3.15.75 \(\int \frac {400+388 x+75 x^2+2 x^3+(-160-156 x-28 x^2) \log (\frac {4}{4+x})+(16+16 x+3 x^2) \log ^2(\frac {4}{4+x})}{(200 x+158 x^2+31 x^3+x^4+(-80 x-60 x^2-10 x^3) \log (\frac {4}{4+x})+(8 x+6 x^2+x^3) \log ^2(\frac {4}{4+x})) \log ^2(\frac {25+x-10 \log (\frac {4}{4+x})+\log ^2(\frac {4}{4+x})}{4 x^2+2 x^3})} \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{\log \left (\frac {x+\left (5-\log \left (\frac {4}{4+x}\right )\right )^2}{2 x^2 (2+x)}\right )} \]

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Rubi [F]  time = 44.21, 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 {400+388 x+75 x^2+2 x^3+\left (-160-156 x-28 x^2\right ) \log \left (\frac {4}{4+x}\right )+\left (16+16 x+3 x^2\right ) \log ^2\left (\frac {4}{4+x}\right )}{\left (200 x+158 x^2+31 x^3+x^4+\left (-80 x-60 x^2-10 x^3\right ) \log \left (\frac {4}{4+x}\right )+\left (8 x+6 x^2+x^3\right ) \log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{4 x^2+2 x^3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(400 + 388*x + 75*x^2 + 2*x^3 + (-160 - 156*x - 28*x^2)*Log[4/(4 + x)] + (16 + 16*x + 3*x^2)*Log[4/(4 + x)
]^2)/((200*x + 158*x^2 + 31*x^3 + x^4 + (-80*x - 60*x^2 - 10*x^3)*Log[4/(4 + x)] + (8*x + 6*x^2 + x^3)*Log[4/(
4 + x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(4*x^2 + 2*x^3)]^2),x]

[Out]

