3.10.16 \(\int \frac {38+8 e^{2 x}+29 x+8 x^2+e^x (35+10 x)+(38+8 e^{2 x}+35 x+8 x^2+e^x (35+16 x)) \log (\frac {361 x^4+64 e^{2 x} x^4+304 x^5+64 x^6+e^x (304 x^4+128 x^5)}{16+4 e^{2 x}+16 x+4 x^2+e^x (16+8 x)})}{38+8 e^{2 x}+35 x+8 x^2+e^x (35+16 x)} \, dx\)

Optimal. Leaf size=25 \[ x \left (-3+\log \left (x^4 \left (-4-\frac {3}{2 \left (2+e^x+x\right )}\right )^2\right )\right ) \]

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Rubi [A]  time = 1.52, antiderivative size = 34, normalized size of antiderivative = 1.36, number of steps used = 15, number of rules used = 4, integrand size = 153, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6688, 6742, 2548, 12} \begin {gather*} x \log \left (\frac {x^4 \left (8 x+8 e^x+19\right )^2}{4 \left (x+e^x+2\right )^2}\right )-3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(38 + 8*E^(2*x) + 29*x + 8*x^2 + E^x*(35 + 10*x) + (38 + 8*E^(2*x) + 35*x + 8*x^2 + E^x*(35 + 16*x))*Log[(
361*x^4 + 64*E^(2*x)*x^4 + 304*x^5 + 64*x^6 + E^x*(304*x^4 + 128*x^5))/(16 + 4*E^(2*x) + 16*x + 4*x^2 + E^x*(1
6 + 8*x))])/(38 + 8*E^(2*x) + 35*x + 8*x^2 + E^x*(35 + 16*x)),x]

[Out]

-3*x + x*Log[(x^4*(19 + 8*E^x + 8*x)^2)/(4*(2 + E^x + x)^2)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {38+8 e^{2 x}+29 x+8 x^2+5 e^x (7+2 x)+\left (38+8 e^{2 x}+35 x+8 x^2+e^x (35+16 x)\right ) \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )}{38+8 e^{2 x}+35 x+8 x^2+e^x (35+16 x)} \, dx\\ &=\int \left (1+\frac {2 x (1+x)}{2+e^x+x}-\frac {2 x (11+8 x)}{19+8 e^x+8 x}+\log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )\right ) \, dx\\ &=x+2 \int \frac {x (1+x)}{2+e^x+x} \, dx-2 \int \frac {x (11+8 x)}{19+8 e^x+8 x} \, dx+\int \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right ) \, dx\\ &=x+x \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )+2 \int \left (\frac {x}{2+e^x+x}+\frac {x^2}{2+e^x+x}\right ) \, dx-2 \int \left (\frac {11 x}{19+8 e^x+8 x}+\frac {8 x^2}{19+8 e^x+8 x}\right ) \, dx-\int \frac {2 \left (76+16 e^{2 x}+67 x+16 x^2+e^x (70+29 x)\right )}{\left (2+e^x+x\right ) \left (19+8 e^x+8 x\right )} \, dx\\ &=x+x \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )+2 \int \frac {x}{2+e^x+x} \, dx+2 \int \frac {x^2}{2+e^x+x} \, dx-2 \int \frac {76+16 e^{2 x}+67 x+16 x^2+e^x (70+29 x)}{\left (2+e^x+x\right ) \left (19+8 e^x+8 x\right )} \, dx-16 \int \frac {x^2}{19+8 e^x+8 x} \, dx-22 \int \frac {x}{19+8 e^x+8 x} \, dx\\ &=x+x \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )+2 \int \frac {x}{2+e^x+x} \, dx+2 \int \frac {x^2}{2+e^x+x} \, dx-2 \int \left (2+\frac {x (1+x)}{2+e^x+x}-\frac {x (11+8 x)}{19+8 e^x+8 x}\right ) \, dx-16 \int \frac {x^2}{19+8 e^x+8 x} \, dx-22 \int \frac {x}{19+8 e^x+8 x} \, dx\\ &=-3 x+x \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )+2 \int \frac {x}{2+e^x+x} \, dx+2 \int \frac {x^2}{2+e^x+x} \, dx-2 \int \frac {x (1+x)}{2+e^x+x} \, dx+2 \int \frac {x (11+8 x)}{19+8 e^x+8 x} \, dx-16 \int \frac {x^2}{19+8 e^x+8 x} \, dx-22 \int \frac {x}{19+8 e^x+8 x} \, dx\\ &=-3 x+x \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )+2 \int \frac {x}{2+e^x+x} \, dx+2 \int \frac {x^2}{2+e^x+x} \, dx-2 \int \left (\frac {x}{2+e^x+x}+\frac {x^2}{2+e^x+x}\right ) \, dx+2 \int \left (\frac {11 x}{19+8 e^x+8 x}+\frac {8 x^2}{19+8 e^x+8 x}\right ) \, dx-16 \int \frac {x^2}{19+8 e^x+8 x} \, dx-22 \int \frac {x}{19+8 e^x+8 x} \, dx\\ &=-3 x+x \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.36, size = 34, normalized size = 1.36 \begin {gather*} -3 x+x \log \left (\frac {x^4 \left (19+8 e^x+8 x\right )^2}{4 \left (2+e^x+x\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(38 + 8*E^(2*x) + 29*x + 8*x^2 + E^x*(35 + 10*x) + (38 + 8*E^(2*x) + 35*x + 8*x^2 + E^x*(35 + 16*x))
*Log[(361*x^4 + 64*E^(2*x)*x^4 + 304*x^5 + 64*x^6 + E^x*(304*x^4 + 128*x^5))/(16 + 4*E^(2*x) + 16*x + 4*x^2 +
E^x*(16 + 8*x))])/(38 + 8*E^(2*x) + 35*x + 8*x^2 + E^x*(35 + 16*x)),x]

