3.77.76 \(\int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+(4 x^3+4 e^4 x^3+4 x^4) \log (1+e^4+x)}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10})+(1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8)) \log (1+e^4+x)+(640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 (640 x^3-1920 x^4+1920 x^5-640 x^6)) \log ^2(1+e^4+x)+(160 x^2-160 x^3-160 x^4+160 x^5+e^4 (160 x^2-320 x^3+160 x^4)) \log ^3(1+e^4+x)+(20 x-20 x^3+e^4 (20 x-20 x^2)) \log ^4(1+e^4+x)+(1+e^4+x) \log ^5(1+e^4+x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^4}{\left (4 \left (x-x^2\right )+\log \left (1+e^4+x\right )\right )^4} \]

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Rubi [F]  time = 2.00, 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 {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x^4 + 16*x^5 + 16*E^4*x^5 + 16*x^6 + (4*x^3 + 4*E^4*x^3 + 4*x^4)*Log[1 + E^4 + x])/(1024*x^5 - 4096*x^
6 + 5120*x^7 - 5120*x^9 + 4096*x^10 - 1024*x^11 + E^4*(1024*x^5 - 5120*x^6 + 10240*x^7 - 10240*x^8 + 5120*x^9
- 1024*x^10) + (1280*x^4 - 3840*x^5 + 2560*x^6 + 2560*x^7 - 3840*x^8 + 1280*x^9 + E^4*(1280*x^4 - 5120*x^5 + 7
680*x^6 - 5120*x^7 + 1280*x^8))*Log[1 + E^4 + x] + (640*x^3 - 1280*x^4 + 1280*x^6 - 640*x^7 + E^4*(640*x^3 - 1
920*x^4 + 1920*x^5 - 640*x^6))*Log[1 + E^4 + x]^2 + (160*x^2 - 160*x^3 - 160*x^4 + 160*x^5 + E^4*(160*x^2 - 32
0*x^3 + 160*x^4))*Log[1 + E^4 + x]^3 + (20*x - 20*x^3 + E^4*(20*x - 20*x^2))*Log[1 + E^4 + x]^4 + (1 + E^4 + x
)*Log[1 + E^4 + x]^5),x]

[Out]