2*Defer[Int][1/((25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x
)]^2)/(x^2*(4 + 2*x))]^2), x] + 50*Defer[Int][1/(x*(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)*Log[(25 + x
- 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(x^2*(4 + 2*x))]^2), x] + 23*Defer[Int][1/((2 + x)*(25 + x - 10*Log[4
/(4 + x)] + Log[4/(4 + x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(x^2*(4 + 2*x))]^2), x] - 10
*Defer[Int][1/((4 + x)*(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4
/(4 + x)]^2)/(x^2*(4 + 2*x))]^2), x] - 20*Defer[Int][Log[4/(4 + x)]/(x*(25 + x - 10*Log[4/(4 + x)] + Log[4/(4
+ x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(x^2*(4 + 2*x))]^2), x] - 10*Defer[Int][Log[4/(4
+ x)]/((2 + x)*(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)
]^2)/(x^2*(4 + 2*x))]^2), x] + 2*Defer[Int][Log[4/(4 + x)]/((4 + x)*(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x
)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(x^2*(4 + 2*x))]^2), x] + 2*Defer[Int][Log[4/(4 + x)
]^2/(x*(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(x^
2*(4 + 2*x))]^2), x] + Defer[Int][Log[4/(4 + x)]^2/((2 + x)*(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)*Lo
g[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(x^2*(4 + 2*x))]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {400+388 x+75 x^2+2 x^3+\left (-160-156 x-28 x^2\right ) \log \left (\frac {4}{4+x}\right )+\left (16+16 x+3 x^2\right ) \log ^2\left (\frac {4}{4+x}\right )}{x \left (8+6 x+x^2\right ) \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )} \, dx\\ &=\int \left (\frac {-400-388 x-75 x^2-2 x^3+160 \log \left (\frac {4}{4+x}\right )+156 x \log \left (\frac {4}{4+x}\right )+28 x^2 \log \left (\frac {4}{4+x}\right )-16 \log ^2\left (\frac {4}{4+x}\right )-16 x \log ^2\left (\frac {4}{4+x}\right )-3 x^2 \log ^2\left (\frac {4}{4+x}\right )}{4 (2+x) \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )}+\frac {400+388 x+75 x^2+2 x^3-160 \log \left (\frac {4}{4+x}\right )-156 x \log \left (\frac {4}{4+x}\right )-28 x^2 \log \left (\frac {4}{4+x}\right )+16 \log ^2\left (\frac {4}{4+x}\right )+16 x \log ^2\left (\frac {4}{4+x}\right )+3 x^2 \log ^2\left (\frac {4}{4+x}\right )}{8 x \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )}+\frac {400+388 x+75 x^2+2 x^3-160 \log \left (\frac {4}{4+x}\right )-156 x \log \left (\frac {4}{4+x}\right )-28 x^2 \log \left (\frac {4}{4+x}\right )+16 \log ^2\left (\frac {4}{4+x}\right )+16 x \log ^2\left (\frac {4}{4+x}\right )+3 x^2 \log ^2\left (\frac {4}{4+x}\right )}{8 (4+x) \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )}\right ) \, dx\\ &=\frac {1}{8} \int \frac {400+388 x+75 x^2+2 x^3-160 \log \left (\frac {4}{4+x}\right )-156 x \log \left (\frac {4}{4+x}\right )-28 x^2 \log \left (\frac {4}{4+x}\right )+16 \log ^2\left (\frac {4}{4+x}\right )+16 x \log ^2\left (\frac {4}{4+x}\right )+3 x^2 \log ^2\left (\frac {4}{4+x}\right )}{x \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )} \, dx+\frac {1}{8} \int \frac {400+388 x+75 x^2+2 x^3-160 \log \left (\frac {4}{4+x}\right )-156 x \log \left (\frac {4}{4+x}\right )-28 x^2 \log \left (\frac {4}{4+x}\right )+16 \log ^2\left (\frac {4}{4+x}\right )+16 x \log ^2\left (\frac {4}{4+x}\right )+3 x^2 \log ^2\left (\frac {4}{4+x}\right )}{(4+x) \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )} \, dx+\frac {1}{4} \int \frac {-400-388 x-75 x^2-2 x^3+160 \log \left (\frac {4}{4+x}\right )+156 x \log \left (\frac {4}{4+x}\right )+28 x^2 \log \left (\frac {4}{4+x}\right )-16 \log ^2\left (\frac {4}{4+x}\right )-16 x \log ^2\left (\frac {4}{4+x}\right )-3 x^2 \log ^2\left (\frac {4}{4+x}\right )}{(2+x) \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )} \, dx\\ &=\frac {1}{8} \int \frac {400+388 x+75 x^2+2 x^3-4 \left (40+39 x+7 x^2\right ) \log \left (\frac {4}{4+x}\right )+\left (16+16 x+3 x^2\right ) \log ^2\left (\frac {4}{4+x}\right )}{x \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )} \, dx+\frac {1}{8} \int \frac {400+388 x+75 x^2+2 x^3-4 \left (40+39 x+7 x^2\right ) \log \left (\frac {4}{4+x}\right )+\left (16+16 x+3 x^2\right ) \log ^2\left (\frac {4}{4+x}\right )}{(4+x) \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )} \, dx+\frac {1}{4} \int \frac {-400-388 x-75 x^2-2 x^3+4 \left (40+39 x+7 x^2\right ) \log \left (\frac {4}{4+x}\right )-\left (16+16 x+3 x^2\right ) \log ^2\left (\frac {4}{4+x}\right )}{(2+x) \left (25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )\right ) \log ^2\left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{x^2 (4+2 x)}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 38, normalized size = 1.23 \begin {gather*} \frac {1}{\log \left (\frac {25+x-10 \log \left (\frac {4}{4+x}\right )+\log ^2\left (\frac {4}{4+x}\right )}{2 x^2 (2+x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(400 + 388*x + 75*x^2 + 2*x^3 + (-160 - 156*x - 28*x^2)*Log[4/(4 + x)] + (16 + 16*x + 3*x^2)*Log[4/(
4 + x)]^2)/((200*x + 158*x^2 + 31*x^3 + x^4 + (-80*x - 60*x^2 - 10*x^3)*Log[4/(4 + x)] + (8*x + 6*x^2 + x^3)*L
og[4/(4 + x)]^2)*Log[(25 + x - 10*Log[4/(4 + x)] + Log[4/(4 + x)]^2)/(4*x^2 + 2*x^3)]^2),x]

[Out]

