Optimal. Leaf size=95 \[ \frac {\left (1-a^2 x^2\right )^p (1-a x)^{1-2 p} \left (c-a^2 c x^2\right )^{-p}}{a (1-2 p)}+\frac {\left (1-a^2 x^2\right )^p (1-a x)^{-2 p} \left (c-a^2 c x^2\right )^{-p}}{a p} \]
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6143, 6140, 43} \[ \frac {\left (1-a^2 x^2\right )^p (1-a x)^{1-2 p} \left (c-a^2 c x^2\right )^{-p}}{a (1-2 p)}+\frac {\left (1-a^2 x^2\right )^p (1-a x)^{-2 p} \left (c-a^2 c x^2\right )^{-p}}{a p} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{2 (1+p) \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx &=\left (\left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}\right ) \int e^{2 (1+p) \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{-p} \, dx\\ &=\left (\left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}\right ) \int (1-a x)^{-1-2 p} (1+a x) \, dx\\ &=\left (\left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}\right ) \int \left (2 (1-a x)^{-1-2 p}-(1-a x)^{-2 p}\right ) \, dx\\ &=\frac {(1-a x)^{1-2 p} \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p} \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}}{a p}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.61 \[ \frac {(1-a x)^{-2 p} (a p x+p-1) \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}}{a p (2 p-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 81, normalized size = 0.85 \[ -\frac {{\left (a^{2} p x^{2} - a x - p + 1\right )} \left (\frac {a x + 1}{a x - 1}\right )^{p + 1}}{{\left (2 \, a p^{2} - a p + {\left (2 \, a^{2} p^{2} - a^{2} p\right )} x\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}} \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{p + 1}}{{\left (-a^{2} c x^{2} + c\right )}^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 60, normalized size = 0.63 \[ -\frac {\left (a x -1\right ) \left (a p x +p -1\right ) {\mathrm e}^{2 \left (1+p \right ) \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{-p}}{a p \left (2 p -1\right ) \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 41, normalized size = 0.43 \[ -\frac {a p x + p - 1}{{\left (2 \, p^{2} - p\right )} {\left (a x - 1\right )}^{2 \, p} a \left (-c\right )^{p}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 86, normalized size = 0.91 \[ -\frac {p\,{\left (a\,x+1\right )}^p-{\left (a\,x+1\right )}^p+a\,p\,x\,{\left (a\,x+1\right )}^p}{a\,p\,{\left (c-a^2\,c\,x^2\right )}^p\,{\left (1-a\,x\right )}^p-2\,a\,p^2\,{\left (c-a^2\,c\,x^2\right )}^p\,{\left (1-a\,x\right )}^p} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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