-4*(1 + E^4)^3*Defer[Int][(-4*x + 4*x^2 - Log[1 + E^4 + x])^(-5), x] + 4*(1 + E^4)^2*Defer[Int][x/(-4*x + 4*x^
2 - Log[1 + E^4 + x])^5, x] - 4*(1 + E^4)*Defer[Int][x^2/(-4*x + 4*x^2 - Log[1 + E^4 + x])^5, x] + 4*Defer[Int
][x^3/(-4*x + 4*x^2 - Log[1 + E^4 + x])^5, x] + 16*Defer[Int][x^4/(-4*x + 4*x^2 - Log[1 + E^4 + x])^5, x] - 32
*Defer[Int][x^5/(-4*x + 4*x^2 - Log[1 + E^4 + x])^5, x] + 4*(1 + E^4)^4*Defer[Int][1/((1 + E^4 + x)*(-4*x + 4*
x^2 - Log[1 + E^4 + x])^5), x] + 4*Defer[Int][x^3/(-4*x + 4*x^2 - Log[1 + E^4 + x])^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x^4+\left (16+16 e^4\right ) x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx\\ &=\int \frac {4 x^3 \left (-x \left (-1+4 \left (1+e^4\right ) x+4 x^2\right )-\left (1+e^4+x\right ) \log \left (1+e^4+x\right )\right )}{\left (1+e^4+x\right ) \left (4 (-1+x) x-\log \left (1+e^4+x\right )\right )^5} \, dx\\ &=4 \int \frac {x^3 \left (-x \left (-1+4 \left (1+e^4\right ) x+4 x^2\right )-\left (1+e^4+x\right ) \log \left (1+e^4+x\right )\right )}{\left (1+e^4+x\right ) \left (4 (-1+x) x-\log \left (1+e^4+x\right )\right )^5} \, dx\\ &=4 \int \left (\frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4}+\frac {x^4 \left (-5-4 e^4+4 \left (1+2 e^4\right ) x+8 x^2\right )}{\left (1+e^4+x\right ) \left (4 x-4 x^2+\log \left (1+e^4+x\right )\right )^5}\right ) \, dx\\ &=4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4} \, dx+4 \int \frac {x^4 \left (-5-4 e^4+4 \left (1+2 e^4\right ) x+8 x^2\right )}{\left (1+e^4+x\right ) \left (4 x-4 x^2+\log \left (1+e^4+x\right )\right )^5} \, dx\\ &=4 \int \left (-\frac {\left (1+e^4\right )^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {\left (1+e^4\right )^2 x}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}-\frac {\left (1+e^4\right ) x^2}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {4 x^4}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}-\frac {8 x^5}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {\left (1+e^4\right )^4}{\left (1+e^4+x\right ) \left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}\right ) \, dx+4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4} \, dx\\ &=4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx+4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4} \, dx+16 \int \frac {x^4}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx-32 \int \frac {x^5}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx-\left (4 \left (1+e^4\right )\right ) \int \frac {x^2}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx+\left (4 \left (1+e^4\right )^2\right ) \int \frac {x}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx-\left (4 \left (1+e^4\right )^3\right ) \int \frac {1}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx+\left (4 \left (1+e^4\right )^4\right ) \int \frac {1}{\left (1+e^4+x\right ) \left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 4.27, size = 20, normalized size = 0.87 \begin {gather*} \frac {x^4}{\left (-4 (-1+x) x+\log \left (1+e^4+x\right )\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^4 + 16*x^5 + 16*E^4*x^5 + 16*x^6 + (4*x^3 + 4*E^4*x^3 + 4*x^4)*Log[1 + E^4 + x])/(1024*x^5 - 4
096*x^6 + 5120*x^7 - 5120*x^9 + 4096*x^10 - 1024*x^11 + E^4*(1024*x^5 - 5120*x^6 + 10240*x^7 - 10240*x^8 + 512
0*x^9 - 1024*x^10) + (1280*x^4 - 3840*x^5 + 2560*x^6 + 2560*x^7 - 3840*x^8 + 1280*x^9 + E^4*(1280*x^4 - 5120*x
^5 + 7680*x^6 - 5120*x^7 + 1280*x^8))*Log[1 + E^4 + x] + (640*x^3 - 1280*x^4 + 1280*x^6 - 640*x^7 + E^4*(640*x
^3 - 1920*x^4 + 1920*x^5 - 640*x^6))*Log[1 + E^4 + x]^2 + (160*x^2 - 160*x^3 - 160*x^4 + 160*x^5 + E^4*(160*x^
2 - 320*x^3 + 160*x^4))*Log[1 + E^4 + x]^3 + (20*x - 20*x^3 + E^4*(20*x - 20*x^2))*Log[1 + E^4 + x]^4 + (1 + E
^4 + x)*Log[1 + E^4 + x]^5),x]

[Out]

x^4/(-4*(-1 + x)*x + Log[1 + E^4 + x])^4

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fricas [B]  time = 1.46, size = 106, normalized size = 4.61 \begin {gather*} \frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} + 1536 \, x^{6} - 1024 \, x^{5} + 256 \, x^{4} - 16 \, {\left (x^{2} - x\right )} \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4} + 96 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (x + e^{4} + 1\right )^{2} - 256 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} \log \left (x + e^{4} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(4)+4*x^4+4*x^3)*log(exp(4)+x+1)+16*x^5*exp(4)+16*x^6+16*x^5-4*x^4)/((exp(4)+x+1)*log(exp
(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x^3+20*x)*log(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160
*x^4-160*x^3+160*x^2)*log(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3)*exp(4)-640*x^7+1280*x^6-1280*x^4
+640*x^3)*log(exp(4)+x+1)^2+((1280*x^8-5120*x^7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+
2560*x^6-3840*x^5+1280*x^4)*log(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+10240*x^7-5120*x^6+1024*x^5)*exp(4)
-1024*x^11+4096*x^10-5120*x^9+5120*x^7-4096*x^6+1024*x^5),x, algorithm="fricas")

