Optimal. Leaf size=119 \[ \frac {1}{2} e^{\text {sech}^{-1}\left (a x^p\right )} x^2+\frac {p x^{2-p}}{2 a (2-p)}+\frac {p x^{2-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (-1+\frac {2}{p}\right );\frac {1}{2} \left (1+\frac {2}{p}\right );a^2 x^{2 p}\right )}{2 a (2-p)} \]
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Rubi [A]
time = 0.04, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6470, 30, 265,
371} \begin {gather*} \frac {p x^{2-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (\frac {2}{p}-1\right );\frac {1}{2} \left (1+\frac {2}{p}\right );a^2 x^{2 p}\right )}{2 a (2-p)}+\frac {p x^{2-p}}{2 a (2-p)}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^p\right )} \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 \, dx &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^p\right )} x^2+\frac {p \int x^{1-p} \, dx}{2 a}+\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{1-p}}{\sqrt {1-a x^p} \sqrt {1+a x^p}} \, dx}{2 a}\\ &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^p\right )} x^2+\frac {p x^{2-p}}{2 a (2-p)}+\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{1-p}}{\sqrt {1-a^2 x^{2 p}}} \, dx}{2 a}\\ &=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^p\right )} x^2+\frac {p x^{2-p}}{2 a (2-p)}+\frac {p x^{2-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (-1+\frac {2}{p}\right );\frac {1}{2} \left (1+\frac {2}{p}\right );a^2 x^{2 p}\right )}{2 a (2-p)}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 159, normalized size = 1.34 \begin {gather*} \frac {x^{2-p} \left (-1-\sqrt {\frac {1-a x^p}{1+a x^p}}-a x^p \sqrt {\frac {1-a x^p}{1+a x^p}}+\frac {a^2 p x^{2 p} \sqrt {\frac {1-a x^p}{1+a x^p}} \sqrt {1-a^2 x^{2 p}} \, _2F_1\left (\frac {1}{2},\frac {1}{2}+\frac {1}{p};\frac {3}{2}+\frac {1}{p};a^2 x^{2 p}\right )}{(2+p) \left (-1+a x^p\right )}\right )}{a (-2+p)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.41, size = 0, normalized size = 0.00 \[\int \left (\frac {x^{-p}}{a}+\sqrt {\frac {x^{-p}}{a}-1}\, \sqrt {\frac {x^{-p}}{a}+1}\right ) x\, dx\]
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 x^{- p}\, dx + \int a x \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\,\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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