Optimal. Leaf size=151 \[ -\frac {3 x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1}+\frac {4 x^{m+1} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1}-\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}+\frac {4 x^m \, _2F_1\left (\frac {3}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m} \]
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Rubi [A] time = 1.23, antiderivative size = 151, 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 = {6172, 6742, 364, 850, 808} \[ -\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}+\frac {4 x^m \, _2F_1\left (\frac {3}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}-\frac {3 x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1}+\frac {4 x^{m+1} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 364
Rule 808
Rule 850
Rule 6172
Rule 6742
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.36, size = 228, normalized size = 1.51 \[ \frac {x^{m+1} \left (3 (m+1) \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {1-a x} \sqrt {\frac {a x+1}{a^2}} F_1\left (m;-\frac {1}{2},\frac {1}{2};m+1;-a x,a x\right )-2 (m+1) \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {1-a x} \sqrt {\frac {a x+1}{a^2}} F_1\left (m;-\frac {1}{2},\frac {3}{2};m+1;-a x,a x\right )+m \sqrt {a x-1} \sqrt {a x+1} \sqrt {x^2-\frac {1}{a^2}} \, _2F_1\left (-\frac {1}{2},-\frac {m}{2}-\frac {1}{2};\frac {1}{2}-\frac {m}{2};\frac {1}{a^2 x^2}\right )\right )}{m (m+1) \sqrt {a x-1} \sqrt {a x+1} \sqrt {x^2-\frac {1}{a^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 \[ \int \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^m}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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