Optimal. Leaf size=127 \[ \frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \left (1-\frac{1}{a x}\right )^{p-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n}{2}+p+1} \text{Hypergeometric2F1}\left (-2 p-1,\frac{1}{2} (n-2 p),-2 p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{2 p+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.149586, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6192, 6196, 132} \[ \frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \left (1-\frac{1}{a x}\right )^{p-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n}{2}+p+1} \, _2F_1\left (-2 p-1,\frac{1}{2} (n-2 p);-2 p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{2 p+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6192
Rule 6196
Rule 132
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^p x^{2 p} \, dx\\ &=-\left (\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \operatorname{Subst}\left (\int x^{-2-2 p} \left (1-\frac{x}{a}\right )^{-\frac{n}{2}+p} \left (1+\frac{x}{a}\right )^{\frac{n}{2}+p} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}+p} \left (1+\frac{1}{a x}\right )^{1+\frac{n}{2}+p} x \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-2 p,\frac{1}{2} (n-2 p);-2 p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{1+2 p}\\ \end{align*}
Mathematica [A] time = 0.188116, size = 83, normalized size = 0.65 \[ -\frac{\left (a^2 x^2-1\right ) \left (e^{2 \coth ^{-1}(a x)}-1\right ) \left (c-a^2 c x^2\right )^p e^{(n-2) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,-\frac{n}{2}-p,-\frac{n}{2}+p+2,e^{-2 \coth ^{-1}(a x)}\right )}{a (n-2 (p+1))} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.378, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} e^{n \operatorname{acoth}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]