3.10.60 \(\int \frac {-16 x^2-8 x^3-17 x^4-8 x^5-x^6+(-100-10 x-140 x^2-10 x^3+10 x^4) \log ^2(2)+(-25-25 x^2) \log ^4(2)+(-8 x^2-2 x^3-8 x^4-2 x^5+(-25+10 x-25 x^2+10 x^3) \log ^2(2)) \log (1+x^2)+(-x^2-x^4) \log ^2(1+x^2)}{16 x^2+8 x^3+17 x^4+8 x^5+x^6+(-40 x-10 x^2-40 x^3-10 x^4) \log ^2(2)+(25+25 x^2) \log ^4(2)+(8 x^2+2 x^3+8 x^4+2 x^5+(-10 x-10 x^3) \log ^2(2)) \log (1+x^2)+(x^2+x^4) \log ^2(1+x^2)} \, dx\)

Optimal. Leaf size=34 \[ -x+\frac {x}{\frac {x}{5}+\frac {\log ^2(2)}{-4-x-\log \left (1+x^2\right )}} \]

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Rubi [F]  time = 2.25, 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 {-16 x^2-8 x^3-17 x^4-8 x^5-x^6+\left (-100-10 x-140 x^2-10 x^3+10 x^4\right ) \log ^2(2)+\left (-25-25 x^2\right ) \log ^4(2)+\left (-8 x^2-2 x^3-8 x^4-2 x^5+\left (-25+10 x-25 x^2+10 x^3\right ) \log ^2(2)\right ) \log \left (1+x^2\right )+\left (-x^2-x^4\right ) \log ^2\left (1+x^2\right )}{16 x^2+8 x^3+17 x^4+8 x^5+x^6+\left (-40 x-10 x^2-40 x^3-10 x^4\right ) \log ^2(2)+\left (25+25 x^2\right ) \log ^4(2)+\left (8 x^2+2 x^3+8 x^4+2 x^5+\left (-10 x-10 x^3\right ) \log ^2(2)\right ) \log \left (1+x^2\right )+\left (x^2+x^4\right ) \log ^2\left (1+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16*x^2 - 8*x^3 - 17*x^4 - 8*x^5 - x^6 + (-100 - 10*x - 140*x^2 - 10*x^3 + 10*x^4)*Log[2]^2 + (-25 - 25*x
^2)*Log[2]^4 + (-8*x^2 - 2*x^3 - 8*x^4 - 2*x^5 + (-25 + 10*x - 25*x^2 + 10*x^3)*Log[2]^2)*Log[1 + x^2] + (-x^2
- x^4)*Log[1 + x^2]^2)/(16*x^2 + 8*x^3 + 17*x^4 + 8*x^5 + x^6 + (-40*x - 10*x^2 - 40*x^3 - 10*x^4)*Log[2]^2 +
(25 + 25*x^2)*Log[2]^4 + (8*x^2 + 2*x^3 + 8*x^4 + 2*x^5 + (-10*x - 10*x^3)*Log[2]^2)*Log[1 + x^2] + (x^2 + x^
4)*Log[1 + x^2]^2),x]

[Out]