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

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fricas [A]  time = 0.50, size = 39, normalized size = 1.26 \begin {gather*} \frac {1}{\log \left (\frac {\log \left (\frac {4}{x + 4}\right )^{2} + x - 10 \, \log \left (\frac {4}{x + 4}\right ) + 25}{2 \, {\left (x^{3} + 2 \, x^{2}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+16*x+16)*log(4/(4+x))^2+(-28*x^2-156*x-160)*log(4/(4+x))+2*x^3+75*x^2+388*x+400)/((x^3+6*x^2
+8*x)*log(4/(4+x))^2+(-10*x^3-60*x^2-80*x)*log(4/(4+x))+x^4+31*x^3+158*x^2+200*x)/log((log(4/(4+x))^2-10*log(4
/(4+x))+x+25)/(2*x^3+4*x^2))^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 11.59, size = 5401, normalized size = 174.23 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+16*x+16)*log(4/(4+x))^2+(-28*x^2-156*x-160)*log(4/(4+x))+2*x^3+75*x^2+388*x+400)/((x^3+6*x^2
+8*x)*log(4/(4+x))^2+(-10*x^3-60*x^2-80*x)*log(4/(4+x))+x^4+31*x^3+158*x^2+200*x)/log((log(4/(4+x))^2-10*log(4
/(4+x))+x+25)/(2*x^3+4*x^2))^2,x, algorithm="giac")

[Out]

