20+e^x\right ) x^2 \log (4)+100 x^3 \log (4)-2 x \left (e^x-25 \log (4)\right ) \log (4)-10 \log ^2(4)}{\left (10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 \left (1-e^x+25 \log (4)\right )\right )^2} \, dx\\ &=\frac {3}{8} \int \left (\frac {-x^3-x \log (4)-\log (16)}{x \left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )}+\frac {(1-5 x) \left (-11 x^5+5 x^6-12 x^3 \log (4)+10 x^4 \log (4)-2 \log ^2(4)-x \log ^2(4)-2 x^2 \log (4) (1-\log (32))\right )}{x \left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}\right ) \, dx\\ &=\frac {3}{8} \int \frac {-x^3-x \log (4)-\log (16)}{x \left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )} \, dx+\frac {3}{8} \int \frac {(1-5 x) \left (-11 x^5+5 x^6-12 x^3 \log (4)+10 x^4 \log (4)-2 \log ^2(4)-x \log ^2(4)-2 x^2 \log (4) (1-\log (32))\right )}{x \left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx\\ &=\frac {3}{8} \int \left (\frac {x^2}{-e^x x^2-10 x^3+25 x^4+\log (4)-10 x \log (4)+x^2 (1+25 \log (4))}+\frac {\log (4)}{-e^x x^2-10 x^3+25 x^4+\log (4)-10 x \log (4)+x^2 (1+25 \log (4))}+\frac {\log (16)}{x \left (-e^x x^2-10 x^3+25 x^4+\log (4)-10 x \log (4)+x^2 (1+25 \log (4))\right )}\right ) \, dx+\frac {3}{8} \int \left (\frac {60 x^5}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}-\frac {25 x^6}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}+\frac {x^4 (-11-50 \log (4))}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}+\frac {70 x^3 \log (4)}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}+\frac {9 \log ^2(4)}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}-\frac {2 \log ^2(4)}{x \left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}+\frac {2 x^2 \log (4) (-1-5 \log (32))}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}+\frac {2 x \log (4) (-1+\log (1024))}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2}\right ) \, dx\\ &=\frac {3}{8} \int \frac {x^2}{-e^x x^2-10 x^3+25 x^4+\log (4)-10 x \log (4)+x^2 (1+25 \log (4))} \, dx-\frac {75}{8} \int \frac {x^6}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx+\frac {45}{2} \int \frac {x^5}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx+\frac {1}{8} (3 \log (4)) \int \frac {1}{-e^x x^2-10 x^3+25 x^4+\log (4)-10 x \log (4)+x^2 (1+25 \log (4))} \, dx+\frac {1}{4} (105 \log (4)) \int \frac {x^3}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx-\frac {1}{4} \left (3 \log ^2(4)\right ) \int \frac {1}{x \left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx+\frac {1}{8} \left (27 \log ^2(4)\right ) \int \frac {1}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx-\frac {1}{8} (3 (11+50 \log (4))) \int \frac {x^4}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx+\frac {1}{8} (3 \log (16)) \int \frac {1}{x \left (-e^x x^2-10 x^3+25 x^4+\log (4)-10 x \log (4)+x^2 (1+25 \log (4))\right )} \, dx-\frac {1}{4} (3 \log (4) (1+5 \log (32))) \int \frac {x^2}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx-\frac {1}{4} (3 \log (4) (1-\log (1024))) \int \frac {x}{\left (e^x x^2+10 x^3-25 x^4-\log (4)+10 x \log (4)-x^2 (1+25 \log (4))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 45, normalized size = 1.50 \begin {gather*} -\frac {3 \left (x^2+\log (4)\right )}{8 \left (-10 x^3+25 x^4+\log (4)-10 x \log (4)+x^2 \left (1-e^x+25 \log (4)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-30*x^4 + 150*x^5 + (-60*x^2 + 300*x^3)*Log[4] + (-30 + 150*x)*Log[4]^2 + E^x*(-3*x^4 + (-6*x - 3*x
^2)*Log[4]))/(8*x^4 + 8*E^(2*x)*x^4 - 160*x^5 + 1200*x^6 - 4000*x^7 + 5000*x^8 + (16*x^2 - 320*x^3 + 2400*x^4
- 8000*x^5 + 10000*x^6)*Log[4] + (8 - 160*x + 1200*x^2 - 4000*x^3 + 5000*x^4)*Log[4]^2 + E^x*(-16*x^4 + 160*x^
5 - 400*x^6 + (-16*x^2 + 160*x^3 - 400*x^4)*Log[4])),x]