-x - 50*Log[2]^2*Defer[Int][(4*x + x^2 - 5*Log[2]^2 + x*Log[1 + x^2])^(-2), x] + (25*I)*Log[2]^2*Defer[Int][1/
((I - x)*(4*x + x^2 - 5*Log[2]^2 + x*Log[1 + x^2])^2), x] - 125*Log[2]^4*Defer[Int][1/(x*(4*x + x^2 - 5*Log[2]
^2 + x*Log[1 + x^2])^2), x] - 25*Log[2]^2*Defer[Int][x/(4*x + x^2 - 5*Log[2]^2 + x*Log[1 + x^2])^2, x] + (25*I
)*Log[2]^2*Defer[Int][1/((I + x)*(4*x + x^2 - 5*Log[2]^2 + x*Log[1 + x^2])^2), x] - 25*Log[2]^2*Defer[Int][1/(
x*(4*x + x^2 - 5*Log[2]^2 + x*Log[1 + x^2])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8 x^5-x^6-10 x \log ^2(2)-x^4 \left (17-10 \log ^2(2)\right )-25 \log ^2(2) \left (4+\log ^2(2)\right )-2 x^3 \left (4+5 \log ^2(2)\right )-x^2 \left (16+140 \log ^2(2)+25 \log ^4(2)\right )-\left (1+x^2\right ) \left (8 x^2+2 x^3+25 \log ^2(2)-10 x \log ^2(2)\right ) \log \left (1+x^2\right )-\left (x^2+x^4\right ) \log ^2\left (1+x^2\right )}{\left (1+x^2\right ) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx\\ &=\int \left (-1+\frac {25 \log ^2(2) \left (-2 x^3-x^4-5 \log ^2(2)-x^2 \left (1+5 \log ^2(2)\right )\right )}{x \left (1+x^2\right ) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2}-\frac {25 \log ^2(2)}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )}\right ) \, dx\\ &=-x+\left (25 \log ^2(2)\right ) \int \frac {-2 x^3-x^4-5 \log ^2(2)-x^2 \left (1+5 \log ^2(2)\right )}{x \left (1+x^2\right ) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx-\left (25 \log ^2(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )} \, dx\\ &=-x-\left (25 \log ^2(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )} \, dx+\left (25 \log ^2(2)\right ) \int \left (-\frac {2}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2}-\frac {x}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2}+\frac {2}{\left (1+x^2\right ) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2}-\frac {5 \log ^2(2)}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2}\right ) \, dx\\ &=-x-\left (25 \log ^2(2)\right ) \int \frac {x}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx-\left (25 \log ^2(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )} \, dx-\left (50 \log ^2(2)\right ) \int \frac {1}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx+\left (50 \log ^2(2)\right ) \int \frac {1}{\left (1+x^2\right ) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx-\left (125 \log ^4(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx\\ &=-x-\left (25 \log ^2(2)\right ) \int \frac {x}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx-\left (25 \log ^2(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )} \, dx-\left (50 \log ^2(2)\right ) \int \frac {1}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx+\left (50 \log ^2(2)\right ) \int \left (\frac {i}{2 (i-x) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2}+\frac {i}{2 (i+x) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2}\right ) \, dx-\left (125 \log ^4(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx\\ &=-x+\left (25 i \log ^2(2)\right ) \int \frac {1}{(i-x) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx+\left (25 i \log ^2(2)\right ) \int \frac {1}{(i+x) \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx-\left (25 \log ^2(2)\right ) \int \frac {x}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx-\left (25 \log ^2(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )} \, dx-\left (50 \log ^2(2)\right ) \int \frac {1}{\left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx-\left (125 \log ^4(2)\right ) \int \frac {1}{x \left (4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 33, normalized size = 0.97 \begin {gather*} -x+\frac {25 \log ^2(2)}{4 x+x^2-5 \log ^2(2)+x \log \left (1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x^2 - 8*x^3 - 17*x^4 - 8*x^5 - x^6 + (-100 - 10*x - 140*x^2 - 10*x^3 + 10*x^4)*Log[2]^2 + (-25
- 25*x^2)*Log[2]^4 + (-8*x^2 - 2*x^3 - 8*x^4 - 2*x^5 + (-25 + 10*x - 25*x^2 + 10*x^3)*Log[2]^2)*Log[1 + x^2] +
(-x^2 - x^4)*Log[1 + x^2]^2)/(16*x^2 + 8*x^3 + 17*x^4 + 8*x^5 + x^6 + (-40*x - 10*x^2 - 40*x^3 - 10*x^4)*Log[
2]^2 + (25 + 25*x^2)*Log[2]^4 + (8*x^2 + 2*x^3 + 8*x^4 + 2*x^5 + (-10*x - 10*x^3)*Log[2]^2)*Log[1 + x^2] + (x^
2 + x^4)*Log[1 + x^2]^2),x]

[Out]

-x + (25*Log[2]^2)/(4*x + x^2 - 5*Log[2]^2 + x*Log[1 + x^2])