[Out]

-3*x + x*Log[(x^4*(19 + 8*E^x + 8*x)^2)/(4*(2 + E^x + x)^2)]

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fricas [B]  time = 0.49, size = 70, normalized size = 2.80 \begin {gather*} x \log \left (\frac {64 \, x^{6} + 304 \, x^{5} + 64 \, x^{4} e^{\left (2 \, x\right )} + 361 \, x^{4} + 16 \, {\left (8 \, x^{5} + 19 \, x^{4}\right )} e^{x}}{4 \, {\left (x^{2} + 2 \, {\left (x + 2\right )} e^{x} + 4 \, x + e^{\left (2 \, x\right )} + 4\right )}}\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^2+(16*x+35)*exp(x)+8*x^2+35*x+38)*log((64*exp(x)^2*x^4+(128*x^5+304*x^4)*exp(x)+64*x^6+30
4*x^5+361*x^4)/(4*exp(x)^2+(8*x+16)*exp(x)+4*x^2+16*x+16))+8*exp(x)^2+(10*x+35)*exp(x)+8*x^2+29*x+38)/(8*exp(x
)^2+(16*x+35)*exp(x)+8*x^2+35*x+38),x, algorithm="fricas")

[Out]

x*log(1/4*(64*x^6 + 304*x^5 + 64*x^4*e^(2*x) + 361*x^4 + 16*(8*x^5 + 19*x^4)*e^x)/(x^2 + 2*(x + 2)*e^x + 4*x +
e^(2*x) + 4)) - 3*x

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giac [B]  time = 7.08, size = 71, normalized size = 2.84 \begin {gather*} x \log \left (\frac {64 \, x^{6} + 128 \, x^{5} e^{x} + 304 \, x^{5} + 64 \, x^{4} e^{\left (2 \, x\right )} + 304 \, x^{4} e^{x} + 361 \, x^{4}}{4 \, {\left (x^{2} + 2 \, x e^{x} + 4 \, x + e^{\left (2 \, x\right )} + 4 \, e^{x} + 4\right )}}\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^2+(16*x+35)*exp(x)+8*x^2+35*x+38)*log((64*exp(x)^2*x^4+(128*x^5+304*x^4)*exp(x)+64*x^6+30
4*x^5+361*x^4)/(4*exp(x)^2+(8*x+16)*exp(x)+4*x^2+16*x+16))+8*exp(x)^2+(10*x+35)*exp(x)+8*x^2+29*x+38)/(8*exp(x
)^2+(16*x+35)*exp(x)+8*x^2+35*x+38),x, algorithm="giac")

[Out]

x*log(1/4*(64*x^6 + 128*x^5*e^x + 304*x^5 + 64*x^4*e^(2*x) + 304*x^4*e^x + 361*x^4)/(x^2 + 2*x*e^x + 4*x + e^(
2*x) + 4*e^x + 4)) - 3*x

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maple [B]  time = 0.41, size = 74, normalized size = 2.96

