3.74.69 \(\int \frac {2+4 x+e^x (-2-6 x-4 x^2)+(2048+1280 x+136 x^2+4 x^3+e^x (2048+1280 x+136 x^2+4 x^3)) \log (1+2 x)+(320+708 x+139 x^2+6 x^3+e^x (-192-636 x-535 x^2-63 x^3-2 x^4)) \log ^2(1+2 x)}{1+2 x+e^{2 x} (1+2 x)+e^x (2+4 x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {(2+x) \left (2+(16+x)^2 \log ^2(1+2 x)\right )}{1+e^x} \]

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Rubi [F]  time = 8.60, 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 {2+4 x+e^x \left (-2-6 x-4 x^2\right )+\left (2048+1280 x+136 x^2+4 x^3+e^x \left (2048+1280 x+136 x^2+4 x^3\right )\right ) \log (1+2 x)+\left (320+708 x+139 x^2+6 x^3+e^x \left (-192-636 x-535 x^2-63 x^3-2 x^4\right )\right ) \log ^2(1+2 x)}{1+2 x+e^{2 x} (1+2 x)+e^x (2+4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 4*x + E^x*(-2 - 6*x - 4*x^2) + (2048 + 1280*x + 136*x^2 + 4*x^3 + E^x*(2048 + 1280*x + 136*x^2 + 4*x^
3))*Log[1 + 2*x] + (320 + 708*x + 139*x^2 + 6*x^3 + E^x*(-192 - 636*x - 535*x^2 - 63*x^3 - 2*x^4))*Log[1 + 2*x
]^2)/(1 + 2*x + E^(2*x)*(1 + 2*x) + E^x*(2 + 4*x)),x]

[Out]

