Optimal. Leaf size=20 \[ \frac {x^{1-2 n}}{(a+b)^2 (1-2 n)} \]
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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}
\begin {gather*} \frac {x^{1-2 n}}{(1-2 n) (a+b)^2} \end {gather*}
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 \begin {gather*} \frac {x^{1-2 n}}{(a+b)^2 (1-2 n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 1.05
method | result | size |
gosper | \(-\frac {x \,x^{-2 n}}{\left (-1+2 n \right ) \left (a +b \right )^{2}}\) | \(21\) |
risch | \(-\frac {x \,x^{-2 n}}{\left (a^{2}+2 a b +b^{2}\right ) \left (-1+2 n \right )}\) | \(29\) |
norman | \(-\frac {x \,{\mathrm e}^{-2 n \ln \left (x \right )}}{\left (2 a n +2 b n -a -b \right ) \left (a +b \right )}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 40, normalized size = 2.00 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.18, size = 36, normalized size = 1.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (15) = 30\).
time = 0.38, size = 82, normalized size = 4.10 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\frac {x^{1-2\,n}}{{\left (a+b\right )}^2\,\left (2\,n-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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