Optimal. Leaf size=105 \[ \frac {p x^{1-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (\frac {1}{p}-1\right );\frac {p+1}{2 p};a^2 x^{2 p}\right )}{a (1-p)}+\frac {p x^{1-p}}{a (1-p)}+x e^{\text {sech}^{-1}\left (a x^p\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6330, 30, 259, 364} \[ \frac {p x^{1-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (\frac {1}{p}-1\right );\frac {p+1}{2 p};a^2 x^{2 p}\right )}{a (1-p)}+\frac {p x^{1-p}}{a (1-p)}+x e^{\text {sech}^{-1}\left (a x^p\right )} \]
Antiderivative was successfully verified.
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Rule 30
Rule 259
Rule 364
Rule 6330
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^p\right )} \, dx &=e^{\text {sech}^{-1}\left (a x^p\right )} x+\frac {p \int x^{-p} \, dx}{a}+\frac {\left (p \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int \frac {x^{-p}}{\sqrt {1-a x^p} \sqrt {1+a x^p}} \, dx}{a}\\ &=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^{-p}}{\sqrt {1-a^2 x^{2 p}}} \, dx}{a}\\ &=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}{2} \left (-1+\frac {1}{p}\right );\frac {1+p}{2 p};a^2 x^{2 p}\right )}{a (1-p)}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 139, normalized size = 1.32 \[ \frac {x \left (-\frac {a^2 p x^p \sqrt {\frac {1-a x^p}{a x^p+1}} \sqrt {1-a^2 x^{2 p}} \, _2F_1\left (\frac {1}{2},\frac {p+1}{2 p};\frac {1}{2} \left (3+\frac {1}{p}\right );a^2 x^{2 p}\right )}{(p+1) \left (a x^p-1\right )}+\left (a+x^{-p}\right ) \sqrt {\frac {1-a x^p}{a x^p+1}}+x^{-p}\right )}{a-a p} \]
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 \[ \int \sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {x^{-p}}{a}+\sqrt {\frac {x^{-p}}{a}-1}\, \sqrt {\frac {x^{-p}}{a}+1}\, dx \]
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 \[ \int \sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p} \,d x \]
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 \[ \frac {\int x^{- p}\, dx + \int a \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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