[Out]

x^4/(256*x^8 - 1024*x^7 + 1536*x^6 - 1024*x^5 + 256*x^4 - 16*(x^2 - x)*log(x + e^4 + 1)^3 + log(x + e^4 + 1)^4
+ 96*(x^4 - 2*x^3 + x^2)*log(x + e^4 + 1)^2 - 256*(x^6 - 3*x^5 + 3*x^4 - x^3)*log(x + e^4 + 1))

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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(((4*x^3*exp(4)+4*x^4+4*x^3)*log(exp(4)+x+1)+16*x^5*exp(4)+16*x^6+16*x^5-4*x^4)/((exp(4)+x+1)*log(exp
(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x^3+20*x)*log(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160
*x^4-160*x^3+160*x^2)*log(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3)*exp(4)-640*x^7+1280*x^6-1280*x^4
+640*x^3)*log(exp(4)+x+1)^2+((1280*x^8-5120*x^7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+
2560*x^6-3840*x^5+1280*x^4)*log(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+10240*x^7-5120*x^6+1024*x^5)*exp(4)
-1024*x^11+4096*x^10-5120*x^9+5120*x^7-4096*x^6+1024*x^5),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.10, size = 24, normalized size = 1.04

method	result	size
risch	\(\frac {x^{4}}{\left (4 x^{2}-4 x -\ln \left ({\mathrm e}^{4}+x +1\right )\right )^{4}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3*exp(4)+4*x^4+4*x^3)*ln(exp(4)+x+1)+16*x^5*exp(4)+16*x^6+16*x^5-4*x^4)/((exp(4)+x+1)*ln(exp(4)+x+1)
^5+((-20*x^2+20*x)*exp(4)-20*x^3+20*x)*ln(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160*x^4-160*
x^3+160*x^2)*ln(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3)*exp(4)-640*x^7+1280*x^6-1280*x^4+640*x^3)*
ln(exp(4)+x+1)^2+((1280*x^8-5120*x^7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+2560*x^6-38
40*x^5+1280*x^4)*ln(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+10240*x^7-5120*x^6+1024*x^5)*exp(4)-1024*x^11+4
096*x^10-5120*x^9+5120*x^7-4096*x^6+1024*x^5),x,method=_RETURNVERBOSE)

[Out]

x^4/(4*x^2-4*x-ln(exp(4)+x+1))^4

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maxima [B]  time = 0.62, size = 106, normalized size = 4.61 \begin {gather*} \frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} + 1536 \, x^{6} - 1024 \, x^{5} + 256 \, x^{4} - 16 \, {\left (x^{2} - x\right )} \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4} + 96 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (x + e^{4} + 1\right )^{2} - 256 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} \log \left (x + e^{4} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(4)+4*x^4+4*x^3)*log(exp(4)+x+1)+16*x^5*exp(4)+16*x^6+16*x^5-4*x^4)/((exp(4)+x+1)*log(exp
(4)+x+1)^5+((-20*x^2+20*x)*exp(4)-20*x^3+20*x)*log(exp(4)+x+1)^4+((160*x^4-320*x^3+160*x^2)*exp(4)+160*x^5-160
*x^4-160*x^3+160*x^2)*log(exp(4)+x+1)^3+((-640*x^6+1920*x^5-1920*x^4+640*x^3)*exp(4)-640*x^7+1280*x^6-1280*x^4
+640*x^3)*log(exp(4)+x+1)^2+((1280*x^8-5120*x^7+7680*x^6-5120*x^5+1280*x^4)*exp(4)+1280*x^9-3840*x^8+2560*x^7+
2560*x^6-3840*x^5+1280*x^4)*log(exp(4)+x+1)+(-1024*x^10+5120*x^9-10240*x^8+10240*x^7-5120*x^6+1024*x^5)*exp(4)
-1024*x^11+4096*x^10-5120*x^9+5120*x^7-4096*x^6+1024*x^5),x, algorithm="maxima")

