Optimal. Leaf size=36 \[ -\frac {x^{1+m}}{1+m}+\frac {2 x^{1+m} \, _2F_1(1,1+m;2+m;a x)}{1+m} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6261, 81, 66}
\begin {gather*} \frac {2 x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{m+1}-\frac {x^{m+1}}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 81
Rule 6261
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^m \, dx &=\int \frac {x^m (1+a x)}{1-a x} \, dx\\ &=-\frac {x^{1+m}}{1+m}+2 \int \frac {x^m}{1-a x} \, dx\\ &=-\frac {x^{1+m}}{1+m}+\frac {2 x^{1+m} \, _2F_1(1,1+m;2+m;a x)}{1+m}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.72 \begin {gather*} \frac {x^{1+m} (-1+2 \, _2F_1(1,1+m;2+m;a x))}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
5.
time = 0.78, size = 184, normalized size = 5.11
method | result | size |
meijerg | \(-\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (-3-m \right )}{\left (3+m \right ) \left (1+m \right ) a^{2}}+\frac {x^{1+m} \left (-a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \Phi \left (a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{a^{2}}\right )}{2}-\frac {\left (-a^{2}\right )^{-\frac {m}{2}} \left (-\frac {2 x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (-m -2\right )}{\left (2+m \right ) m}-x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \Phi \left (a^{2} x^{2}, 1, \frac {m}{2}\right )\right )}{a}+\frac {x^{1+m} \left (\frac {1}{2}+\frac {m}{2}\right ) \Phi \left (a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}\) | \(184\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (26) = 52\).
time = 2.02, size = 99, normalized size = 2.75 \begin {gather*} \frac {a m x^{2} x^{m} \Phi \left (a x, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {2 a x^{2} x^{m} \Phi \left (a x, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {m x x^{m} \Phi \left (a x, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac {x x^{m} \Phi \left (a x, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.03 \begin {gather*} \int -\frac {x^m\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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