Optimal. Leaf size=54 \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1(1-2 p,-2 p;2-2 p;-a x)}{1-2 p} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6160, 6150, 64} \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1(1-2 p,-2 p;2-2 p;-a x)}{1-2 p} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 64
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{2 p \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int e^{2 p \tanh ^{-1}(a x)} x^{-2 p} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (c-\frac {c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} (1+a x)^{2 p} \, dx\\ &=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \, _2F_1(1-2 p,-2 p;2-2 p;-a x)}{1-2 p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 54, normalized size = 1.00 \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1(1-2 p,-2 p;2-2 p;-a x)}{1-2 p} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {a x + 1}{a x - 1}\right )^{p} \left (\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{2 p \arctanh \left (a x \right )} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{2\,p\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} e^{2 p \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________