Optimal. Leaf size=20 \[ \frac{x^{1-2 n}}{(1-2 n) (a+b)^2} \]
[Out]
________________________________________________________________________________________
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]
[Out]
Rule 6
Rule 12
Rule 30
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]