3.273 \(\int \frac{1}{(2-\log (1+x^2))^5} \, dx\)

Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\frac{1}{\left (2-\log \left (x^2+1\right )\right )^5},x\right ) \]

[Out]

Defer[Int][(2 - Log[1 + x^2])^(-5), x]

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Rubi [A]  time = 0.0038601, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (2-\log \left (1+x^2\right )\right )^5} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(2 - Log[1 + x^2])^(-5),x]

[Out]

Defer[Int][(2 - Log[1 + x^2])^(-5), x]

Rubi steps

\begin{align*} \int \frac{1}{\left (2-\log \left (1+x^2\right )\right )^5} \, dx &=\int \frac{1}{\left (2-\log \left (1+x^2\right )\right )^5} \, dx\\ \end{align*}

Mathematica [A]  time = 2.81093, size = 0, normalized size = 0. \[ \int \frac{1}{\left (2-\log \left (1+x^2\right )\right )^5} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(2 - Log[1 + x^2])^(-5),x]

[Out]

Integrate[(2 - Log[1 + x^2])^(-5), x]

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Maple [A]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int \left ( 2-\ln \left ({x}^{2}+1 \right ) \right ) ^{-5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2-ln(x^2+1))^5,x)

[Out]

int(1/(2-ln(x^2+1))^5,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{32 \, x^{8} + 56 \, x^{6} + 120 \, x^{4} +{\left (x^{8} - 10 \, x^{4} - 24 \, x^{2} - 15\right )} \log \left (x^{2} + 1\right )^{3} - 2 \,{\left (2 \, x^{8} - x^{6} - 33 \, x^{4} - 75 \, x^{2} - 45\right )} \log \left (x^{2} + 1\right )^{2} + 216 \, x^{2} + 4 \,{\left (3 \, x^{8} - 2 \, x^{6} - 38 \, x^{4} - 78 \, x^{2} - 45\right )} \log \left (x^{2} + 1\right ) + 120}{384 \,{\left (x^{7} \log \left (x^{2} + 1\right )^{4} - 8 \, x^{7} \log \left (x^{2} + 1\right )^{3} + 24 \, x^{7} \log \left (x^{2} + 1\right )^{2} - 32 \, x^{7} \log \left (x^{2} + 1\right ) + 16 \, x^{7}\right )}} - \int \frac{x^{8} + 30 \, x^{4} + 120 \, x^{2} + 105}{384 \,{\left (x^{8} \log \left (x^{2} + 1\right ) - 2 \, x^{8}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2-log(x^2+1))^5,x, algorithm="maxima")

[Out]

1/384*(32*x^8 + 56*x^6 + 120*x^4 + (x^8 - 10*x^4 - 24*x^2 - 15)*log(x^2 + 1)^3 - 2*(2*x^8 - x^6 - 33*x^4 - 75*
x^2 - 45)*log(x^2 + 1)^2 + 216*x^2 + 4*(3*x^8 - 2*x^6 - 38*x^4 - 78*x^2 - 45)*log(x^2 + 1) + 120)/(x^7*log(x^2
+ 1)^4 - 8*x^7*log(x^2 + 1)^3 + 24*x^7*log(x^2 + 1)^2 - 32*x^7*log(x^2 + 1) + 16*x^7) - integrate(1/384*(x^8
+ 30*x^4 + 120*x^2 + 105)/(x^8*log(x^2 + 1) - 2*x^8), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{\log \left (x^{2} + 1\right )^{5} - 10 \, \log \left (x^{2} + 1\right )^{4} + 40 \, \log \left (x^{2} + 1\right )^{3} - 80 \, \log \left (x^{2} + 1\right )^{2} + 80 \, \log \left (x^{2} + 1\right ) - 32}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2-log(x^2+1))^5,x, algorithm="fricas")

[Out]

integral(-1/(log(x^2 + 1)^5 - 10*log(x^2 + 1)^4 + 40*log(x^2 + 1)^3 - 80*log(x^2 + 1)^2 + 80*log(x^2 + 1) - 32
), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{120 x^{2}}{x^{8} \log{\left (x^{2} + 1 \right )} - 2 x^{8}}\, dx + \int \frac{30 x^{4}}{x^{8} \log{\left (x^{2} + 1 \right )} - 2 x^{8}}\, dx + \int \frac{x^{8}}{x^{8} \log{\left (x^{2} + 1 \right )} - 2 x^{8}}\, dx + \int \frac{105}{x^{8} \log{\left (x^{2} + 1 \right )} - 2 x^{8}}\, dx}{384} + \frac{\frac{2 x^{8}}{3} + \frac{7 x^{6}}{6} + \frac{5 x^{4}}{2} + \frac{9 x^{2}}{2} + \left (\frac{x^{8}}{48} - \frac{5 x^{4}}{24} - \frac{x^{2}}{2} - \frac{5}{16}\right ) \log{\left (x^{2} + 1 \right )}^{3} + \left (- \frac{x^{8}}{12} + \frac{x^{6}}{24} + \frac{11 x^{4}}{8} + \frac{25 x^{2}}{8} + \frac{15}{8}\right ) \log{\left (x^{2} + 1 \right )}^{2} + \left (\frac{x^{8}}{4} - \frac{x^{6}}{6} - \frac{19 x^{4}}{6} - \frac{13 x^{2}}{2} - \frac{15}{4}\right ) \log{\left (x^{2} + 1 \right )} + \frac{5}{2}}{8 x^{7} \log{\left (x^{2} + 1 \right )}^{4} - 64 x^{7} \log{\left (x^{2} + 1 \right )}^{3} + 192 x^{7} \log{\left (x^{2} + 1 \right )}^{2} - 256 x^{7} \log{\left (x^{2} + 1 \right )} + 128 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2-ln(x**2+1))**5,x)

[Out]

-(Integral(120*x**2/(x**8*log(x**2 + 1) - 2*x**8), x) + Integral(30*x**4/(x**8*log(x**2 + 1) - 2*x**8), x) + I
ntegral(x**8/(x**8*log(x**2 + 1) - 2*x**8), x) + Integral(105/(x**8*log(x**2 + 1) - 2*x**8), x))/384 + (2*x**8
/3 + 7*x**6/6 + 5*x**4/2 + 9*x**2/2 + (x**8/48 - 5*x**4/24 - x**2/2 - 5/16)*log(x**2 + 1)**3 + (-x**8/12 + x**
6/24 + 11*x**4/8 + 25*x**2/8 + 15/8)*log(x**2 + 1)**2 + (x**8/4 - x**6/6 - 19*x**4/6 - 13*x**2/2 - 15/4)*log(x
**2 + 1) + 5/2)/(8*x**7*log(x**2 + 1)**4 - 64*x**7*log(x**2 + 1)**3 + 192*x**7*log(x**2 + 1)**2 - 256*x**7*log
(x**2 + 1) + 128*x**7)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (\log \left (x^{2} + 1\right ) - 2\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2-log(x^2+1))^5,x, algorithm="giac")

[Out]

integrate(-1/(log(x^2 + 1) - 2)^5, x)