Optimal. Leaf size=20 \[ \frac {x^{1-2 n}}{(1-2 n) (a+b)^2} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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.
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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*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ \frac {x^{1-2 n}}{(1-2 n) (a+b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 36, normalized size = 1.80 \[ \frac {x}{{\left (a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} n\right )} x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x^{n} + b x^{n}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 1.05 \[ -\frac {x \,x^{-2 n}}{\left (2 n -1\right ) \left (a +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 40, normalized size = 2.00 \[ -\frac {x}{{\left (a^{2} {\left (2 \, n - 1\right )} + 2 \, a b {\left (2 \, n - 1\right )} + b^{2} {\left (2 \, n - 1\right )}\right )} x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 21, normalized size = 1.05 \[ -\frac {x^{1-2\,n}}{{\left (a+b\right )}^2\,\left (2\,n-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 82, normalized size = 4.10 \[ \begin {cases} - \frac {x}{2 a^{2} n x^{2 n} - a^{2} x^{2 n} + 4 a b n x^{2 n} - 2 a b x^{2 n} + 2 b^{2} n x^{2 n} - b^{2} x^{2 n}} & \text {for}\: n \neq \frac {1}{2} \\\frac {\log {\relax (x )}}{a^{2} + 2 a b + b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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