Optimal. Leaf size=73 \[ -a x \, _2F_1\left (\frac {1}{2},\frac {1}{2}-p;\frac {3}{2};a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \, _2F_1\left (1,\frac {1}{2}+p;\frac {3}{2}+p;1-a^2 x^2\right )}{1+2 p} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6284, 778, 272,
67, 251} \begin {gather*} -\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}} \, _2F_1\left (1,p+\frac {1}{2};p+\frac {3}{2};1-a^2 x^2\right )}{2 p+1}-a x \, _2F_1\left (\frac {1}{2},\frac {1}{2}-p;\frac {3}{2};a^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 67
Rule 251
Rule 272
Rule 778
Rule 6284
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p}{x} \, dx &=\int \frac {(1-a x) \left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{x} \, dx\\ &=-\left (a \int \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\right )+\int \frac {\left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{x} \, dx\\ &=-a x \, _2F_1\left (\frac {1}{2},\frac {1}{2}-p;\frac {3}{2};a^2 x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\left (1-a^2 x\right )^{-\frac {1}{2}+p}}{x} \, dx,x,x^2\right )\\ &=-a x \, _2F_1\left (\frac {1}{2},\frac {1}{2}-p;\frac {3}{2};a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \, _2F_1\left (1,\frac {1}{2}+p;\frac {3}{2}+p;1-a^2 x^2\right )}{1+2 p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 75, normalized size = 1.03 \begin {gather*} -a x \, _2F_1\left (\frac {1}{2},\frac {1}{2}-p;\frac {3}{2};a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \, _2F_1\left (1,\frac {1}{2}+p;\frac {3}{2}+p;1-a^2 x^2\right )}{2 \left (\frac {1}{2}+p\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} x^{2}+1\right )^{p} \sqrt {-a^{2} x^{2}+1}}{\left (a x +1\right ) x}\, dx\]
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 [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{x \left (a x + 1\right )}\, dx \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 {{\left (1-a^2\,x^2\right )}^p\,\sqrt {1-a^2\,x^2}}{x\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________