[Out]

(-3*(x^2 + Log[4]))/(8*(-10*x^3 + 25*x^4 + Log[4] - 10*x*Log[4] + x^2*(1 - E^x + 25*Log[4])))

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fricas [A]  time = 0.60, size = 47, normalized size = 1.57 \begin {gather*} -\frac {3 \, {\left (x^{2} + 2 \, \log \relax (2)\right )}}{8 \, {\left (25 \, x^{4} - 10 \, x^{3} - x^{2} e^{x} + x^{2} + 2 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-3*x^2-6*x)*log(2)-3*x^4)*exp(x)+4*(150*x-30)*log(2)^2+2*(300*x^3-60*x^2)*log(2)+150*x^5-30*x^4
)/(8*exp(x)^2*x^4+(2*(-400*x^4+160*x^3-16*x^2)*log(2)-400*x^6+160*x^5-16*x^4)*exp(x)+4*(5000*x^4-4000*x^3+1200
*x^2-160*x+8)*log(2)^2+2*(10000*x^6-8000*x^5+2400*x^4-320*x^3+16*x^2)*log(2)+5000*x^8-4000*x^7+1200*x^6-160*x^
5+8*x^4),x, algorithm="fricas")

[Out]

-3/8*(x^2 + 2*log(2))/(25*x^4 - 10*x^3 - x^2*e^x + x^2 + 2*(25*x^2 - 10*x + 1)*log(2))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-3*x^2-6*x)*log(2)-3*x^4)*exp(x)+4*(150*x-30)*log(2)^2+2*(300*x^3-60*x^2)*log(2)+150*x^5-30*x^4
)/(8*exp(x)^2*x^4+(2*(-400*x^4+160*x^3-16*x^2)*log(2)-400*x^6+160*x^5-16*x^4)*exp(x)+4*(5000*x^4-4000*x^3+1200
*x^2-160*x+8)*log(2)^2+2*(10000*x^6-8000*x^5+2400*x^4-320*x^3+16*x^2)*log(2)+5000*x^8-4000*x^7+1200*x^6-160*x^
5+8*x^4),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.46, size = 50, normalized size = 1.67

method	result	size
risch	\(-\frac {3 \left (x^{2}+2 \ln \relax (2)\right )}{8 \left (25 x^{4}+50 x^{2} \ln \relax (2)-10 x^{3}-{\mathrm e}^{x} x^{2}-20 x \ln \relax (2)+x^{2}+2 \ln \relax (2)\right )}\)	\(50\)
norman	\(\frac {-\frac {3 x^{2}}{8}-\frac {3 \ln \relax (2)}{4}}{25 x^{4}+50 x^{2} \ln \relax (2)-10 x^{3}-{\mathrm e}^{x} x^{2}-20 x \ln \relax (2)+x^{2}+2 \ln \relax (2)}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*(-3*x^2-6*x)*ln(2)-3*x^4)*exp(x)+4*(150*x-30)*ln(2)^2+2*(300*x^3-60*x^2)*ln(2)+150*x^5-30*x^4)/(8*exp(
x)^2*x^4+(2*(-400*x^4+160*x^3-16*x^2)*ln(2)-400*x^6+160*x^5-16*x^4)*exp(x)+4*(5000*x^4-4000*x^3+1200*x^2-160*x
+8)*ln(2)^2+2*(10000*x^6-8000*x^5+2400*x^4-320*x^3+16*x^2)*ln(2)+5000*x^8-4000*x^7+1200*x^6-160*x^5+8*x^4),x,m
ethod=_RETURNVERBOSE)

[Out]

-3/8*(x^2+2*ln(2))/(25*x^4+50*x^2*ln(2)-10*x^3-exp(x)*x^2-20*x*ln(2)+x^2+2*ln(2))

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maxima [A]  time = 0.51, size = 49, normalized size = 1.63 \begin {gather*} -\frac {3 \, {\left (x^{2} + 2 \, \log \relax (2)\right )}}{8 \, {\left (25 \, x^{4} - 10 \, x^{3} + x^{2} {\left (50 \, \log \relax (2) + 1\right )} - x^{2} e^{x} - 20 \, x \log \relax (2) + 2 \, \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-3*x^2-6*x)*log(2)-3*x^4)*exp(x)+4*(150*x-30)*log(2)^2+2*(300*x^3-60*x^2)*log(2)+150*x^5-30*x^4
)/(8*exp(x)^2*x^4+(2*(-400*x^4+160*x^3-16*x^2)*log(2)-400*x^6+160*x^5-16*x^4)*exp(x)+4*(5000*x^4-4000*x^3+1200
*x^2-160*x+8)*log(2)^2+2*(10000*x^6-8000*x^5+2400*x^4-320*x^3+16*x^2)*log(2)+5000*x^8-4000*x^7+1200*x^6-160*x^
5+8*x^4),x, algorithm="maxima")

