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

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 = 0.0217089, 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 \[ \frac{x^{1-2 n}}{(1-2 n) (a+b)^2} \]

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))

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Rubi in Sympy [A]  time = 3.69257, size = 15, normalized size = 0.75 \[ \frac{x^{- 2 n + 1}}{\left (a + b\right )^{2} \left (- 2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*x**n+b*x**n)**2,x)

[Out]

x**(-2*n + 1)/((a + b)**2*(-2*n + 1))

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Mathematica [A]  time = 0.005337, 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 = 0.003, size = 21, normalized size = 1.1 \[ -{\frac{x}{ \left ( -1+2\,n \right ) \left ({x}^{n} \right ) ^{2} \left ( a+b \right ) ^{2}}} \]

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 = 1.46628, size = 51, normalized size = 2.55 \[ -\frac{x x^{-2 \, n}}{a^{2}{\left (2 \, n - 1\right )} + 2 \, a b{\left (2 \, n - 1\right )} + b^{2}{\left (2 \, n - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^n + b*x^n)^(-2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235885, size = 49, normalized size = 2.45 \[ \frac{x}{{\left (a^{2} + 2 \, a b + b^{2} - 2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} n\right )} x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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 = 2.67464, size = 82, normalized size = 4.1 \[ \begin{cases} - \frac{x}{2 a^{2} n x^{2 n} - a^{2} x^{2 n} + 4 a b n x^{2 n} - 2 a b x^{2 n} + 2 b^{2} n x^{2 n} - b^{2} x^{2 n}} & \text{for}\: n \neq \frac{1}{2} \\\frac{\log{\left (x \right )}}{a^{2} + 2 a b + b^{2}} & \text{otherwise} \end{cases} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^n + b*x^n)^(-2),x, algorithm="giac")

[Out]

integrate((a*x^n + b*x^n)^(-2), x)