Optimal. Leaf size=51 \[ \frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{a x}+1\right )^{2 p+1} \left (c-a^2 c x^2\right )^p}{2 p+1} \]
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Rubi [A] time = 0.12, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6192, 6196, 37} \[ \frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{a x}+1\right )^{2 p+1} \left (c-a^2 c x^2\right )^p}{2 p+1} \]
Antiderivative was successfully verified.
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Rule 37
Rule 6192
Rule 6196
Rubi steps
\begin {align*} \int e^{2 p \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p x^{2 p} \, dx\\ &=-\left (\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \operatorname {Subst}\left (\int x^{-2-2 p} \left (1+\frac {x}{a}\right )^{2 p} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1+2 p} x \left (c-a^2 c x^2\right )^p}{1+2 p}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 36, normalized size = 0.71 \[ \frac {(a x+1) \left (c-a^2 c x^2\right )^p e^{2 p \coth ^{-1}(a x)}}{2 a p+a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 42, normalized size = 0.82 \[ \frac {{\left (a x - 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{p}}{2 \, a p + a} \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 38, normalized size = 0.75 \[ \frac {\left (a x +1\right ) {\mathrm e}^{2 p \,\mathrm {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{p}}{a \left (1+2 p \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 36, normalized size = 0.71 \[ \frac {{\left (a \left (-c\right )^{p} x - \left (-c\right )^{p}\right )} {\left (a x - 1\right )}^{2 \, p}}{a {\left (2 \, p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 59, normalized size = 1.16 \[ \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x+1\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^p}{a\,\left (2\,p+1\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^p} \]
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 \[ \begin {cases} - \frac {i x}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x e^{i \pi p} & \text {for}\: a = 0 \\\int \frac {e^{- \operatorname {acoth}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a x \left (- a^{2} c x^{2} + c\right )^{p} e^{2 p \operatorname {acoth}{\left (a x \right )}}}{2 a p + a} + \frac {\left (- a^{2} c x^{2} + c\right )^{p} e^{2 p \operatorname {acoth}{\left (a x \right )}}}{2 a p + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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