Optimal. Leaf size=133 \[ \frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac {p x^{1+m-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \, _2F_1\left (\frac {1}{2},\frac {1+m-p}{2 p};\frac {1+m+p}{2 p};a^2 x^{2 p}\right )}{a (1+m) (1+m-p)} \]
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Rubi [A]
time = 0.06, 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 = {6470, 30, 265,
371} \begin {gather*} \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} \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^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 = 3.78, size = 212, normalized size = 1.59 \begin {gather*} \frac {2^{\frac {1+m}{p}} \left (\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^p\right )}}\right )^{\frac {1+m+p}{p}} \left (1+e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )^{\frac {1+m+p}{p}} x^{1+m} \left (a x^p\right )^{-\frac {1+m}{p}} \left ((1+m+3 p) \, _2F_1\left (\frac {1+m+p}{2 p},\frac {1+m+p}{p};\frac {1+m+3 p}{2 p};-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )-e^{2 \text {sech}^{-1}\left (a x^p\right )} (1+m+p) \, _2F_1\left (\frac {1+m+p}{p},\frac {1+m+3 p}{2 p};\frac {1+m+5 p}{2 p};-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )\right )}{(1+m+p) (1+m+3 p)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.42, 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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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