-(12*x^2*log(2)^2*log(4/(x + 4))^2 - 12*x^2*log(2)*log(x + 4)*log(4/(x + 4))^2 + 3*x^2*log(x + 4)^2*log(4/(x +
4))^2 + 12*x^3*log(2)^2 - 12*x^3*log(2)*log(x + 4) + 3*x^3*log(x + 4)^2 - 120*x^2*log(2)^2*log(4/(x + 4)) + 1
20*x^2*log(2)*log(x + 4)*log(4/(x + 4)) - 30*x^2*log(x + 4)^2*log(4/(x + 4)) + 2*x^3*log(4/(x + 4))^2 - 56*x^2
*log(2)*log(4/(x + 4))^2 + 64*x*log(2)^2*log(4/(x + 4))^2 + 28*x^2*log(x + 4)*log(4/(x + 4))^2 - 64*x*log(2)*l
og(x + 4)*log(4/(x + 4))^2 + 16*x*log(x + 4)^2*log(4/(x + 4))^2 + 2*x^4 - 56*x^3*log(2) + 364*x^2*log(2)^2 + 2
8*x^3*log(x + 4) - 364*x^2*log(2)*log(x + 4) + 91*x^2*log(x + 4)^2 - 20*x^3*log(4/(x + 4)) + 560*x^2*log(2)*lo
g(4/(x + 4)) - 640*x*log(2)^2*log(4/(x + 4)) - 280*x^2*log(x + 4)*log(4/(x + 4)) + 640*x*log(2)*log(x + 4)*log
(4/(x + 4)) - 160*x*log(x + 4)^2*log(4/(x + 4)) + 75*x^2*log(4/(x + 4))^2 - 312*x*log(2)*log(4/(x + 4))^2 + 64
*log(2)^2*log(4/(x + 4))^2 + 156*x*log(x + 4)*log(4/(x + 4))^2 - 64*log(2)*log(x + 4)*log(4/(x + 4))^2 + 16*lo
g(x + 4)^2*log(4/(x + 4))^2 + 125*x^3 - 1712*x^2*log(2) + 1664*x*log(2)^2 + 856*x^2*log(x + 4) - 1664*x*log(2)
*log(x + 4) + 416*x*log(x + 4)^2 - 750*x^2*log(4/(x + 4)) + 3120*x*log(2)*log(4/(x + 4)) - 640*log(2)^2*log(4/
(x + 4)) - 1560*x*log(x + 4)*log(4/(x + 4)) + 640*log(2)*log(x + 4)*log(4/(x + 4)) - 160*log(x + 4)^2*log(4/(x
+ 4)) + 388*x*log(4/(x + 4))^2 - 320*log(2)*log(4/(x + 4))^2 + 160*log(x + 4)*log(4/(x + 4))^2 + 2263*x^2 - 8
120*x*log(2) + 1600*log(2)^2 + 4060*x*log(x + 4) - 1600*log(2)*log(x + 4) + 400*log(x + 4)^2 - 3880*x*log(4/(x
+ 4)) + 3200*log(2)*log(4/(x + 4)) - 1600*log(x + 4)*log(4/(x + 4)) + 400*log(4/(x + 4))^2 + 10100*x - 8000*l
og(2) + 4000*log(x + 4) - 4000*log(4/(x + 4)) + 10000)/(12*x^2*log(2)^3*log(4/(x + 4))^2 + 12*x^2*log(2)^2*log
(x^3 + 2*x^2)*log(4/(x + 4))^2 - 12*x^2*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x +
4))^2 - 12*x^2*log(2)^2*log(x + 4)*log(4/(x + 4))^2 - 12*x^2*log(2)*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4)
)^2 + 12*x^2*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4))^2 + 3*x^2*log
(2)*log(x + 4)^2*log(4/(x + 4))^2 + 3*x^2*log(x^3 + 2*x^2)*log(x + 4)^2*log(4/(x + 4))^2 - 3*x^2*log(log(4/(x
+ 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)^2*log(4/(x + 4))^2 + 8*x^3*log(2)^3 + 8*x^3*log(2)^2*log(x^3
+ 2*x^2) - 8*x^3*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) - 8*x^3*log(2)^2*log(x + 4) - 8*x
^3*log(2)*log(x^3 + 2*x^2)*log(x + 4) + 8*x^3*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x
+ 4) + 2*x^3*log(2)*log(x + 4)^2 + 2*x^3*log(x^3 + 2*x^2)*log(x + 4)^2 - 2*x^3*log(log(4/(x + 4))^2 + x - 10*l
og(4/(x + 4)) + 25)*log(x + 4)^2 - 112*x^2*log(2)^3*log(4/(x + 4)) - 112*x^2*log(2)^2*log(x^3 + 2*x^2)*log(4/(
x + 4)) + 112*x^2*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4)) + 112*x^2*log(2)^
2*log(x + 4)*log(4/(x + 4)) + 112*x^2*log(2)*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4)) - 112*x^2*log(2)*log(l
og(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4)) - 28*x^2*log(2)*log(x + 4)^2*log(4/(x
+ 4)) - 28*x^2*log(x^3 + 2*x^2)*log(x + 4)^2*log(4/(x + 4)) + 28*x^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x +
4)) + 25)*log(x + 4)^2*log(4/(x + 4)) + 3*x^3*log(2)*log(4/(x + 4))^2 - 60*x^2*log(2)^2*log(4/(x + 4))^2 + 64*
x*log(2)^3*log(4/(x + 4))^2 + 3*x^3*log(x^3 + 2*x^2)*log(4/(x + 4))^2 - 60*x^2*log(2)*log(x^3 + 2*x^2)*log(4/(
x + 4))^2 + 64*x*log(2)^2*log(x^3 + 2*x^2)*log(4/(x + 4))^2 - 3*x^3*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4
)) + 25)*log(4/(x + 4))^2 + 60*x^2*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4))^2
- 64*x*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4))^2 + 30*x^2*log(2)*log(x + 4)
*log(4/(x + 4))^2 - 64*x*log(2)^2*log(x + 4)*log(4/(x + 4))^2 + 30*x^2*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x +
4))^2 - 64*x*log(2)*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4))^2 - 30*x^2*log(log(4/(x + 4))^2 + x - 10*log(4/
(x + 4)) + 25)*log(x + 4)*log(4/(x + 4))^2 + 64*x*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*lo
g(x + 4)*log(4/(x + 4))^2 + 16*x*log(2)*log(x + 4)^2*log(4/(x + 4))^2 + 16*x*log(x^3 + 2*x^2)*log(x + 4)^2*log
(4/(x + 4))^2 - 16*x*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)^2*log(4/(x + 4))^2 + 2*x^4*
log(2) - 40*x^3*log(2)^2 + 300*x^2*log(2)^3 + 2*x^4*log(x^3 + 2*x^2) - 40*x^3*log(2)*log(x^3 + 2*x^2) + 300*x^
2*log(2)^2*log(x^3 + 2*x^2) - 2*x^4*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) + 40*x^3*log(2)*log(log
(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) - 300*x^2*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) +
25) + 20*x^3*log(2)*log(x + 4) - 300*x^2*log(2)^2*log(x + 4) + 20*x^3*log(x^3 + 2*x^2)*log(x + 4) - 300*x^2*lo
g(2)*log(x^3 + 2*x^2)*log(x + 4) - 20*x^3*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4) + 300*
x^2*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4) + 75*x^2*log(2)*log(x + 4)^2 + 75*x^2
*log(x^3 + 2*x^2)*log(x + 4)^2 - 75*x^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)^2 - 28*x
^3*log(2)*log(4/(x + 4)) + 560*x^2*log(2)^2*log(4/(x + 4)) - 624*x*log(2)^3*log(4/(x + 4)) - 28*x^3*log(x^3 +
2*x^2)*log(4/(x + 4)) + 560*x^2*log(2)*log(x^3 + 2*x^2)*log(4/(x + 4)) - 624*x*log(2)^2*log(x^3 + 2*x^2)*log(4
/(x + 4)) + 28*x^3*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4)) - 560*x^2*log(2)*log(log(
4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4)) + 624*x*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4
/(x + 4)) + 25)*log(4/(x + 4)) - 280*x^2*log(2)*log(x + 4)*log(4/(x + 4)) + 624*x*log(2)^2*log(x + 4)*log(4/(x
+ 4)) - 280*x^2*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4)) + 624*x*log(2)*log(x^3 + 2*x^2)*log(x + 4)*log(4/(
