Optimal. Leaf size=277 \[ -\frac {a^3 2^{\frac {n+1}{2}} x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{(n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6192, 6195, 89, 79, 69} \[ -\frac {a^3 2^{\frac {n+1}{2}} x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{(n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 79
Rule 89
Rule 6192
Rule 6195
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int x^2 \left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \left (\frac {n}{a}+\frac {(1+n) x}{a^2}\right ) \, dx,x,\frac {1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2} (-1+n)} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {2^{\frac {1+n}{2}} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} x^3 \, _2F_1\left (\frac {1-n}{2},\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 127, normalized size = 0.46 \[ \frac {e^{n \coth ^{-1}(a x)} \left (a x \sqrt {1-\frac {1}{a^2 x^2}} (a n x-1)-2 (n-1) \left (a^2 x^2-1\right ) e^{\coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-e^{2 \coth ^{-1}(a x)}\right )\right )}{a c (n-1) (n+1) x \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{2} x^{5} - 2 \, a^{2} c^{2} x^{3} + c^{2} x}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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