Optimal. Leaf size=105 \[ -\frac {(1-x)^{1+n} (1+x)^{1-n}}{3 x^3}+\frac {n (1-x)^{1+n} (1+x)^{1-n}}{3 x^2}-\frac {2 \left (1+2 n^2\right ) (1-x)^{1+n} (1+x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac {1-x}{1+x}\right )}{3 (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {105, 156, 12,
133} \begin {gather*} -\frac {2 \left (2 n^2+1\right ) (1-x)^{n+1} (x+1)^{-n-1} \, _2F_1\left (2,n+1;n+2;\frac {1-x}{x+1}\right )}{3 (n+1)}-\frac {(1-x)^{n+1} (x+1)^{1-n}}{3 x^3}+\frac {n (1-x)^{n+1} (x+1)^{1-n}}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 105
Rule 133
Rule 156
Rubi steps
\begin {align*} \int \frac {(1-x)^n (1+x)^{-n}}{x^4} \, dx &=-\frac {(1-x)^{1+n} (1+x)^{1-n}}{3 x^3}-\frac {1}{3} \int \frac {(1-x)^n (2 n-x) (1+x)^{-n}}{x^3} \, dx\\ &=-\frac {(1-x)^{1+n} (1+x)^{1-n}}{3 x^3}+\frac {n (1-x)^{1+n} (1+x)^{1-n}}{3 x^2}+\frac {1}{6} \int \frac {\left (2+4 n^2\right ) (1-x)^n (1+x)^{-n}}{x^2} \, dx\\ &=-\frac {(1-x)^{1+n} (1+x)^{1-n}}{3 x^3}+\frac {n (1-x)^{1+n} (1+x)^{1-n}}{3 x^2}+\frac {1}{3} \left (1+2 n^2\right ) \int \frac {(1-x)^n (1+x)^{-n}}{x^2} \, dx\\ &=-\frac {(1-x)^{1+n} (1+x)^{1-n}}{3 x^3}+\frac {n (1-x)^{1+n} (1+x)^{1-n}}{3 x^2}-\frac {2 \left (1+2 n^2\right ) (1-x)^{1+n} (1+x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac {1-x}{1+x}\right )}{3 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 77, normalized size = 0.73 \begin {gather*} -\frac {(1-x)^{1+n} (1+x)^{-1-n} \left (-\left ((1+n) (1+x)^2 (-1+n x)\right )+2 \left (1+2 n^2\right ) x^3 \, _2F_1\left (2,1+n;2+n;\frac {1-x}{1+x}\right )\right )}{3 (1+n) x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (1-x \right )^{n} \left (1+x \right )^{-n}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1 - x\right )^{n} \left (x + 1\right )^{- n}}{x^{4}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-x\right )}^n}{x^4\,{\left (x+1\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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