3.472 \(\int \frac{1}{x^7 (1+x^2)^3} \, dx\)

Optimal. Leaf size=52 \[ -\frac{2}{x^2+1}-\frac{3}{x^2}-\frac{1}{4 \left (x^2+1\right )^2}+\frac{3}{4 x^4}-\frac{1}{6 x^6}+5 \log \left (x^2+1\right )-10 \log (x) \]

[Out]

-1/(6*x^6) + 3/(4*x^4) - 3/x^2 - 1/(4*(1 + x^2)^2) - 2/(1 + x^2) - 10*Log[x] + 5*Log[1 + x^2]

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Rubi [A]  time = 0.0280003, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 44} \[ -\frac{2}{x^2+1}-\frac{3}{x^2}-\frac{1}{4 \left (x^2+1\right )^2}+\frac{3}{4 x^4}-\frac{1}{6 x^6}+5 \log \left (x^2+1\right )-10 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 + x^2)^3),x]

[Out]

-1/(6*x^6) + 3/(4*x^4) - 3/x^2 - 1/(4*(1 + x^2)^2) - 2/(1 + x^2) - 10*Log[x] + 5*Log[1 + x^2]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^7 \left (1+x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 (1+x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{3}{x^3}+\frac{6}{x^2}-\frac{10}{x}+\frac{1}{(1+x)^3}+\frac{4}{(1+x)^2}+\frac{10}{1+x}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{3}{4 x^4}-\frac{3}{x^2}-\frac{1}{4 \left (1+x^2\right )^2}-\frac{2}{1+x^2}-10 \log (x)+5 \log \left (1+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0262952, size = 49, normalized size = 0.94 \[ -\frac{60 x^8+90 x^6+20 x^4-5 x^2+2}{12 x^6 \left (x^2+1\right )^2}+5 \log \left (x^2+1\right )-10 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 + x^2)^3),x]

[Out]

-(2 - 5*x^2 + 20*x^4 + 90*x^6 + 60*x^8)/(12*x^6*(1 + x^2)^2) - 10*Log[x] + 5*Log[1 + x^2]

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Maple [A]  time = 0.011, size = 47, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,{x}^{6}}}+{\frac{3}{4\,{x}^{4}}}-3\,{x}^{-2}-{\frac{1}{4\, \left ({x}^{2}+1 \right ) ^{2}}}-2\, \left ({x}^{2}+1 \right ) ^{-1}-10\,\ln \left ( x \right ) +5\,\ln \left ({x}^{2}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^2+1)^3,x)

[Out]

-1/6/x^6+3/4/x^4-3/x^2-1/4/(x^2+1)^2-2/(x^2+1)-10*ln(x)+5*ln(x^2+1)

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Maxima [A]  time = 0.942041, size = 72, normalized size = 1.38 \begin{align*} -\frac{60 \, x^{8} + 90 \, x^{6} + 20 \, x^{4} - 5 \, x^{2} + 2}{12 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )}} + 5 \, \log \left (x^{2} + 1\right ) - 5 \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^2+1)^3,x, algorithm="maxima")

[Out]

-1/12*(60*x^8 + 90*x^6 + 20*x^4 - 5*x^2 + 2)/(x^10 + 2*x^8 + x^6) + 5*log(x^2 + 1) - 5*log(x^2)

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Fricas [A]  time = 2.2616, size = 189, normalized size = 3.63 \begin{align*} -\frac{60 \, x^{8} + 90 \, x^{6} + 20 \, x^{4} - 5 \, x^{2} - 60 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )} \log \left (x^{2} + 1\right ) + 120 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )} \log \left (x\right ) + 2}{12 \,{\left (x^{10} + 2 \, x^{8} + x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^2+1)^3,x, algorithm="fricas")

[Out]

-1/12*(60*x^8 + 90*x^6 + 20*x^4 - 5*x^2 - 60*(x^10 + 2*x^8 + x^6)*log(x^2 + 1) + 120*(x^10 + 2*x^8 + x^6)*log(
x) + 2)/(x^10 + 2*x^8 + x^6)

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Sympy [A]  time = 0.165768, size = 49, normalized size = 0.94 \begin{align*} - 10 \log{\left (x \right )} + 5 \log{\left (x^{2} + 1 \right )} - \frac{60 x^{8} + 90 x^{6} + 20 x^{4} - 5 x^{2} + 2}{12 x^{10} + 24 x^{8} + 12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**2+1)**3,x)

[Out]

-10*log(x) + 5*log(x**2 + 1) - (60*x**8 + 90*x**6 + 20*x**4 - 5*x**2 + 2)/(12*x**10 + 24*x**8 + 12*x**6)

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Giac [A]  time = 1.07521, size = 78, normalized size = 1.5 \begin{align*} -\frac{30 \, x^{4} + 68 \, x^{2} + 39}{4 \,{\left (x^{2} + 1\right )}^{2}} + \frac{110 \, x^{6} - 36 \, x^{4} + 9 \, x^{2} - 2}{12 \, x^{6}} + 5 \, \log \left (x^{2} + 1\right ) - 5 \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^2+1)^3,x, algorithm="giac")

[Out]

-1/4*(30*x^4 + 68*x^2 + 39)/(x^2 + 1)^2 + 1/12*(110*x^6 - 36*x^4 + 9*x^2 - 2)/x^6 + 5*log(x^2 + 1) - 5*log(x^2
)