x + 4)) + 280*x^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4)) - 624*x*log(2)*
log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4)) - 156*x*log(2)*log(x + 4)^2*log(4
/(x + 4)) - 156*x*log(x^3 + 2*x^2)*log(x + 4)^2*log(4/(x + 4)) + 156*x*log(log(4/(x + 4))^2 + x - 10*log(4/(x
+ 4)) + 25)*log(x + 4)^2*log(4/(x + 4)) + 91*x^2*log(2)*log(4/(x + 4))^2 - 320*x*log(2)^2*log(4/(x + 4))^2 + 6
4*log(2)^3*log(4/(x + 4))^2 + 91*x^2*log(x^3 + 2*x^2)*log(4/(x + 4))^2 - 320*x*log(2)*log(x^3 + 2*x^2)*log(4/(
x + 4))^2 + 64*log(2)^2*log(x^3 + 2*x^2)*log(4/(x + 4))^2 - 91*x^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)
) + 25)*log(4/(x + 4))^2 + 320*x*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4))^2 -
64*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4))^2 + 160*x*log(2)*log(x + 4)*log(
4/(x + 4))^2 - 64*log(2)^2*log(x + 4)*log(4/(x + 4))^2 + 160*x*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4))^2 -
64*log(2)*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4))^2 - 160*x*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) +
25)*log(x + 4)*log(4/(x + 4))^2 + 64*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)*log(
4/(x + 4))^2 + 16*log(2)*log(x + 4)^2*log(4/(x + 4))^2 + 16*log(x^3 + 2*x^2)*log(x + 4)^2*log(4/(x + 4))^2 - 1
6*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)^2*log(4/(x + 4))^2 + 125*x^3*log(2) - 1500*x^2
*log(2)^2 + 1552*x*log(2)^3 + 125*x^3*log(x^3 + 2*x^2) - 1500*x^2*log(2)*log(x^3 + 2*x^2) + 1552*x*log(2)^2*lo
g(x^3 + 2*x^2) - 125*x^3*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) + 1500*x^2*log(2)*log(log(4/(x + 4
))^2 + x - 10*log(4/(x + 4)) + 25) - 1552*x*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) + 750*
x^2*log(2)*log(x + 4) - 1552*x*log(2)^2*log(x + 4) + 750*x^2*log(x^3 + 2*x^2)*log(x + 4) - 1552*x*log(2)*log(x
^3 + 2*x^2)*log(x + 4) - 750*x^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4) + 1552*x*log(2)
*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4) + 388*x*log(2)*log(x + 4)^2 + 388*x*log(x^3 + 2
*x^2)*log(x + 4)^2 - 388*x*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)^2 - 856*x^2*log(2)*lo
g(4/(x + 4)) + 3120*x*log(2)^2*log(4/(x + 4)) - 640*log(2)^3*log(4/(x + 4)) - 856*x^2*log(x^3 + 2*x^2)*log(4/(
x + 4)) + 3120*x*log(2)*log(x^3 + 2*x^2)*log(4/(x + 4)) - 640*log(2)^2*log(x^3 + 2*x^2)*log(4/(x + 4)) + 856*x
^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4)) - 3120*x*log(2)*log(log(4/(x + 4))^2 + x
- 10*log(4/(x + 4)) + 25)*log(4/(x + 4)) + 640*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log
(4/(x + 4)) - 1560*x*log(2)*log(x + 4)*log(4/(x + 4)) + 640*log(2)^2*log(x + 4)*log(4/(x + 4)) - 1560*x*log(x^
3 + 2*x^2)*log(x + 4)*log(4/(x + 4)) + 640*log(2)*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4)) + 1560*x*log(log(
4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4)) - 640*log(2)*log(log(4/(x + 4))^2 + x - 1
0*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4)) - 160*log(2)*log(x + 4)^2*log(4/(x + 4)) - 160*log(x^3 + 2*x^
2)*log(x + 4)^2*log(4/(x + 4)) + 160*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)^2*log(4/(x
+ 4)) + 416*x*log(2)*log(4/(x + 4))^2 - 320*log(2)^2*log(4/(x + 4))^2 + 416*x*log(x^3 + 2*x^2)*log(4/(x + 4))^
2 - 320*log(2)*log(x^3 + 2*x^2)*log(4/(x + 4))^2 - 416*x*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*lo
g(4/(x + 4))^2 + 320*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4))^2 + 160*log(2)*l
og(x + 4)*log(4/(x + 4))^2 + 160*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4))^2 - 160*log(log(4/(x + 4))^2 + x -
10*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4))^2 + 2263*x^2*log(2) - 7760*x*log(2)^2 + 1600*log(2)^3 + 226
3*x^2*log(x^3 + 2*x^2) - 7760*x*log(2)*log(x^3 + 2*x^2) + 1600*log(2)^2*log(x^3 + 2*x^2) - 2263*x^2*log(log(4/
(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) + 7760*x*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) -
1600*log(2)^2*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) + 3880*x*log(2)*log(x + 4) - 1600*log(2)^2*lo
g(x + 4) + 3880*x*log(x^3 + 2*x^2)*log(x + 4) - 1600*log(2)*log(x^3 + 2*x^2)*log(x + 4) - 3880*x*log(log(4/(x
+ 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4) + 1600*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) +
25)*log(x + 4) + 400*log(2)*log(x + 4)^2 + 400*log(x^3 + 2*x^2)*log(x + 4)^2 - 400*log(log(4/(x + 4))^2 + x -
10*log(4/(x + 4)) + 25)*log(x + 4)^2 - 4060*x*log(2)*log(4/(x + 4)) + 3200*log(2)^2*log(4/(x + 4)) - 4060*x*lo
g(x^3 + 2*x^2)*log(4/(x + 4)) + 3200*log(2)*log(x^3 + 2*x^2)*log(4/(x + 4)) + 4060*x*log(log(4/(x + 4))^2 + x
- 10*log(4/(x + 4)) + 25)*log(4/(x + 4)) - 3200*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(
4/(x + 4)) - 1600*log(2)*log(x + 4)*log(4/(x + 4)) - 1600*log(x^3 + 2*x^2)*log(x + 4)*log(4/(x + 4)) + 1600*lo
g(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(x + 4)*log(4/(x + 4)) + 400*log(2)*log(4/(x + 4))^2 + 400
*log(x^3 + 2*x^2)*log(4/(x + 4))^2 - 400*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4))^2 +
10100*x*log(2) - 8000*log(2)^2 + 10100*x*log(x^3 + 2*x^2) - 8000*log(2)*log(x^3 + 2*x^2) - 10100*x*log(log(4/
(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) + 8000*log(2)*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) + 25) + 40
00*log(2)*log(x + 4) + 4000*log(x^3 + 2*x^2)*log(x + 4) - 4000*log(log(4/(x + 4))^2 + x - 10*log(4/(x + 4)) +
25)*log(x + 4) - 4000*log(2)*log(4/(x + 4)) - 4000*log(x^3 + 2*x^2)*log(4/(x + 4)) + 4000*log(log(4/(x + 4))^2
+ x - 10*log(4/(x + 4)) + 25)*log(4/(x + 4)) + 10000*log(2) + 10000*log(x^3 + 2*x^2) - 10000*log(log(4/(x + 4
))^2 + x - 10*log(4/(x + 4)) + 25))

