Optimal. Leaf size=60 \[ \frac {x (1-a x)^{-p} F_1\left (1-p;\frac {1}{2}-p,-\frac {1}{2};2-p;a x,-a x\right ) \left (c-\frac {c}{a x}\right )^p}{1-p} \]
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Rubi [A] time = 0.10, 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 = {6134, 6129, 133} \[ \frac {x (1-a x)^{-p} F_1\left (1-p;\frac {1}{2}-p,-\frac {1}{2};2-p;a x,-a x\right ) \left (c-\frac {c}{a x}\right )^p}{1-p} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6129
Rule 6134
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 = 1.40, size = 0, normalized size = 0.00 \[ \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \left (\frac {a c x - c}{a x}\right )^{p}}{a x - 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, 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 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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