Optimal. Leaf size=64 \[ -\frac {2 x^{-1+m}}{a^2 (1-m)}+\frac {x^{1+m}}{1+m}+\frac {2 x^m \, _2F_1\left (-\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};-\frac {1}{a^2 x^2}\right )}{a m} \]
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Rubi [A]
time = 0.23, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6473, 6874,
346, 371} \begin {gather*} \frac {2 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 {2 x^{m-1}}{a^2 (1-m)}+\frac {x^{m+1}}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 346
Rule 371
Rule 6473
Rule 6874
Rubi steps
\begin {align*} \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx &=\int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 x^m \, dx\\ &=\int \left (\frac {2 x^{-2+m}}{a^2}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} x^{-1+m}}{a}+x^m\right ) \, dx\\ &=-\frac {2 x^{-1+m}}{a^2 (1-m)}+\frac {x^{1+m}}{1+m}+\frac {2 \int \sqrt {1+\frac {1}{a^2 x^2}} x^{-1+m} \, dx}{a}\\ &=-\frac {2 x^{-1+m}}{a^2 (1-m)}+\frac {x^{1+m}}{1+m}-\frac {\left (2 \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int x^{-1-m} \sqrt {1+\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {2 x^{-1+m}}{a^2 (1-m)}+\frac {x^{1+m}}{1+m}+\frac {2 x^m \, _2F_1\left (-\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};-\frac {1}{a^2 x^2}\right )}{a m}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 57, normalized size = 0.89 \begin {gather*} x^m \left (\frac {2}{a^2 (-1+m) x}+\frac {x}{1+m}+\frac {2 \, _2F_1\left (-\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};-\frac {1}{a^2 x^2}\right )}{a m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (\frac {1}{a x}+\sqrt {1+\frac {1}{a^{2} x^{2}}}\right )^{2} x^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 [A]
time = 4.17, size = 71, normalized size = 1.11 \begin {gather*} \begin {cases} \frac {x^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases} - \frac {x^{m} \Gamma \left (- \frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {m}{2} \\ 1 - \frac {m}{2} \end {matrix}\middle | {\frac {e^{i \pi }}{a^{2} x^{2}}} \right )}}{a \Gamma \left (1 - \frac {m}{2}\right )} + \frac {2 \left (\begin {cases} \frac {x^{m}}{m x - x} & \text {for}\: m \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{a^{2}} \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 = -0.02 \begin {gather*} \int x^m\,{\left (\sqrt {\frac {1}{a^2\,x^2}+1}+\frac {1}{a\,x}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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