[Out]

x^4/(256*x^8 - 1024*x^7 + 1536*x^6 - 1024*x^5 + 256*x^4 - 16*(x^2 - x)*log(x + e^4 + 1)^3 + log(x + e^4 + 1)^4
+ 96*(x^4 - 2*x^3 + x^2)*log(x + e^4 + 1)^2 - 256*(x^6 - 3*x^5 + 3*x^4 - x^3)*log(x + e^4 + 1))

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + exp(4) + 1)*(4*x^3*exp(4) + 4*x^3 + 4*x^4) + 16*x^5*exp(4) - 4*x^4 + 16*x^5 + 16*x^6)/(log(x + ex
p(4) + 1)^4*(20*x + exp(4)*(20*x - 20*x^2) - 20*x^3) + log(x + exp(4) + 1)^3*(exp(4)*(160*x^2 - 320*x^3 + 160*
x^4) + 160*x^2 - 160*x^3 - 160*x^4 + 160*x^5) + exp(4)*(1024*x^5 - 5120*x^6 + 10240*x^7 - 10240*x^8 + 5120*x^9
- 1024*x^10) + log(x + exp(4) + 1)^2*(640*x^3 - 1280*x^4 + 1280*x^6 - 640*x^7 + exp(4)*(640*x^3 - 1920*x^4 +
1920*x^5 - 640*x^6)) + log(x + exp(4) + 1)^5*(x + exp(4) + 1) + 1024*x^5 - 4096*x^6 + 5120*x^7 - 5120*x^9 + 40
96*x^10 - 1024*x^11 + log(x + exp(4) + 1)*(exp(4)*(1280*x^4 - 5120*x^5 + 7680*x^6 - 5120*x^7 + 1280*x^8) + 128
0*x^4 - 3840*x^5 + 2560*x^6 + 2560*x^7 - 3840*x^8 + 1280*x^9)),x)

[Out]

\text{Hanged}

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sympy [B]  time = 0.45, size = 110, normalized size = 4.78 \begin {gather*} \frac {x^{4}}{256 x^{8} - 1024 x^{7} + 1536 x^{6} - 1024 x^{5} + 256 x^{4} + \left (- 16 x^{2} + 16 x\right ) \log {\left (x + 1 + e^{4} \right )}^{3} + \left (96 x^{4} - 192 x^{3} + 96 x^{2}\right ) \log {\left (x + 1 + e^{4} \right )}^{2} + \left (- 256 x^{6} + 768 x^{5} - 768 x^{4} + 256 x^{3}\right ) \log {\left (x + 1 + e^{4} \right )} + \log {\left (x + 1 + e^{4} \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3*exp(4)+4*x**4+4*x**3)*ln(exp(4)+x+1)+16*x**5*exp(4)+16*x**6+16*x**5-4*x**4)/((exp(4)+x+1)*l
n(exp(4)+x+1)**5+((-20*x**2+20*x)*exp(4)-20*x**3+20*x)*ln(exp(4)+x+1)**4+((160*x**4-320*x**3+160*x**2)*exp(4)+
160*x**5-160*x**4-160*x**3+160*x**2)*ln(exp(4)+x+1)**3+((-640*x**6+1920*x**5-1920*x**4+640*x**3)*exp(4)-640*x*
*7+1280*x**6-1280*x**4+640*x**3)*ln(exp(4)+x+1)**2+((1280*x**8-5120*x**7+7680*x**6-5120*x**5+1280*x**4)*exp(4)
+1280*x**9-3840*x**8+2560*x**7+2560*x**6-3840*x**5+1280*x**4)*ln(exp(4)+x+1)+(-1024*x**10+5120*x**9-10240*x**8
+10240*x**7-5120*x**6+1024*x**5)*exp(4)-1024*x**11+4096*x**10-5120*x**9+5120*x**7-4096*x**6+1024*x**5),x)

[Out]

x**4/(256*x**8 - 1024*x**7 + 1536*x**6 - 1024*x**5 + 256*x**4 + (-16*x**2 + 16*x)*log(x + 1 + exp(4))**3 + (96
*x**4 - 192*x**3 + 96*x**2)*log(x + 1 + exp(4))**2 + (-256*x**6 + 768*x**5 - 768*x**4 + 256*x**3)*log(x + 1 +
exp(4)) + log(x + 1 + exp(4))**4)

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