[Out]

-3/8*(x^2 + 2*log(2))/(25*x^4 - 10*x^3 + x^2*(50*log(2) + 1) - x^2*e^x - 20*x*log(2) + 2*log(2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (3\,x^2+6\,x\right )+3\,x^4\right )-4\,{\ln \relax (2)}^2\,\left (150\,x-30\right )+2\,\ln \relax (2)\,\left (60\,x^2-300\,x^3\right )+30\,x^4-150\,x^5}{4\,{\ln \relax (2)}^2\,\left (5000\,x^4-4000\,x^3+1200\,x^2-160\,x+8\right )-{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (400\,x^4-160\,x^3+16\,x^2\right )+16\,x^4-160\,x^5+400\,x^6\right )+2\,\ln \relax (2)\,\left (10000\,x^6-8000\,x^5+2400\,x^4-320\,x^3+16\,x^2\right )+8\,x^4\,{\mathrm {e}}^{2\,x}+8\,x^4-160\,x^5+1200\,x^6-4000\,x^7+5000\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(2*log(2)*(6*x + 3*x^2) + 3*x^4) - 4*log(2)^2*(150*x - 30) + 2*log(2)*(60*x^2 - 300*x^3) + 30*x^4
- 150*x^5)/(4*log(2)^2*(1200*x^2 - 160*x - 4000*x^3 + 5000*x^4 + 8) - exp(x)*(2*log(2)*(16*x^2 - 160*x^3 + 40
0*x^4) + 16*x^4 - 160*x^5 + 400*x^6) + 2*log(2)*(16*x^2 - 320*x^3 + 2400*x^4 - 8000*x^5 + 10000*x^6) + 8*x^4*e
xp(2*x) + 8*x^4 - 160*x^5 + 1200*x^6 - 4000*x^7 + 5000*x^8),x)

[Out]

int(-(exp(x)*(2*log(2)*(6*x + 3*x^2) + 3*x^4) - 4*log(2)^2*(150*x - 30) + 2*log(2)*(60*x^2 - 300*x^3) + 30*x^4
- 150*x^5)/(4*log(2)^2*(1200*x^2 - 160*x - 4000*x^3 + 5000*x^4 + 8) - exp(x)*(2*log(2)*(16*x^2 - 160*x^3 + 40
0*x^4) + 16*x^4 - 160*x^5 + 400*x^6) + 2*log(2)*(16*x^2 - 320*x^3 + 2400*x^4 - 8000*x^5 + 10000*x^6) + 8*x^4*e
xp(2*x) + 8*x^4 - 160*x^5 + 1200*x^6 - 4000*x^7 + 5000*x^8), x)

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sympy [B]  time = 0.30, size = 53, normalized size = 1.77 \begin {gather*} \frac {3 x^{2} + 6 \log {\relax (2 )}}{- 200 x^{4} + 80 x^{3} + 8 x^{2} e^{x} - 400 x^{2} \log {\relax (2 )} - 8 x^{2} + 160 x \log {\relax (2 )} - 16 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-3*x**2-6*x)*ln(2)-3*x**4)*exp(x)+4*(150*x-30)*ln(2)**2+2*(300*x**3-60*x**2)*ln(2)+150*x**5-30*
x**4)/(8*exp(x)**2*x**4+(2*(-400*x**4+160*x**3-16*x**2)*ln(2)-400*x**6+160*x**5-16*x**4)*exp(x)+4*(5000*x**4-4
000*x**3+1200*x**2-160*x+8)*ln(2)**2+2*(10000*x**6-8000*x**5+2400*x**4-320*x**3+16*x**2)*ln(2)+5000*x**8-4000*
x**7+1200*x**6-160*x**5+8*x**4),x)

[Out]

(3*x**2 + 6*log(2))/(-200*x**4 + 80*x**3 + 8*x**2*exp(x) - 400*x**2*log(2) - 8*x**2 + 160*x*log(2) - 16*log(2)
)

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