3.32.97 \(\int (4 e^{-3 x}+e^x)^{-1+20 x} (-240 e^{-3 x} x+20 e^x x+(80 e^{-3 x}+20 e^x) \log (4 e^{-3 x}+e^x)) \, dx\)

Optimal. Leaf size=15 \[ \left (4 e^{-3 x}+e^x\right )^{20 x} \]

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Rubi [F]  time = 1.58, 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 \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4/E^(3*x) + E^x)^(-1 + 20*x)*((-240*x)/E^(3*x) + 20*E^x*x + (80/E^(3*x) + 20*E^x)*Log[4/E^(3*x) + E^x]),x
]

[Out]

20*Log[(4 + E^(4*x))/E^(3*x)]*Defer[Int][((4 + E^(4*x))/E^(3*x))^(20*x), x] - 240*Defer[Int][(((4 + E^(4*x))/E
^(3*x))^(-1 + 20*x)*x)/E^(3*x), x] + 20*Defer[Int][E^x*((4 + E^(4*x))/E^(3*x))^(-1 + 20*x)*x, x] - 20*Defer[In
t][Defer[Int][((4 + E^(4*x))/E^(3*x))^(20*x), x], x] + 40*Defer[Int][(E^x*Defer[Int][((4 + E^(4*x))/E^(3*x))^(
20*x), x])/(-2 + 2*E^x - E^(2*x)), x] + 80*Defer[Int][Defer[Int][((4 + E^(4*x))/E^(3*x))^(20*x), x]/(2 - 2*E^x
+ E^(2*x)), x] + 80*Defer[Int][Defer[Int][((4 + E^(4*x))/E^(3*x))^(20*x), x]/(2 + 2*E^x + E^(2*x)), x] + 40*D
efer[Int][(E^x*Defer[Int][((4 + E^(4*x))/E^(3*x))^(20*x), x])/(2 + 2*E^x + E^(2*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx\\ &=\int \left (-240 e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x+20 e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x+20 e^{-3 x} \left (4+e^{4 x}\right ) \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+20 \int e^{-3 x} \left (4+e^{4 x}\right ) \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right ) \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx-20 \int \frac {\left (-12+e^{4 x}\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{4+e^{4 x}} \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+\left (20 \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx-20 \int \left (\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx-\frac {16 \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{4+e^{4 x}}\right ) \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+\left (20 \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx-20 \int \left (\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\right ) \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+320 \int \frac {\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{4+e^{4 x}} \, dx+\left (20 \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx-20 \int \left (\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\right ) \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+320 \int \left (\frac {\left (2-e^x\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{8 \left (2-2 e^x+e^{2 x}\right )}+\frac {\left (2+e^x\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{8 \left (2+2 e^x+e^{2 x}\right )}\right ) \, dx+\left (20 \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx-20 \int \left (\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\right ) \, dx+40 \int \frac {\left (2-e^x\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2-2 e^x+e^{2 x}} \, dx+40 \int \frac {\left (2+e^x\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2+2 e^x+e^{2 x}} \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+\left (20 \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx-20 \int \left (\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\right ) \, dx+40 \int \left (\frac {e^x \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{-2+2 e^x-e^{2 x}}+\frac {2 \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2-2 e^x+e^{2 x}}\right ) \, dx+40 \int \left (\frac {2 \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2+2 e^x+e^{2 x}}+\frac {e^x \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2+2 e^x+e^{2 x}}\right ) \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+\left (20 \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\\ &=20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx-20 \int \left (\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\right ) \, dx+40 \int \frac {e^x \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{-2+2 e^x-e^{2 x}} \, dx+40 \int \frac {e^x \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2+2 e^x+e^{2 x}} \, dx+80 \int \frac {\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2-2 e^x+e^{2 x}} \, dx+80 \int \frac {\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx}{2+2 e^x+e^{2 x}} \, dx-240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{-1+20 x} x \, dx+\left (20 \log \left (e^{-3 x} \left (4+e^{4 x}\right )\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 15, normalized size = 1.00 \begin {gather*} \left (4 e^{-3 x}+e^x\right )^{20 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4/E^(3*x) + E^x)^(-1 + 20*x)*((-240*x)/E^(3*x) + 20*E^x*x + (80/E^(3*x) + 20*E^x)*Log[4/E^(3*x) + E
^x]),x]

[Out]

(4/E^(3*x) + E^x)^(20*x)

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fricas [A]  time = 0.60, size = 15, normalized size = 1.00 \begin {gather*} \left ({\left (e^{\left (4 \, x\right )} + 4\right )} e^{\left (-3 \, x\right )}\right )^{20 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*exp(x)+80*exp(-3*x))*log(exp(x)+4*exp(-3*x))+20*exp(x)*x-240*x*exp(-3*x))*exp(20*x*log(exp(x)+4
*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x, algorithm="fricas")

