Optimal. Leaf size=107 \[ -\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} x^{m-1} \, _2F_1\left (\frac {1}{2},\frac {m-1}{4};\frac {m+3}{4};a^2 x^4\right )}{a \left (1-m^2\right )}-\frac {2 x^{m-1}}{a \left (1-m^2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^2\right )}}{m+1} \]
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Rubi [A] time = 0.06, 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 = {6335, 30, 259, 364} \[ -\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} x^{m-1} \, _2F_1\left (\frac {1}{2},\frac {m-1}{4};\frac {m+3}{4};a^2 x^4\right )}{a \left (1-m^2\right )}-\frac {2 x^{m-1}}{a \left (1-m^2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^2\right )}}{m+1} \]
Antiderivative was successfully verified.
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Rule 30
Rule 259
Rule 364
Rule 6335
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 = 2.39, size = 159, normalized size = 1.49 \[ \frac {2^{\frac {m+1}{2}} x^{m+1} \left (a x^2\right )^{\frac {1}{2} (-m-1)} e^{\text {sech}^{-1}\left (a x^2\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{e^{2 \text {sech}^{-1}\left (a x^2\right )}+1}\right )^{\frac {m+1}{2}} \left ((m+7) \, _2F_1\left (1,\frac {1-m}{4};\frac {m+7}{4};-e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )-(m+3) e^{2 \text {sech}^{-1}\left (a x^2\right )} \, _2F_1\left (1,\frac {5-m}{4};\frac {m+11}{4};-e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )\right )}{(m+3) (m+7)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{2} x^{m} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + x^{m}}{a x^{2}}, x\right ) \]
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 = 0.07, 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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,\left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right ) \,d x \]
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 \[ \frac {\int \frac {x^{m}}{x^{2}}\, dx + \int a x^{m} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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