Optimal. Leaf size=107 \[ -\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x}+\frac {p x^{-1-p}}{a (1+p)}+\frac {p x^{-1-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \, _2F_1\left (\frac {1}{2},-\frac {1+p}{2 p};-\frac {1-p}{2 p};a^2 x^{2 p}\right )}{a (1+p)} \]
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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 {p x^{-p-1} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \, _2F_1\left (\frac {1}{2},-\frac {p+1}{2 p};-\frac {1-p}{2 p};a^2 x^{2 p}\right )}{a (p+1)}+\frac {p x^{-p-1}}{a (p+1)}-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 265
Rule 371
Rule 6470
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^2} \, dx &=-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x}-\frac {p \int x^{-2-p} \, dx}{a}-\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{-2-p}}{\sqrt {1-a x^p} \sqrt {1+a x^p}} \, dx}{a}\\ &=-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x}+\frac {p x^{-1-p}}{a (1+p)}-\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{-2-p}}{\sqrt {1-a^2 x^{2 p}}} \, dx}{a}\\ &=-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x}+\frac {p x^{-1-p}}{a (1+p)}+\frac {p x^{-1-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \, _2F_1\left (\frac {1}{2},-\frac {1+p}{2 p};-\frac {1-p}{2 p};a^2 x^{2 p}\right )}{a (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 156, normalized size = 1.46 \begin {gather*} x^{-1-p} \left (-\frac {1}{a+a p}-\frac {\sqrt {\frac {1-a x^p}{1+a x^p}} \left (1+a x^p\right )}{a (1+p)}+\frac {a 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+p}{2 p};\frac {3}{2}-\frac {1}{2 p};a^2 x^{2 p}\right )}{(-1+p) (1+p) \left (-1+a x^p\right )}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.46, size = 0, normalized size = 0.00 \[\int \frac {\frac {x^{-p}}{a}+\sqrt {\frac {x^{-p}}{a}-1}\, \sqrt {\frac {x^{-p}}{a}+1}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 \frac {x^{- p}}{x^{2}}\, dx + \int \frac {a \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}}{x^{2}}\, 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 \frac {\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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