3.3021 \(\int \frac{x}{(1+(x^n)^{\frac{1}{n}})^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{x^2 \left (x^n\right )^{-2/n}}{\left (x^n\right )^{\frac{1}{n}}+1}+x^2 \left (x^n\right )^{-2/n} \log \left (\left (x^n\right )^{\frac{1}{n}}+1\right ) \]

[Out]

x^2/((x^n)^(2/n)*(1 + (x^n)^n^(-1))) + (x^2*Log[1 + (x^n)^n^(-1)])/(x^n)^(2/n)

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Rubi [A]  time = 0.0113031, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {368, 43} \[ \frac{x^2 \left (x^n\right )^{-2/n}}{\left (x^n\right )^{\frac{1}{n}}+1}+x^2 \left (x^n\right )^{-2/n} \log \left (\left (x^n\right )^{\frac{1}{n}}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + (x^n)^n^(-1))^2,x]

[Out]

x^2/((x^n)^(2/n)*(1 + (x^n)^n^(-1))) + (x^2*Log[1 + (x^n)^n^(-1)])/(x^n)^(2/n)

Rule 368

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0179509, size = 35, normalized size = 0.73 \[ x^2 \left (x^n\right )^{-2/n} \left (\frac{1}{\left (x^n\right )^{\frac{1}{n}}+1}+\log \left (\left (x^n\right )^{\frac{1}{n}}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + (x^n)^n^(-1))^2,x]

[Out]

(x^2*((1 + (x^n)^n^(-1))^(-1) + Log[1 + (x^n)^n^(-1)]))/(x^n)^(2/n)

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Maple [A]  time = 0.037, size = 75, normalized size = 1.6 \begin{align*}{\frac{{x}^{2}}{1+\sqrt [n]{{x}^{n}}}}-x{{\rm e}^{-{\frac{\ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) }{n}}}}+\ln \left ( 1+{{\rm e}^{-{\frac{n\ln \left ( x \right ) -\ln \left ({x}^{n} \right ) }{n}}}}x \right ){{\rm e}^{-2\,{\frac{\ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) }{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1+(x^n)^(1/n))^2,x)

[Out]

x^2/(1+(x^n)^(1/n))-x*exp(-1/n*(ln(x^n)-n*ln(x)))+ln(1+exp(-(n*ln(x)-ln(x^n))/n)*x)*exp(-2/n*(ln(x^n)-n*ln(x))
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2/((x^n)^(1/n) + 1) - integrate(x/((x^n)^(1/n) + 1), x)

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Fricas [A]  time = 1.44312, size = 46, normalized size = 0.96 \begin{align*} \frac{{\left (x + 1\right )} \log \left (x + 1\right ) + 1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x + 1)*log(x + 1) + 1)/(x + 1)

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Sympy [A]  time = 0.086545, size = 19, normalized size = 0.4 \begin{align*} \log{\left (\left (x^{n}\right )^{\frac{1}{n}} + 1 \right )} + \frac{1}{\left (x^{n}\right )^{\frac{1}{n}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(1+(x**n)**(1/n))**2,x)

[Out]

log((x**n)**(1/n) + 1) + 1/((x**n)**(1/n) + 1)

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Giac [A]  time = 1.12246, size = 15, normalized size = 0.31 \begin{align*} \frac{1}{x + 1} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + 1) + log(abs(x + 1))