Optimal. Leaf size=41 \[ \frac {(1-a x)^{1-2 p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p}}{a p} \]
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Rubi [A]
time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6275, 45}
\begin {gather*} \frac {(1-a x)^{1-2 p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p}}{a p} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6275
Rubi steps
\begin {align*} \int e^{2 (1+p) \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{-p} \, dx &=\int (1-a x)^{-1-2 p} (1+a x) \, dx\\ &=\int \left (2 (1-a x)^{-1-2 p}-(1-a x)^{-2 p}\right ) \, dx\\ &=\frac {(1-a x)^{1-2 p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p}}{a p}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.76 \begin {gather*} \frac {(1-a x)^{-2 p} (-1+p+a p x)}{a p (-1+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 59, normalized size = 1.44
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (a p x +p -1\right ) {\mathrm e}^{2 \left (1+p \right ) \arctanh \left (a x \right )} \left (-a^{2} x^{2}+1\right )^{-p}}{a p \left (2 p -1\right ) \left (a x +1\right )}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 34, normalized size = 0.83 \begin {gather*} \frac {a p x + p - 1}{{\left (2 \, p^{2} - p\right )} {\left (-a x + 1\right )}^{2 \, p} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 81, normalized size = 1.98 \begin {gather*} -\frac {{\left (a^{2} p x^{2} - a x - p + 1\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{p + 1}}{{\left (2 \, a p^{2} - a p + {\left (2 \, a^{2} p^{2} - a^{2} p\right )} x\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}} \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 \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{- p} e^{2 p \operatorname {atanh}{\left (a x \right )}} e^{2 \operatorname {atanh}{\left (a x \right )}}\, 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 [B]
time = 1.12, size = 84, normalized size = 2.05 \begin {gather*} \frac {p\,{\left (a\,x+1\right )}^p-{\left (a\,x+1\right )}^p+a\,p\,x\,{\left (a\,x+1\right )}^p}{2\,a\,p^2\,{\left (1-a^2\,x^2\right )}^p\,{\left (1-a\,x\right )}^p-a\,p\,{\left (1-a^2\,x^2\right )}^p\,{\left (1-a\,x\right )}^p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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