Optimal. Leaf size=93 \[ \frac {(4-p) \left (c-\frac {c}{a x}\right )^p \, _2F_1\left (1,p;p+1;1-\frac {1}{a x}\right )}{a p}-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{p-1}}{a (1-p)}+c x \left (c-\frac {c}{a x}\right )^{p-1} \]
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Rubi [A] time = 0.11, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6133, 25, 514, 375, 89, 79, 65} \[ \frac {(4-p) \left (c-\frac {c}{a x}\right )^p \, _2F_1\left (1,p;p+1;1-\frac {1}{a x}\right )}{a p}-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{p-1}}{a (1-p)}+c x \left (c-\frac {c}{a x}\right )^{p-1} \]
Antiderivative was successfully verified.
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Rule 25
Rule 65
Rule 79
Rule 89
Rule 375
Rule 514
Rule 6133
Rubi steps
\begin {align*} \int e^{4 \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)^2}{(1-a x)^2} \, dx\\ &=\frac {c^2 \int \frac {\left (c-\frac {c}{a x}\right )^{-2+p} (1+a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \left (a+\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{-2+p} \, dx}{a^2}\\ &=-\frac {c^2 \operatorname {Subst}\left (\int \frac {(a+x)^2 \left (c-\frac {c x}{a}\right )^{-2+p}}{x^2} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=c \left (c-\frac {c}{a x}\right )^{-1+p} x-\frac {c \operatorname {Subst}\left (\int \frac {(a c (4-p)+c x) \left (c-\frac {c x}{a}\right )^{-2+p}}{x} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{-1+p}}{a (1-p)}+c \left (c-\frac {c}{a x}\right )^{-1+p} x-\frac {(c (4-p)) \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{-1+p}}{x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{-1+p}}{a (1-p)}+c \left (c-\frac {c}{a x}\right )^{-1+p} x+\frac {(4-p) \left (c-\frac {c}{a x}\right )^p \, _2F_1\left (1,p;1+p;1-\frac {1}{a x}\right )}{a p}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 0.87 \[ \frac {\left (c-\frac {c}{a x}\right )^p \left (a p x (p (a x-1)-a x+5)-\left (p^2-5 p+4\right ) (a x-1) \, _2F_1\left (1,p;p+1;1-\frac {1}{a x}\right )\right )}{a (p-1) p (a x-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \left (\frac {a c x - c}{a x}\right )^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a x}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{4} \left (c -\frac {c}{a x}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a x}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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