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (3 x^{2}+16 x +16\right ) \ln \left (\frac {4}{4+x}\right )^{2}+\left (-28 x^{2}-156 x -160\right ) \ln \left (\frac {4}{4+x}\right )+2 x^{3}+75 x^{2}+388 x +400}{\left (\left (x^{3}+6 x^{2}+8 x \right ) \ln \left (\frac {4}{4+x}\right )^{2}+\left (-10 x^{3}-60 x^{2}-80 x \right ) \ln \left (\frac {4}{4+x}\right )+x^{4}+31 x^{3}+158 x^{2}+200 x \right ) \ln \left (\frac {\ln \left (\frac {4}{4+x}\right )^{2}-10 \ln \left (\frac {4}{4+x}\right )+x +25}{2 x^{3}+4 x^{2}}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2+16*x+16)*ln(4/(4+x))^2+(-28*x^2-156*x-160)*ln(4/(4+x))+2*x^3+75*x^2+388*x+400)/((x^3+6*x^2+8*x)*ln
(4/(4+x))^2+(-10*x^3-60*x^2-80*x)*ln(4/(4+x))+x^4+31*x^3+158*x^2+200*x)/ln((ln(4/(4+x))^2-10*ln(4/(4+x))+x+25)
/(2*x^3+4*x^2))^2,x)

