3.326 \(\int \frac{1}{(a x^n+b x^n)^2} \, dx\)

Optimal. Leaf size=20 \[ \frac{x^{1-2 n}}{(1-2 n) (a+b)^2} \]

[Out]

x^(1 - 2*n)/((a + b)^2*(1 - 2*n))

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Rubi [A]  time = 0.0082571, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {6, 12, 30} \[ \frac{x^{1-2 n}}{(1-2 n) (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^n + b*x^n)^(-2),x]

[Out]

x^(1 - 2*n)/((a + b)^2*(1 - 2*n))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0032169, size = 20, normalized size = 1. \[ \frac{x^{1-2 n}}{(1-2 n) (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^n + b*x^n)^(-2),x]

[Out]

x^(1 - 2*n)/((a + b)^2*(1 - 2*n))

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Maple [A]  time = 0.001, size = 21, normalized size = 1.1 \begin{align*} -{\frac{x}{ \left ( -1+2\,n \right ) \left ({x}^{n} \right ) ^{2} \left ( a+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^n+b*x^n)^2,x)

[Out]

-x/(-1+2*n)/(x^n)^2/(a+b)^2

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Maxima [A]  time = 1.05347, size = 54, normalized size = 2.7 \begin{align*} -\frac{x}{{\left (a^{2}{\left (2 \, n - 1\right )} + 2 \, a b{\left (2 \, n - 1\right )} + b^{2}{\left (2 \, n - 1\right )}\right )} x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/((a^2*(2*n - 1) + 2*a*b*(2*n - 1) + b^2*(2*n - 1))*x^(2*n))

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Fricas [A]  time = 0.916805, size = 80, normalized size = 4. \begin{align*} \frac{x}{{\left (a^{2} + 2 \, a b + b^{2} - 2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} n\right )} x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/((a^2 + 2*a*b + b^2 - 2*(a^2 + 2*a*b + b^2)*n)*x^(2*n))

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Sympy [A]  time = 1.38359, size = 82, normalized size = 4.1 \begin{align*} \begin{cases} - \frac{x}{2 a^{2} n x^{2 n} - a^{2} x^{2 n} + 4 a b n x^{2 n} - 2 a b x^{2 n} + 2 b^{2} n x^{2 n} - b^{2} x^{2 n}} & \text{for}\: n \neq \frac{1}{2} \\\frac{\log{\left (x \right )}}{a^{2} + 2 a b + b^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x/(2*a**2*n*x**(2*n) - a**2*x**(2*n) + 4*a*b*n*x**(2*n) - 2*a*b*x**(2*n) + 2*b**2*n*x**(2*n) - b**
2*x**(2*n)), Ne(n, 1/2)), (log(x)/(a**2 + 2*a*b + b**2), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^n + b*x^n)^(-2), x)