Optimal. Leaf size=87 \[ -\frac {x^{-p}}{a p}-\frac {x^{-p} \sqrt {1-a x^p}}{a p \sqrt {\frac {1}{1+a x^p}}}-\frac {\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \text {ArcSin}\left (a x^p\right )}{p} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 106, normalized size of antiderivative = 1.22, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6469, 265, 352,
248, 283, 222} \begin {gather*} -\frac {x^{-p} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \sqrt {1-a^2 x^{2 p}}}{a p}-\frac {x^{-p}}{a p}-\frac {\sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \csc ^{-1}\left (\frac {x^{-p}}{a}\right )}{p} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 222
Rule 248
Rule 265
Rule 283
Rule 352
Rule 6469
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx &=-\frac {x^{-p}}{a p}+\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int x^{-1-p} \sqrt {1-a x^p} \sqrt {1+a x^p} \, dx}{a}\\ &=-\frac {x^{-p}}{a p}+\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \int x^{-1-p} \sqrt {1-a^2 x^{2 p}} \, dx}{a}\\ &=-\frac {x^{-p}}{a p}-\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \text {Subst}\left (\int \sqrt {1-\frac {a^2}{x^2}} \, dx,x,x^{-p}\right )}{a p}\\ &=-\frac {x^{-p}}{a p}+\frac {\left (\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \text {Subst}\left (\int \frac {\sqrt {1-a^2 x^2}}{x^2} \, dx,x,x^p\right )}{a p}\\ &=-\frac {x^{-p}}{a p}-\frac {x^{-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \sqrt {1-a^2 x^{2 p}}}{a p}-\frac {\left (a \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,x^p\right )}{p}\\ &=-\frac {x^{-p}}{a p}-\frac {x^{-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \sqrt {1-a^2 x^{2 p}}}{a p}-\frac {\sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \sin ^{-1}\left (a x^p\right )}{p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 96, normalized size = 1.10 \begin {gather*} -\frac {i \left (-i x^{-p}-i \left (a+x^{-p}\right ) \sqrt {\frac {1-a x^p}{1+a x^p}}+a \log \left (-2 i a x^p+2 \sqrt {\frac {1-a x^p}{1+a x^p}} \left (1+a x^p\right )\right )\right )}{a p} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.62, size = 116, normalized size = 1.33
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \left (\arctan \left (\frac {\mathrm {csgn}\left (a \right ) a \,x^{p}}{\sqrt {1-a^{2} x^{2 p}}}\right ) a \,x^{p}+\mathrm {csgn}\left (a \right ) \sqrt {1-a^{2} x^{2 p}}\right ) \mathrm {csgn}\left (a \right )}{\sqrt {1-a^{2} x^{2 p}}}-\frac {x^{-p}}{a}}{p}\) | \(116\) |
default | \(\frac {-\frac {\sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \left (\arctan \left (\frac {\mathrm {csgn}\left (a \right ) a \,x^{p}}{\sqrt {1-a^{2} x^{2 p}}}\right ) a \,x^{p}+\mathrm {csgn}\left (a \right ) \sqrt {1-a^{2} x^{2 p}}\right ) \mathrm {csgn}\left (a \right )}{\sqrt {1-a^{2} x^{2 p}}}-\frac {x^{-p}}{a}}{p}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.61, size = 102, normalized size = 1.17 \begin {gather*} -\frac {a x^{p} \sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}} - a x^{p} \arctan \left (\sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}}\right ) + 1}{a p x^{p}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{- p}}{x}\, dx + \int \frac {a \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}}{x}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________