[Out]

((e^(4*x) + 4)*e^(-3*x))^(20*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {20 \, {\left (12 \, x e^{\left (-3 \, x\right )} - x e^{x} - {\left (4 \, e^{\left (-3 \, x\right )} + e^{x}\right )} \log \left (4 \, e^{\left (-3 \, x\right )} + e^{x}\right )\right )} {\left (4 \, e^{\left (-3 \, x\right )} + e^{x}\right )}^{20 \, x}}{4 \, e^{\left (-3 \, x\right )} + e^{x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*exp(x)+80*exp(-3*x))*log(exp(x)+4*exp(-3*x))+20*exp(x)*x-240*x*exp(-3*x))*exp(20*x*log(exp(x)+4
*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x, algorithm="giac")

[Out]

integrate(-20*(12*x*e^(-3*x) - x*e^x - (4*e^(-3*x) + e^x)*log(4*e^(-3*x) + e^x))*(4*e^(-3*x) + e^x)^(20*x)/(4*
e^(-3*x) + e^x), x)

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maple [C]  time = 0.18, size = 270, normalized size = 18.00

method	result	size
risch	\({\mathrm e}^{10 x \left (i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-2 i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )^{3}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right )^{3}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-3 x}\right )+i \pi \mathrm {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{4 x}+4\right )\right )-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-3 x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{4 x}+4\right )\right )-6 \ln \left ({\mathrm e}^{x}\right )+2 \ln \left ({\mathrm e}^{4 x}+4\right )\right )}\)	\(270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*exp(x)+80*exp(-3*x))*ln(exp(x)+4*exp(-3*x))+20*exp(x)*x-240*x*exp(-3*x))*exp(20*x*ln(exp(x)+4*exp(-3*
x)))/(exp(x)+4*exp(-3*x)),x,method=_RETURNVERBOSE)

[Out]

exp(10*x*(I*Pi*csgn(I*exp(2*x))^3-2*I*Pi*csgn(I*exp(2*x))^2*csgn(I*exp(x))+I*Pi*csgn(I*exp(2*x))*csgn(I*exp(x)
)^2+I*Pi*csgn(I*exp(2*x))*csgn(I*exp(x))*csgn(I*exp(3*x))-I*Pi*csgn(I*exp(2*x))*csgn(I*exp(3*x))^2-I*Pi*csgn(I
*exp(x))*csgn(I*exp(3*x))^2+I*Pi*csgn(I*exp(3*x))^3-I*Pi*csgn(I*exp(-3*x)*(exp(4*x)+4))^3+I*Pi*csgn(I*exp(-3*x
)*(exp(4*x)+4))^2*csgn(I*exp(-3*x))+I*Pi*csgn(I*exp(-3*x)*(exp(4*x)+4))^2*csgn(I*(exp(4*x)+4))-I*Pi*csgn(I*exp
(-3*x)*(exp(4*x)+4))*csgn(I*exp(-3*x))*csgn(I*(exp(4*x)+4))-6*ln(exp(x))+2*ln(exp(4*x)+4)))

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maxima [B]  time = 1.00, size = 35, normalized size = 2.33 \begin {gather*} e^{\left (-60 \, x^{2} + 20 \, x \log \left (e^{\left (2 \, x\right )} + 2 \, e^{x} + 2\right ) + 20 \, x \log \left (e^{\left (2 \, x\right )} - 2 \, e^{x} + 2\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*exp(x)+80*exp(-3*x))*log(exp(x)+4*exp(-3*x))+20*exp(x)*x-240*x*exp(-3*x))*exp(20*x*log(exp(x)+4
*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x, algorithm="maxima")

[Out]

e^(-60*x^2 + 20*x*log(e^(2*x) + 2*e^x + 2) + 20*x*log(e^(2*x) - 2*e^x + 2))

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mupad [B]  time = 0.17, size = 13, normalized size = 0.87 \begin {gather*} {\left (4\,{\mathrm {e}}^{-3\,x}+{\mathrm {e}}^x\right )}^{20\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(20*x*log(4*exp(-3*x) + exp(x)))*(20*x*exp(x) - 240*x*exp(-3*x) + log(4*exp(-3*x) + exp(x))*(80*exp(-3
*x) + 20*exp(x))))/(4*exp(-3*x) + exp(x)),x)

[Out]

(4*exp(-3*x) + exp(x))^(20*x)

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sympy [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(((20*exp(x)+80*exp(-3*x))*ln(exp(x)+4*exp(-3*x))+20*exp(x)*x-240*x*exp(-3*x))*exp(20*x*ln(exp(x)+4*e
xp(-3*x)))/(exp(x)+4*exp(-3*x)),x)

[Out]

Timed out

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