Optimal. Leaf size=60 \[ \frac {\left (c-\frac {c}{a x}\right )^p x (1-a x)^{-p} F_1\left (1-p;\frac {1}{2}-p,-\frac {1}{2};2-p;a x,-a x\right )}{1-p} \]
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Rubi [A]
time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6269, 6264,
138} \begin {gather*} \frac {x (1-a x)^{-p} F_1\left (1-p;\frac {1}{2}-p,-\frac {1}{2};2-p;a x,-a x\right ) \left (c-\frac {c}{a x}\right )^p}{1-p} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx &=\left (\left (c-\frac {c}{a x}\right )^p x^p (1-a x)^{-p}\right ) \int e^{\tanh ^{-1}(a x)} x^{-p} (1-a x)^p \, dx\\ &=\left (\left (c-\frac {c}{a x}\right )^p x^p (1-a x)^{-p}\right ) \int x^{-p} (1-a x)^{-\frac {1}{2}+p} \sqrt {1+a x} \, dx\\ &=\frac {\left (c-\frac {c}{a x}\right )^p x (1-a x)^{-p} F_1\left (1-p;\frac {1}{2}-p,-\frac {1}{2};2-p;a x,-a x\right )}{1-p}\\ \end {align*}
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Mathematica [F]
time = 1.13, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right ) \left (c -\frac {c}{a x}\right )^{p}}{\sqrt {-a^{2} x^{2}+1}}\, dx\]
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 (- c \left (-1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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