Optimal. Leaf size=88 \[ -\frac {2 x \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} (c-a c x)^{\frac {n-4}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2-n} \]
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Rubi [A] time = 0.15, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6176, 6181, 131} \[ -\frac {2 x \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} (c-a c x)^{\frac {n-4}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2-n} \]
Antiderivative was successfully verified.
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Rule 131
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx &=\left (\left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} x^{2-\frac {n}{2}} (c-a c x)^{-2+\frac {n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{-2+\frac {n}{2}} x^{-2+\frac {n}{2}} \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{x}\right )^{-2+\frac {n}{2}} (c-a c x)^{-2+\frac {n}{2}}\right ) \operatorname {Subst}\left (\int \frac {x^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {2 \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x (c-a c x)^{\frac {1}{2} (-4+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2-n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 89, normalized size = 1.01 \[ \frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} (c-a c x)^{n/2} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {2}{a x+1}\right )}{a c^2 (n-2) (a x+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a c x + c\right )}^{\frac {1}{2} \, n - 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )^{-2+\frac {n}{2}}\, dx \]
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 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}-2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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