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fricas [A]  time = 0.62, size = 53, normalized size = 1.56 \begin {gather*} -\frac {x^{3} - 5 \, {\left (x + 5\right )} \log \relax (2)^{2} + x^{2} \log \left (x^{2} + 1\right ) + 4 \, x^{2}}{x^{2} - 5 \, \log \relax (2)^{2} + x \log \left (x^{2} + 1\right ) + 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-x^2)*log(x^2+1)^2+((10*x^3-25*x^2+10*x-25)*log(2)^2-2*x^5-8*x^4-2*x^3-8*x^2)*log(x^2+1)+(-25*
x^2-25)*log(2)^4+(10*x^4-10*x^3-140*x^2-10*x-100)*log(2)^2-x^6-8*x^5-17*x^4-8*x^3-16*x^2)/((x^4+x^2)*log(x^2+1
)^2+((-10*x^3-10*x)*log(2)^2+2*x^5+8*x^4+2*x^3+8*x^2)*log(x^2+1)+(25*x^2+25)*log(2)^4+(-10*x^4-40*x^3-10*x^2-4
0*x)*log(2)^2+x^6+8*x^5+17*x^4+8*x^3+16*x^2),x, algorithm="fricas")

[Out]

-(x^3 - 5*(x + 5)*log(2)^2 + x^2*log(x^2 + 1) + 4*x^2)/(x^2 - 5*log(2)^2 + x*log(x^2 + 1) + 4*x)

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giac [A]  time = 0.60, size = 33, normalized size = 0.97 \begin {gather*} -x + \frac {25 \, \log \relax (2)^{2}}{x^{2} - 5 \, \log \relax (2)^{2} + x \log \left (x^{2} + 1\right ) + 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-x^2)*log(x^2+1)^2+((10*x^3-25*x^2+10*x-25)*log(2)^2-2*x^5-8*x^4-2*x^3-8*x^2)*log(x^2+1)+(-25*
x^2-25)*log(2)^4+(10*x^4-10*x^3-140*x^2-10*x-100)*log(2)^2-x^6-8*x^5-17*x^4-8*x^3-16*x^2)/((x^4+x^2)*log(x^2+1
)^2+((-10*x^3-10*x)*log(2)^2+2*x^5+8*x^4+2*x^3+8*x^2)*log(x^2+1)+(25*x^2+25)*log(2)^4+(-10*x^4-40*x^3-10*x^2-4
0*x)*log(2)^2+x^6+8*x^5+17*x^4+8*x^3+16*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.17, size = 37, normalized size = 1.09

method	result	size
risch	\(-x -\frac {25 \ln \relax (2)^{2}}{5 \ln \relax (2)^{2}-x^{2}-\ln \left (x^{2}+1\right ) x -4 x}\)	\(37\)
norman	\(\frac {x^{3}+\left (-5 \ln \relax (2)^{2}-16\right ) x +\ln \left (x^{2}+1\right ) x^{2}-4 \ln \left (x^{2}+1\right ) x -5 \ln \relax (2)^{2}}{5 \ln \relax (2)^{2}-x^{2}-\ln \left (x^{2}+1\right ) x -4 x}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4-x^2)*ln(x^2+1)^2+((10*x^3-25*x^2+10*x-25)*ln(2)^2-2*x^5-8*x^4-2*x^3-8*x^2)*ln(x^2+1)+(-25*x^2-25)*l
n(2)^4+(10*x^4-10*x^3-140*x^2-10*x-100)*ln(2)^2-x^6-8*x^5-17*x^4-8*x^3-16*x^2)/((x^4+x^2)*ln(x^2+1)^2+((-10*x^
3-10*x)*ln(2)^2+2*x^5+8*x^4+2*x^3+8*x^2)*ln(x^2+1)+(25*x^2+25)*ln(2)^4+(-10*x^4-40*x^3-10*x^2-40*x)*ln(2)^2+x^
6+8*x^5+17*x^4+8*x^3+16*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.98, size = 57, normalized size = 1.68 \begin {gather*} -\frac {x^{3} - 5 \, x \log \relax (2)^{2} + x^{2} \log \left (x^{2} + 1\right ) + 4 \, x^{2} - 25 \, \log \relax (2)^{2}}{x^{2} - 5 \, \log \relax (2)^{2} + x \log \left (x^{2} + 1\right ) + 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-x^2)*log(x^2+1)^2+((10*x^3-25*x^2+10*x-25)*log(2)^2-2*x^5-8*x^4-2*x^3-8*x^2)*log(x^2+1)+(-25*
x^2-25)*log(2)^4+(10*x^4-10*x^3-140*x^2-10*x-100)*log(2)^2-x^6-8*x^5-17*x^4-8*x^3-16*x^2)/((x^4+x^2)*log(x^2+1
)^2+((-10*x^3-10*x)*log(2)^2+2*x^5+8*x^4+2*x^3+8*x^2)*log(x^2+1)+(25*x^2+25)*log(2)^4+(-10*x^4-40*x^3-10*x^2-4
0*x)*log(2)^2+x^6+8*x^5+17*x^4+8*x^3+16*x^2),x, algorithm="maxima")

