Optimal. Leaf size=150 \[ -\frac {3 x^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{1+m}+\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}+\frac {4 x^{1+m} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{1+m}-\frac {4 x^m \, _2F_1\left (\frac {3}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m} \]
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Rubi [A]
time = 0.80, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6307, 6874,
371, 864, 822} \begin {gather*} -\frac {3 x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1}+\frac {4 x^{m+1} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1}+\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}-\frac {4 x^m \, _2F_1\left (\frac {3}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 822
Rule 864
Rule 6307
Rule 6874
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} x^m \, dx &=-\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \left (1-\frac {x}{a}\right )^2}{\left (1+\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \left (-\frac {3 x^{-2-m}}{\sqrt {1-\frac {x^2}{a^2}}}+\frac {x^{-1-m}}{a \sqrt {1-\frac {x^2}{a^2}}}+\frac {4 x^{-2-m}}{\left (1+\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right )\right )\\ &=\left (3 \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )-\left (4 \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m}}{\left (1+\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )-\frac {\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {3 x^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{1+m}+\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}-\left (4 \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \left (1-\frac {x}{a}\right )}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 x^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{1+m}+\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}-\left (4 \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )+\frac {\left (4 \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {3 x^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{1+m}+\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}+\frac {4 x^{1+m} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{1+m}-\frac {4 x^m \, _2F_1\left (\frac {3}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.17, size = 192, normalized size = 1.28 \begin {gather*} \frac {x^{1+m} \left (-3 (1+m) \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {\frac {-1+a x}{a^2}} F_1\left (m;-\frac {1}{2},\frac {1}{2};1+m;a x,-a x\right )+2 (1+m) \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {\frac {-1+a x}{a^2}} F_1\left (m;-\frac {1}{2},\frac {3}{2};1+m;a x,-a x\right )+m \sqrt {1-a x} \sqrt {-\frac {1}{a^2}+x^2} \, _2F_1\left (-\frac {1}{2},-\frac {1}{2}-\frac {m}{2};\frac {1}{2}-\frac {m}{2};\frac {1}{a^2 x^2}\right )\right )}{m (1+m) \sqrt {1-a x} \sqrt {-\frac {1}{a^2}+x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{m} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x^m\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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