4/(1 + E^x) + (2*x)/(1 + E^x) + (2883*Log[1 + 2*x]*Defer[Int][1/((1 + E^x)*(1 + 2*x)), x])/2 + (1213*Defer[Int
][Log[1 + 2*x]/(1 + E^x), x])/2 + 67*Defer[Int][(x*Log[1 + 2*x])/(1 + E^x), x] + 2*Defer[Int][(x^2*Log[1 + 2*x
])/(1 + E^x), x] + 512*Defer[Int][Log[1 + 2*x]^2/(1 + E^x)^2, x] - 192*Defer[Int][Log[1 + 2*x]^2/(1 + E^x), x]
+ 320*Defer[Int][(x*Log[1 + 2*x]^2)/(1 + E^x)^2, x] - 252*Defer[Int][(x*Log[1 + 2*x]^2)/(1 + E^x), x] + 34*De
fer[Int][(x^2*Log[1 + 2*x]^2)/(1 + E^x)^2, x] - 31*Defer[Int][(x^2*Log[1 + 2*x]^2)/(1 + E^x), x] + Defer[Int][
(x^3*Log[1 + 2*x]^2)/(1 + E^x)^2, x] - Defer[Int][(x^3*Log[1 + 2*x]^2)/(1 + E^x), x] - 2883*Defer[Int][Defer[I
nt][1/((1 + E^x)*(1 + 2*x)), x]/(1 + 2*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 (1+2 x) \left (-1+e^x (1+x)\right )+4 \left (1+e^x\right ) (2+x) (16+x)^2 \log (1+2 x)-\left (16+33 x+2 x^2\right ) \left (-20-3 x+e^x \left (12+15 x+x^2\right )\right ) \log ^2(1+2 x)}{\left (1+e^x\right )^2 (1+2 x)} \, dx\\ &=\int \left (\frac {(2+x) \left (2+256 \log ^2(1+2 x)+32 x \log ^2(1+2 x)+x^2 \log ^2(1+2 x)\right )}{\left (1+e^x\right )^2}-\frac {2+6 x+4 x^2-2048 \log (1+2 x)-1280 x \log (1+2 x)-136 x^2 \log (1+2 x)-4 x^3 \log (1+2 x)+192 \log ^2(1+2 x)+636 x \log ^2(1+2 x)+535 x^2 \log ^2(1+2 x)+63 x^3 \log ^2(1+2 x)+2 x^4 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)}\right ) \, dx\\ &=\int \frac {(2+x) \left (2+256 \log ^2(1+2 x)+32 x \log ^2(1+2 x)+x^2 \log ^2(1+2 x)\right )}{\left (1+e^x\right )^2} \, dx-\int \frac {2+6 x+4 x^2-2048 \log (1+2 x)-1280 x \log (1+2 x)-136 x^2 \log (1+2 x)-4 x^3 \log (1+2 x)+192 \log ^2(1+2 x)+636 x \log ^2(1+2 x)+535 x^2 \log ^2(1+2 x)+63 x^3 \log ^2(1+2 x)+2 x^4 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx\\ &=\int \frac {(2+x) \left (2+(16+x)^2 \log ^2(1+2 x)\right )}{\left (1+e^x\right )^2} \, dx-\int \frac {2+6 x+4 x^2-4 (2+x) (16+x)^2 \log (1+2 x)+\left (192+636 x+535 x^2+63 x^3+2 x^4\right ) \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx\\ &=-\int \left (\frac {2}{\left (1+e^x\right ) (1+2 x)}+\frac {6 x}{\left (1+e^x\right ) (1+2 x)}+\frac {4 x^2}{\left (1+e^x\right ) (1+2 x)}-\frac {2048 \log (1+2 x)}{\left (1+e^x\right ) (1+2 x)}-\frac {1280 x \log (1+2 x)}{\left (1+e^x\right ) (1+2 x)}-\frac {136 x^2 \log (1+2 x)}{\left (1+e^x\right ) (1+2 x)}-\frac {4 x^3 \log (1+2 x)}{\left (1+e^x\right ) (1+2 x)}+\frac {192 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)}+\frac {636 x \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)}+\frac {535 x^2 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)}+\frac {63 x^3 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)}+\frac {2 x^4 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)}\right ) \, dx+\int \left (\frac {2 \left (2+256 \log ^2(1+2 x)+32 x \log ^2(1+2 x)+x^2 \log ^2(1+2 x)\right )}{\left (1+e^x\right )^2}+\frac {x \left (2+256 \log ^2(1+2 x)+32 x \log ^2(1+2 x)+x^2 \log ^2(1+2 x)\right )}{\left (1+e^x\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (1+e^x\right ) (1+2 x)} \, dx\right )-2 \int \frac {x^4 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx+2 \int \frac {2+256 \log ^2(1+2 x)+32 x \log ^2(1+2 x)+x^2 \log ^2(1+2 x)}{\left (1+e^x\right )^2} \, dx-4 \int \frac {x^2}{\left (1+e^x\right ) (1+2 x)} \, dx+4 \int \frac {x^3 \log (1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx-6 \int \frac {x}{\left (1+e^x\right ) (1+2 x)} \, dx-63 \int \frac {x^3 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx+136 \int \frac {x^2 \log (1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx-192 \int \frac {\log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx-535 \int \frac {x^2 \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx-636 \int \frac {x \log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx+1280 \int \frac {x \log (1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx+2048 \int \frac {\log (1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx+\int \frac {x \left (2+256 \log ^2(1+2 x)+32 x \log ^2(1+2 x)+x^2 \log ^2(1+2 x)\right )}{\left (1+e^x\right )^2} \, dx\\ &=-\left (2 \int \frac {1}{\left (1+e^x\right ) (1+2 x)} \, dx\right )+2 \int \frac {2+(16+x)^2 \log ^2(1+2 x)}{\left (1+e^x\right )^2} \, dx-2 \int \left (-\frac {\log ^2(1+2 x)}{16 \left (1+e^x\right )}+\frac {x \log ^2(1+2 x)}{8 \left (1+e^x\right )}-\frac {x^2 \log ^2(1+2 x)}{4 \left (1+e^x\right )}+\frac {x^3 \log ^2(1+2 x)}{2 \left (1+e^x\right )}+\frac {\log ^2(1+2 x)}{16 \left (1+e^x\right ) (1+2 x)}\right ) \, dx-4 \int \left (-\frac {1}{4 \left (1+e^x\right )}+\frac {x}{2 \left (1+e^x\right )}+\frac {1}{4 \left (1+e^x\right ) (1+2 x)}\right ) \, dx+4 \int \left (\frac {\log (1+2 x)}{8 \left (1+e^x\right )}-\frac {x \log (1+2 x)}{4 \left (1+e^x\right )}+\frac {x^2 \log (1+2 x)}{2 \left (1+e^x\right )}-\frac {\log (1+2 x)}{8 \left (1+e^x\right ) (1+2 x)}\right ) \, dx-6 \int \left (\frac {1}{2 \left (1+e^x\right )}-\frac {1}{2 \left (1+e^x\right ) (1+2 x)}\right ) \, dx-63 \int \left (\frac {\log ^2(1+2 x)}{8 \left (1+e^x\right )}-\frac {x \log ^2(1+2 x)}{4 \left (1+e^x\right )}+\frac {x^2 \log ^2(1+2 x)}{2 \left (1+e^x\right )}-\frac {\log ^2(1+2 x)}{8 \left (1+e^x\right ) (1+2 x)}\right ) \, dx+136 \int \left (-\frac {\log (1+2 x)}{4 \left (1+e^x\right )}+\frac {x \log (1+2 x)}{2 \left (1+e^x\right )}+\frac {\log (1+2 x)}{4 \left (1+e^x\right ) (1+2 x)}\right ) \, dx-192 \int \frac {\log ^2(1+2 x)}{\left (1+e^x\right ) (1+2 x)} \, dx-535 \int \left (-\frac {\log ^2(1+2 x)}{4 \left (1+e^x\right )}+\frac {x \log ^2(1+2 x)}{2 \left (1+e^x\right )}+\frac {\log ^2(1+2 x)}{4 \left (1+e^x\right ) (1+2 x)}\right ) \, dx-636 \int \left (\frac {\log ^2(1+2 x)}{2 \left (1+e^x\right )}-\frac {\log ^2(1+2 x)}{2 \left (1+e^x\right ) (1+2 x)}\right ) \, dx+1280 \int \left (\frac {\log (1+2 x)}{2 \left (1+e^x\right )}-\frac {\log (1+2 x)}{2 \left (1+e^x\right ) (1+2 x)}\right ) \, dx-2048 \int \frac {2 \int \frac {1}{\left (1+e^x\right ) (1+2 x)} \, dx}{1+2 x} \, dx+(2048 \log (1+2 x)) \int \frac {1}{\left (1+e^x\right ) (1+2 x)} \, dx+\int \frac {x \left (2+(16+x)^2 \log ^2(1+2 x)\right )}{\left (1+e^x\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 27, normalized size = 1.00 \begin {gather*} \frac {(2+x) \left (2+(16+x)^2 \log ^2(1+2 x)\right )}{1+e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 4*x + E^x*(-2 - 6*x - 4*x^2) + (2048 + 1280*x + 136*x^2 + 4*x^3 + E^x*(2048 + 1280*x + 136*x^2
+ 4*x^3))*Log[1 + 2*x] + (320 + 708*x + 139*x^2 + 6*x^3 + E^x*(-192 - 636*x - 535*x^2 - 63*x^3 - 2*x^4))*Log[1
+ 2*x]^2)/(1 + 2*x + E^(2*x)*(1 + 2*x) + E^x*(2 + 4*x)),x]

