Optimal. Leaf size=133 \[ \frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} x^{m-p+1} \, _2F_1\left (\frac {1}{2},\frac {m-p+1}{2 p};\frac {m+p+1}{2 p};a^2 x^{2 p}\right )}{a (m+1) (m-p+1)}+\frac {p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^p\right )}}{m+1} \]
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Rubi [A] time = 0.08, antiderivative size = 133, 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} \[ \frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} x^{m-p+1} \, _2F_1\left (\frac {1}{2},\frac {m-p+1}{2 p};\frac {m+p+1}{2 p};a^2 x^{2 p}\right )}{a (m+1) (m-p+1)}+\frac {p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^p\right )}}{m+1} \]
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^p\right )} x^m \, dx &=\frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p \int x^{m-p} \, dx}{a (1+m)}+\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{m-p}}{\sqrt {1-a x^p} \sqrt {1+a x^p}} \, dx}{a (1+m)}\\ &=\frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{m-p}}{\sqrt {1-a^2 x^{2 p}}} \, dx}{a (1+m)}\\ &=\frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac {p x^{1+m-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \, _2F_1\left (\frac {1}{2},\frac {1+m-p}{2 p};\frac {1+m+p}{2 p};a^2 x^{2 p}\right )}{a (1+m) (1+m-p)}\\ \end {align*}
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Mathematica [A] time = 5.17, size = 186, normalized size = 1.40 \[ \frac {2^{\frac {m+1}{p}} x^{m+1} \left (a x^p\right )^{-\frac {m+1}{p}} e^{\text {sech}^{-1}\left (a x^p\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{e^{2 \text {sech}^{-1}\left (a x^p\right )}+1}\right )^{\frac {m+1}{p}} \left ((m+3 p+1) \, _2F_1\left (1,1-\frac {m+p+1}{2 p};\frac {m+3 p+1}{2 p};-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )-(m+p+1) e^{2 \text {sech}^{-1}\left (a x^p\right )} \, _2F_1\left (1,-\frac {m-3 p+1}{2 p};\frac {m+5 p+1}{2 p};-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )\right )}{(m+p+1) (m+3 p+1)} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} {\left (\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.08, size = 0, normalized size = 0.00 \[ \int \left (\frac {x^{-p}}{a}+\sqrt {\frac {x^{-p}}{a}-1}\, \sqrt {\frac {x^{-p}}{a}+1}\right ) x^{m}\, dx \]
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 \[ \int x^m\,\left (\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}\right ) \,d x \]
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 \[ \frac {\int x^{m} x^{- p}\, dx + \int a x^{m} \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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