[Out]

-(x^3 - 5*x*log(2)^2 + x^2*log(x^2 + 1) + 4*x^2 - 25*log(2)^2)/(x^2 - 5*log(2)^2 + x*log(x^2 + 1) + 4*x)

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mupad [B]  time = 1.85, size = 57, normalized size = 1.68 \begin {gather*} -\frac {4\,x^2-25\,{\ln \relax (2)}^2-5\,x\,{\ln \relax (2)}^2+x^3+x^2\,\ln \left (x^2+1\right )}{4\,x+x\,\ln \left (x^2+1\right )-5\,{\ln \relax (2)}^2+x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)^2*(10*x + 140*x^2 + 10*x^3 - 10*x^4 + 100) + log(x^2 + 1)*(8*x^2 - log(2)^2*(10*x - 25*x^2 + 10*x
^3 - 25) + 2*x^3 + 8*x^4 + 2*x^5) + log(x^2 + 1)^2*(x^2 + x^4) + log(2)^4*(25*x^2 + 25) + 16*x^2 + 8*x^3 + 17*
x^4 + 8*x^5 + x^6)/(log(x^2 + 1)^2*(x^2 + x^4) - log(2)^2*(40*x + 10*x^2 + 40*x^3 + 10*x^4) + log(2)^4*(25*x^2
+ 25) + log(x^2 + 1)*(8*x^2 - log(2)^2*(10*x + 10*x^3) + 2*x^3 + 8*x^4 + 2*x^5) + 16*x^2 + 8*x^3 + 17*x^4 + 8
*x^5 + x^6),x)

[Out]

-(4*x^2 - 25*log(2)^2 - 5*x*log(2)^2 + x^3 + x^2*log(x^2 + 1))/(4*x + x*log(x^2 + 1) - 5*log(2)^2 + x^2)

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sympy [A]  time = 0.27, size = 29, normalized size = 0.85 \begin {gather*} - x + \frac {25 \log {\relax (2 )}^{2}}{x^{2} + x \log {\left (x^{2} + 1 \right )} + 4 x - 5 \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4-x**2)*ln(x**2+1)**2+((10*x**3-25*x**2+10*x-25)*ln(2)**2-2*x**5-8*x**4-2*x**3-8*x**2)*ln(x**2
+1)+(-25*x**2-25)*ln(2)**4+(10*x**4-10*x**3-140*x**2-10*x-100)*ln(2)**2-x**6-8*x**5-17*x**4-8*x**3-16*x**2)/((
x**4+x**2)*ln(x**2+1)**2+((-10*x**3-10*x)*ln(2)**2+2*x**5+8*x**4+2*x**3+8*x**2)*ln(x**2+1)+(25*x**2+25)*ln(2)*
*4+(-10*x**4-40*x**3-10*x**2-40*x)*ln(2)**2+x**6+8*x**5+17*x**4+8*x**3+16*x**2),x)

[Out]

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

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