Optimal. Leaf size=51 \[ \frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1+2 p} x \left (c-a^2 c x^2\right )^p}{1+2 p} \]
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Rubi [A]
time = 0.08, 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 = {6327, 6331, 37}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 6327
Rule 6331
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 ) \text {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 = 0.05, size = 36, normalized size = 0.71 \begin {gather*} \frac {e^{2 p \coth ^{-1}(a x)} (1+a x) \left (c-a^2 c x^2\right )^p}{a+2 a p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 38, normalized size = 0.75
method | result | size |
gosper | \(\frac {\left (a x +1\right ) {\mathrm e}^{2 p \,\mathrm {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{p}}{a \left (1+2 p \right )}\) | \(38\) |
risch | \(\frac {\left (a x +1\right ) \left (a x +1\right )^{p} \left (a x -1\right )^{-p} {\mathrm e}^{\frac {p \left (-i \pi \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{3}+i \pi \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \mathrm {csgn}\left (i \left (a x -1\right )\right )+i \pi \mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \mathrm {csgn}\left (i \left (a x +1\right )\right )-i \pi \,\mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \mathrm {csgn}\left (i \left (a x -1\right )\right ) \mathrm {csgn}\left (i \left (a x +1\right )\right )+i \pi \,\mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \mathrm {csgn}\left (i c \left (a x +1\right ) \left (a x -1\right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \mathrm {csgn}\left (i c \left (a x +1\right ) \left (a x -1\right )\right ) \mathrm {csgn}\left (i c \right )+i \pi \mathrm {csgn}\left (i c \left (a x +1\right ) \left (a x -1\right )\right )^{3}+i \pi \mathrm {csgn}\left (i c \left (a x +1\right ) \left (a x -1\right )\right )^{2} \mathrm {csgn}\left (i c \right )-2 i \pi \mathrm {csgn}\left (i c \left (a x +1\right ) \left (a x -1\right )\right )^{2}+2 i \pi +2 \ln \left (c \right )+2 \ln \left (a x +1\right )+2 \ln \left (a x -1\right )\right )}{2}}}{a \left (1+2 p \right )}\) | \(317\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 34, normalized size = 0.67 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 42, normalized size = 0.82 \begin {gather*} \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} \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*} \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} \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 = 1.34, size = 59, normalized size = 1.16 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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