Optimal. Leaf size=114 \[ \frac {\left (c-\frac {c}{a x}\right )^{p+2} \, _2F_1\left (1,p+2;p+3;\frac {a-\frac {1}{x}}{2 a}\right )}{2 a c^2 (p+2)}-\frac {\left (c-\frac {c}{a x}\right )^{p+2} \, _2F_1\left (1,p+2;p+3;1-\frac {1}{a x}\right )}{a c^2}+\frac {x \left (c-\frac {c}{a x}\right )^{p+2}}{c^2} \]
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Rubi [A] time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6167, 6133, 25, 514, 375, 103, 156, 65, 68} \[ \frac {\left (c-\frac {c}{a x}\right )^{p+2} \, _2F_1\left (1,p+2;p+3;\frac {a-\frac {1}{x}}{2 a}\right )}{2 a c^2 (p+2)}-\frac {\left (c-\frac {c}{a x}\right )^{p+2} \, _2F_1\left (1,p+2;p+3;1-\frac {1}{a x}\right )}{a c^2}+\frac {x \left (c-\frac {c}{a x}\right )^{p+2}}{c^2} \]
Antiderivative was successfully verified.
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Rule 25
Rule 65
Rule 68
Rule 103
Rule 156
Rule 375
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx\\ &=-\int \frac {\left (c-\frac {c}{a x}\right )^p (1-a x)}{1+a x} \, dx\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{1+p} x}{1+a x} \, dx}{c}\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{1+p}}{a+\frac {1}{x}} \, dx}{c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}+\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p} \left (c (2+p)+\frac {c (1+p) x}{a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}-\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p}}{a+x} \, dx,x,\frac {1}{x}\right )}{a c}+\frac {(2+p) \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p}}{x} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=\frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}+\frac {\left (c-\frac {c}{a x}\right )^{2+p} \, _2F_1\left (1,2+p;3+p;\frac {a-\frac {1}{x}}{2 a}\right )}{2 a c^2 (2+p)}-\frac {\left (c-\frac {c}{a x}\right )^{2+p} \, _2F_1\left (1,2+p;3+p;1-\frac {1}{a x}\right )}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 87, normalized size = 0.76 \[ \frac {(a x-1)^2 \left (c-\frac {c}{a x}\right )^p \left (\, _2F_1\left (1,p+2;p+3;\frac {a-\frac {1}{x}}{2 a}\right )+2 (p+2) \left (a x-\, _2F_1\left (1,p+2;p+3;1-\frac {1}{a x}\right )\right )\right )}{2 a^3 (p+2) x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x - 1\right )} \left (\frac {a c x - c}{a x}\right )^{p}}{a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (c -\frac {c}{a x}\right )^{p} \left (a x -1\right )}{a x +1}\, dx \]
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 \[ \int \frac {{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{p}}{a x + 1}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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