[Out]

((2 + x)*(2 + (16 + x)^2*Log[1 + 2*x]^2))/(1 + E^x)

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fricas [A]  time = 0.63, size = 34, normalized size = 1.26 \begin {gather*} \frac {{\left (x^{3} + 34 \, x^{2} + 320 \, x + 512\right )} \log \left (2 \, x + 1\right )^{2} + 2 \, x + 4}{e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4-63*x^3-535*x^2-636*x-192)*exp(x)+6*x^3+139*x^2+708*x+320)*log(2*x+1)^2+((4*x^3+136*x^2+128
0*x+2048)*exp(x)+4*x^3+136*x^2+1280*x+2048)*log(2*x+1)+(-4*x^2-6*x-2)*exp(x)+4*x+2)/((2*x+1)*exp(x)^2+(4*x+2)*
exp(x)+2*x+1),x, algorithm="fricas")

[Out]

((x^3 + 34*x^2 + 320*x + 512)*log(2*x + 1)^2 + 2*x + 4)/(e^x + 1)

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giac [B]  time = 0.19, size = 58, normalized size = 2.15 \begin {gather*} \frac {x^{3} \log \left (2 \, x + 1\right )^{2} + 34 \, x^{2} \log \left (2 \, x + 1\right )^{2} + 320 \, x \log \left (2 \, x + 1\right )^{2} + 512 \, \log \left (2 \, x + 1\right )^{2} + 2 \, x + 4}{e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4-63*x^3-535*x^2-636*x-192)*exp(x)+6*x^3+139*x^2+708*x+320)*log(2*x+1)^2+((4*x^3+136*x^2+128
0*x+2048)*exp(x)+4*x^3+136*x^2+1280*x+2048)*log(2*x+1)+(-4*x^2-6*x-2)*exp(x)+4*x+2)/((2*x+1)*exp(x)^2+(4*x+2)*
exp(x)+2*x+1),x, algorithm="giac")

