Optimal. Leaf size=90 \[ -\frac {2^{p+\frac {1}{2}} \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{-p} F_1\left (\frac {3}{2};\frac {1}{2}-p,2;\frac {5}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{3 a} \]
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Rubi [A] time = 0.08, antiderivative size = 90, 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 = {6182, 6179, 136} \[ -\frac {2^{p+\frac {1}{2}} \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{-p} F_1\left (\frac {3}{2};\frac {1}{2}-p,2;\frac {5}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{3 a} \]
Antiderivative was successfully verified.
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Rule 136
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx &=\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {1}{2}+p} \sqrt {1+\frac {x}{a}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {2^{\frac {1}{2}+p} \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^p F_1\left (\frac {3}{2};\frac {1}{2}-p,2;\frac {5}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{3 a}\\ \end {align*}
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Mathematica [F] time = 1.08, size = 0, normalized size = 0.00 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + 1\right )} \left (\frac {a c x - c}{a x}\right )^{p} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, x\right ) \]
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 \[ \int \frac {{\left (c - \frac {c}{a x}\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (c -\frac {c}{a x}\right )^{p}}{\sqrt {\frac {a x -1}{a x +1}}}\, dx \]
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 \[ \int \frac {{\left (c - \frac {c}{a x}\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^p}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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