Optimal. Leaf size=116 \[ \frac {8 \sqrt {c-a^2 c x^2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \, _2F_1\left (3,\frac {3-n}{2};\frac {5-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.16, antiderivative size = 116, 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 = {6192, 6195, 131} \[ \frac {8 \sqrt {c-a^2 c x^2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \, _2F_1\left (3,\frac {3-n}{2};\frac {5-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 131
Rule 6192
Rule 6195
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=-\frac {\sqrt {c-a^2 c x^2} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2}+\frac {n}{2}}}{x^3} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {8 \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} \sqrt {c-a^2 c x^2} \, _2F_1\left (3,\frac {3-n}{2};\frac {5-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) \sqrt {1-\frac {1}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 101, normalized size = 0.87 \[ -\frac {c x \sqrt {1-\frac {1}{a^2 x^2}} e^{n \coth ^{-1}(a x)} \left (a x \sqrt {1-\frac {1}{a^2 x^2}} (a x+n)+2 (n-1) e^{\coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};e^{2 \coth ^{-1}(a x)}\right )\right )}{2 \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \sqrt {-a^{2} c \,x^{2}+c}\, dx \]
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 \[ \int \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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 \[ \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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