Optimal. Leaf size=64 \[ \frac {2 x^m \, _2F_1\left (-\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};-\frac {1}{a^2 x^2}\right )}{a m}-\frac {2 x^{m-1}}{a^2 (1-m)}+\frac {x^{m+1}}{m+1} \]
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Rubi [A] time = 0.32, 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 = {6338, 6742, 339, 364} \[ \frac {2 x^m \, _2F_1\left (-\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};-\frac {1}{a^2 x^2}\right )}{a m}-\frac {2 x^{m-1}}{a^2 (1-m)}+\frac {x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 339
Rule 364
Rule 6338
Rule 6742
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 ) \operatorname {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 = 0.07, size = 57, normalized size = 0.89 \[ x^m \left (\frac {2 \, _2F_1\left (-\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};-\frac {1}{a^2 x^2}\right )}{a m}+\frac {2}{a^2 (m-1) x}+\frac {x}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, a x x^{m} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + {\left (a^{2} x^{2} + 2\right )} x^{m}}{a^{2} x^{2}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
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 \[ \int \left (\frac {1}{a x}+\sqrt {1+\frac {1}{x^{2} a^{2}}}\right )^{2} x^{m}\, dx \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int x^m\,{\left (\sqrt {\frac {1}{a^2\,x^2}+1}+\frac {1}{a\,x}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.15, size = 71, normalized size = 1.11 \[ \begin {cases} \frac {x^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases} - \frac {x^{m} \Gamma \left (- \frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {m}{2} \\ 1 - \frac {m}{2} \end {matrix}\middle | {\frac {e^{i \pi }}{a^{2} x^{2}}} \right )}}{a \Gamma \left (1 - \frac {m}{2}\right )} + \frac {2 \left (\begin {cases} \frac {x^{m}}{m x - x} & \text {for}\: m \neq 1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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