Optimal. Leaf size=107 \[ -\frac {2 x^{-1+m}}{a \left (1-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}-\frac {2 x^{-1+m} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-1+m);\frac {3+m}{4};a^2 x^4\right )}{a \left (1-m^2\right )} \]
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Rubi [A]
time = 0.04, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6470, 30, 265,
371} \begin {gather*} -\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} x^{m-1} \, _2F_1\left (\frac {1}{2},\frac {m-1}{4};\frac {m+3}{4};a^2 x^4\right )}{a \left (1-m^2\right )}-\frac {2 x^{m-1}}{a \left (1-m^2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^2\right )}}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 265
Rule 371
Rule 6470
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} x^m \, dx &=\frac {e^{\text {sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}+\frac {2 \int x^{-2+m} \, dx}{a (1+m)}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^{-2+m}}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a (1+m)}\\ &=-\frac {2 x^{-1+m}}{a \left (1-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^{-2+m}}{\sqrt {1-a^2 x^4}} \, dx}{a (1+m)}\\ &=-\frac {2 x^{-1+m}}{a \left (1-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^2\right )} x^{1+m}}{1+m}-\frac {2 x^{-1+m} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-1+m);\frac {3+m}{4};a^2 x^4\right )}{a \left (1-m^2\right )}\\ \end {align*}
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Mathematica [A]
time = 1.72, size = 214, normalized size = 2.00 \begin {gather*} \frac {2^{\frac {1+m}{2}} e^{-\frac {1}{2} (1+m) \text {sech}^{-1}\left (a x^2\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^2\right )}}\right )^{\frac {1+m}{2}} \left (1+e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )^{\frac {1+m}{2}} x^{1+m} \left (a x^2\right )^{\frac {1}{2} (-1-m)} \left (e^{\frac {1}{2} (3+m) \text {sech}^{-1}\left (a x^2\right )} (7+m) \, _2F_1\left (\frac {3+m}{4},\frac {3+m}{2};\frac {7+m}{4};-e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )-e^{\frac {1}{2} (7+m) \text {sech}^{-1}\left (a x^2\right )} (3+m) \, _2F_1\left (\frac {3+m}{2},\frac {7+m}{4};\frac {11+m}{4};-e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )\right )}{(3+m) (7+m)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (\frac {1}{a \,x^{2}}+\sqrt {\frac {1}{a \,x^{2}}-1}\, \sqrt {\frac {1}{a \,x^{2}}+1}\right ) x^{m}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{m}}{x^{2}}\, dx + \int a x^{m} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \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 (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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