[Out]

int(((3*x^2+16*x+16)*ln(4/(4+x))^2+(-28*x^2-156*x-160)*ln(4/(4+x))+2*x^3+75*x^2+388*x+400)/((x^3+6*x^2+8*x)*ln
(4/(4+x))^2+(-10*x^3-60*x^2-80*x)*ln(4/(4+x))+x^4+31*x^3+158*x^2+200*x)/ln((ln(4/(4+x))^2-10*ln(4/(4+x))+x+25)
/(2*x^3+4*x^2))^2,x)

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maxima [A]  time = 0.66, size = 49, normalized size = 1.58 \begin {gather*} -\frac {1}{\log \relax (2) - \log \left (4 \, \log \relax (2)^{2} - 2 \, {\left (2 \, \log \relax (2) - 5\right )} \log \left (x + 4\right ) + \log \left (x + 4\right )^{2} + x - 20 \, \log \relax (2) + 25\right ) + \log \left (x + 2\right ) + 2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+16*x+16)*log(4/(4+x))^2+(-28*x^2-156*x-160)*log(4/(4+x))+2*x^3+75*x^2+388*x+400)/((x^3+6*x^2
+8*x)*log(4/(4+x))^2+(-10*x^3-60*x^2-80*x)*log(4/(4+x))+x^4+31*x^3+158*x^2+200*x)/log((log(4/(4+x))^2-10*log(4
/(4+x))+x+25)/(2*x^3+4*x^2))^2,x, algorithm="maxima")

[Out]

-1/(log(2) - log(4*log(2)^2 - 2*(2*log(2) - 5)*log(x + 4) + log(x + 4)^2 + x - 20*log(2) + 25) + log(x + 2) +
2*log(x))

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mupad [B]  time = 2.56, size = 40, normalized size = 1.29 \begin {gather*} \frac {1}{\ln \left (\frac {{\ln \left (\frac {4}{x+4}\right )}^2-10\,\ln \left (\frac {4}{x+4}\right )+x+25}{2\,x^3+4\,x^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((388*x + log(4/(x + 4))^2*(16*x + 3*x^2 + 16) - log(4/(x + 4))*(156*x + 28*x^2 + 160) + 75*x^2 + 2*x^3 + 4
00)/(log((x - 10*log(4/(x + 4)) + log(4/(x + 4))^2 + 25)/(4*x^2 + 2*x^3))^2*(200*x - log(4/(x + 4))*(80*x + 60
*x^2 + 10*x^3) + log(4/(x + 4))^2*(8*x + 6*x^2 + x^3) + 158*x^2 + 31*x^3 + x^4)),x)

[Out]

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

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sympy [A]  time = 1.41, size = 32, normalized size = 1.03 \begin {gather*} \frac {1}{\log {\left (\frac {x + \log {\left (\frac {4}{x + 4} \right )}^{2} - 10 \log {\left (\frac {4}{x + 4} \right )} + 25}{2 x^{3} + 4 x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2+16*x+16)*ln(4/(4+x))**2+(-28*x**2-156*x-160)*ln(4/(4+x))+2*x**3+75*x**2+388*x+400)/((x**3+6
*x**2+8*x)*ln(4/(4+x))**2+(-10*x**3-60*x**2-80*x)*ln(4/(4+x))+x**4+31*x**3+158*x**2+200*x)/ln((ln(4/(4+x))**2-
10*ln(4/(4+x))+x+25)/(2*x**3+4*x**2))**2,x)

[Out]

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

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