Optimal. Leaf size=164 \[ -\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}}-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.34, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6189, 6192, 6195, 131} \[ -\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}}-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 131
Rule 6189
Rule 6192
Rule 6195
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx}{a^2 c}\\ &=-\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\left (\sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}} x} \, dx}{a^2 c \sqrt {c-a^2 c x^2}}\\ &=-\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\left (\sqrt {1-\frac {1}{a^2 x^2}} x\right ) \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} \, dx,x,\frac {1}{x}\right )}{a^2 c \sqrt {c-a^2 c x^2}}\\ &=-\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^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 \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 127, normalized size = 0.77 \[ -\frac {e^{n \coth ^{-1}(a x)} \left (2 (n-1) \left (a^2 x^2-1\right ) e^{\coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};e^{2 \coth ^{-1}(a x)}\right )+a x \sqrt {1-\frac {1}{a^2 x^2}} (a x-n)\right )}{a^4 c (n-1) (n+1) x \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} x^{2}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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