Optimal. Leaf size=119 \[ \frac {p x^{2-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (\frac {2}{p}-1\right );\frac {1}{2} \left (1+\frac {2}{p}\right );a^2 x^{2 p}\right )}{2 a (2-p)}+\frac {p x^{2-p}}{2 a (2-p)}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^p\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, 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 = {6335, 30, 259, 364} \[ \frac {p x^{2-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (\frac {2}{p}-1\right );\frac {1}{2} \left (1+\frac {2}{p}\right );a^2 x^{2 p}\right )}{2 a (2-p)}+\frac {p x^{2-p}}{2 a (2-p)}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^p\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 259
Rule 364
Rule 6335
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.31, size = 159, normalized size = 1.34 \[ \frac {x^{2-p} \left (\frac {a^2 p x^{2 p} \sqrt {\frac {1-a x^p}{a x^p+1}} \sqrt {1-a^2 x^{2 p}} \, _2F_1\left (\frac {1}{2},\frac {1}{2}+\frac {1}{p};\frac {3}{2}+\frac {1}{p};a^2 x^{2 p}\right )}{(p+2) \left (a x^p-1\right )}-a x^p \sqrt {\frac {1-a x^p}{a x^p+1}}-\sqrt {\frac {1-a x^p}{a x^p+1}}-1\right )}{a (p-2)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\left (\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.06, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\left (\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int x x^{- p}\, dx + \int a x \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________