Optimal. Leaf size=105 \[ e^{\text {sech}^{-1}\left (a x^p\right )} x+\frac {p x^{1-p}}{a (1-p)}+\frac {p x^{1-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (-1+\frac {1}{p}\right );\frac {1+p}{2 p};a^2 x^{2 p}\right )}{a (1-p)} \]
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Rubi [A]
time = 0.03, 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 = {6465, 30, 265,
371} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 265
Rule 371
Rule 6465
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 = 0.28, size = 139, normalized size = 1.32 \begin {gather*} \frac {x \left (x^{-p}+\left (a+x^{-p}\right ) \sqrt {\frac {1-a x^p}{1+a x^p}}-\frac {a^2 p x^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 {1}{2} \left (3+\frac {1}{p}\right );a^2 x^{2 p}\right )}{(1+p) \left (-1+a x^p\right )}\right )}{a-a p} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.42, 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 \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^{- p}\, dx + \int a \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 \sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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