method	result	size
norman	\(x \ln \left (\frac {64 \,{\mathrm e}^{2 x} x^{4}+\left (128 x^{5}+304 x^{4}\right ) {\mathrm e}^{x}+64 x^{6}+304 x^{5}+361 x^{4}}{4 \,{\mathrm e}^{2 x}+\left (8 x +16\right ) {\mathrm e}^{x}+4 x^{2}+16 x +16}\right )-3 x\)	\(74\)
risch	\(-3 x +4 x \ln \relax (2)+4 x \ln \relax (x )-2 x \ln \left ({\mathrm e}^{x}+2+x \right )+2 x \ln \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )-\frac {i \pi x \mathrm {csgn}\left (\frac {i x^{4} \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}\right )^{3}}{2}+\frac {i \pi x \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+2+x \right )^{2}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x^{3}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x^{4}\right )^{3}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (\frac {i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{4} \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (\frac {i x^{4} \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{4} \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )^{2}}{2}+i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi x \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )\right ) \mathrm {csgn}\left (i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}\right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}}{\left ({\mathrm e}^{x}+2+x \right )^{2}}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}+\frac {i \pi x \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+2+x \right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+2+x \right )^{2}\right )}{2}-i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+2+x \right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+2+x \right )^{2}\right )^{2}-\frac {i \pi x \mathrm {csgn}\left (i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +{\mathrm e}^{x}+\frac {19}{8}\right )^{2}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}}{2}\)	\(687\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(x)^2+(16*x+35)*exp(x)+8*x^2+35*x+38)*ln((64*exp(x)^2*x^4+(128*x^5+304*x^4)*exp(x)+64*x^6+304*x^5+3
61*x^4)/(4*exp(x)^2+(8*x+16)*exp(x)+4*x^2+16*x+16))+8*exp(x)^2+(10*x+35)*exp(x)+8*x^2+29*x+38)/(8*exp(x)^2+(16
*x+35)*exp(x)+8*x^2+35*x+38),x,method=_RETURNVERBOSE)

[Out]

x*ln((64*exp(x)^2*x^4+(128*x^5+304*x^4)*exp(x)+64*x^6+304*x^5+361*x^4)/(4*exp(x)^2+(8*x+16)*exp(x)+4*x^2+16*x+
16))-3*x

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maxima [A]  time = 0.79, size = 37, normalized size = 1.48 \begin {gather*} -x {\left (2 \, \log \relax (2) + 3\right )} + 2 \, x \log \left (8 \, x + 8 \, e^{x} + 19\right ) - 2 \, x \log \left (x + e^{x} + 2\right ) + 4 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^2+(16*x+35)*exp(x)+8*x^2+35*x+38)*log((64*exp(x)^2*x^4+(128*x^5+304*x^4)*exp(x)+64*x^6+30
4*x^5+361*x^4)/(4*exp(x)^2+(8*x+16)*exp(x)+4*x^2+16*x+16))+8*exp(x)^2+(10*x+35)*exp(x)+8*x^2+29*x+38)/(8*exp(x
)^2+(16*x+35)*exp(x)+8*x^2+35*x+38),x, algorithm="maxima")

[Out]

-x*(2*log(2) + 3) + 2*x*log(8*x + 8*e^x + 19) - 2*x*log(x + e^x + 2) + 4*x*log(x)

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mupad [B]  time = 1.27, size = 71, normalized size = 2.84 \begin {gather*} x\,\left (\ln \left (\frac {{\mathrm {e}}^x\,\left (128\,x^5+304\,x^4\right )+64\,x^4\,{\mathrm {e}}^{2\,x}+361\,x^4+304\,x^5+64\,x^6}{16\,x+4\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (8\,x+16\right )+4\,x^2+16}\right )-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((29*x + 8*exp(2*x) + log((exp(x)*(304*x^4 + 128*x^5) + 64*x^4*exp(2*x) + 361*x^4 + 304*x^5 + 64*x^6)/(16*x
+ 4*exp(2*x) + exp(x)*(8*x + 16) + 4*x^2 + 16))*(35*x + 8*exp(2*x) + exp(x)*(16*x + 35) + 8*x^2 + 38) + exp(x
)*(10*x + 35) + 8*x^2 + 38)/(35*x + 8*exp(2*x) + exp(x)*(16*x + 35) + 8*x^2 + 38),x)

[Out]

x*(log((exp(x)*(304*x^4 + 128*x^5) + 64*x^4*exp(2*x) + 361*x^4 + 304*x^5 + 64*x^6)/(16*x + 4*exp(2*x) + exp(x)
*(8*x + 16) + 4*x^2 + 16)) - 3)

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sympy [B]  time = 0.70, size = 70, normalized size = 2.80 \begin {gather*} x \log {\left (\frac {64 x^{6} + 304 x^{5} + 64 x^{4} e^{2 x} + 361 x^{4} + \left (128 x^{5} + 304 x^{4}\right ) e^{x}}{4 x^{2} + 16 x + \left (8 x + 16\right ) e^{x} + 4 e^{2 x} + 16} \right )} - 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)**2+(16*x+35)*exp(x)+8*x**2+35*x+38)*ln((64*exp(x)**2*x**4+(128*x**5+304*x**4)*exp(x)+64*x
**6+304*x**5+361*x**4)/(4*exp(x)**2+(8*x+16)*exp(x)+4*x**2+16*x+16))+8*exp(x)**2+(10*x+35)*exp(x)+8*x**2+29*x+
38)/(8*exp(x)**2+(16*x+35)*exp(x)+8*x**2+35*x+38),x)

[Out]

x*log((64*x**6 + 304*x**5 + 64*x**4*exp(2*x) + 361*x**4 + (128*x**5 + 304*x**4)*exp(x))/(4*x**2 + 16*x + (8*x
+ 16)*exp(x) + 4*exp(2*x) + 16)) - 3*x

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