[Out]

(x^3*log(2*x + 1)^2 + 34*x^2*log(2*x + 1)^2 + 320*x*log(2*x + 1)^2 + 512*log(2*x + 1)^2 + 2*x + 4)/(e^x + 1)

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maple [A]  time = 0.03, size = 41, normalized size = 1.52

method	result	size
risch	\(\frac {\left (x^{3}+34 x^{2}+320 x +512\right ) \ln \left (2 x +1\right )^{2}}{{\mathrm e}^{x}+1}+\frac {2 x +4}{{\mathrm e}^{x}+1}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^4-63*x^3-535*x^2-636*x-192)*exp(x)+6*x^3+139*x^2+708*x+320)*ln(2*x+1)^2+((4*x^3+136*x^2+1280*x+204
8)*exp(x)+4*x^3+136*x^2+1280*x+2048)*ln(2*x+1)+(-4*x^2-6*x-2)*exp(x)+4*x+2)/((2*x+1)*exp(x)^2+(4*x+2)*exp(x)+2
*x+1),x,method=_RETURNVERBOSE)

[Out]

(x^3+34*x^2+320*x+512)/(exp(x)+1)*ln(2*x+1)^2+2*(2+x)/(exp(x)+1)

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maxima [A]  time = 0.41, size = 34, normalized size = 1.26 \begin {gather*} \frac {{\left (x^{3} + 34 \, x^{2} + 320 \, x + 512\right )} \log \left (2 \, x + 1\right )^{2} + 2 \, x + 4}{e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4-63*x^3-535*x^2-636*x-192)*exp(x)+6*x^3+139*x^2+708*x+320)*log(2*x+1)^2+((4*x^3+136*x^2+128
0*x+2048)*exp(x)+4*x^3+136*x^2+1280*x+2048)*log(2*x+1)+(-4*x^2-6*x-2)*exp(x)+4*x+2)/((2*x+1)*exp(x)^2+(4*x+2)*
exp(x)+2*x+1),x, algorithm="maxima")

[Out]

((x^3 + 34*x^2 + 320*x + 512)*log(2*x + 1)^2 + 2*x + 4)/(e^x + 1)

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mupad [B]  time = 4.78, size = 41, normalized size = 1.52 \begin {gather*} \frac {2\,x+4}{{\mathrm {e}}^x+1}+\frac {{\ln \left (2\,x+1\right )}^2\,\left (x^3+34\,x^2+320\,x+512\right )}{{\mathrm {e}}^x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + log(2*x + 1)^2*(708*x - exp(x)*(636*x + 535*x^2 + 63*x^3 + 2*x^4 + 192) + 139*x^2 + 6*x^3 + 320) +
log(2*x + 1)*(1280*x + 136*x^2 + 4*x^3 + exp(x)*(1280*x + 136*x^2 + 4*x^3 + 2048) + 2048) - exp(x)*(6*x + 4*x^
2 + 2) + 2)/(2*x + exp(x)*(4*x + 2) + exp(2*x)*(2*x + 1) + 1),x)

[Out]

(2*x + 4)/(exp(x) + 1) + (log(2*x + 1)^2*(320*x + 34*x^2 + x^3 + 512))/(exp(x) + 1)

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sympy [B]  time = 0.33, size = 56, normalized size = 2.07 \begin {gather*} \frac {x^{3} \log {\left (2 x + 1 \right )}^{2} + 34 x^{2} \log {\left (2 x + 1 \right )}^{2} + 320 x \log {\left (2 x + 1 \right )}^{2} + 2 x + 512 \log {\left (2 x + 1 \right )}^{2} + 4}{e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**4-63*x**3-535*x**2-636*x-192)*exp(x)+6*x**3+139*x**2+708*x+320)*ln(2*x+1)**2+((4*x**3+136*x
**2+1280*x+2048)*exp(x)+4*x**3+136*x**2+1280*x+2048)*ln(2*x+1)+(-4*x**2-6*x-2)*exp(x)+4*x+2)/((2*x+1)*exp(x)**
2+(4*x+2)*exp(x)+2*x+1),x)

[Out]

(x**3*log(2*x + 1)**2 + 34*x**2*log(2*x + 1)**2 + 320*x*log(2*x + 1)**2 + 2*x + 512*log(2*x + 1)**2 + 4)/(exp(
x) + 1)

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