Optimal. Leaf size=218 \[ \frac{x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (1-2 p),1-p,\frac{1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}+\frac{a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (3-2 p),1-p,\frac{1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}-\frac{a x^2 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (1-p,1-p,2-p,a^2 x^2\right )}{1-p} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.263655, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6159, 6129, 127, 125, 364} \[ \frac{x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (1-2 p),1-p;\frac{1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac{a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (3-2 p),1-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}-\frac{a x^2 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (1-p,1-p;2-p;a^2 x^2\right )}{1-p} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6159
Rule 6129
Rule 127
Rule 125
Rule 364
Rubi steps
\begin{align*} \int e^{-2 \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} (1-a x)^{-p} (1+a x)^{-p}\right ) \int e^{-2 \tanh ^{-1}(a x)} x^{-2 p} (1-a x)^p (1+a x)^p \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} (1-a x)^{1+p} (1+a x)^{-1+p} \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int \left (-2 a x^{1-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}+a^2 x^{2-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}+x^{-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}\right ) \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx-\left (2 a \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{1-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx+\left (a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{2-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p} \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{-1+p} \, dx-\left (2 a \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{1-2 p} \left (1-a^2 x^2\right )^{-1+p} \, dx+\left (a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{2-2 p} \left (1-a^2 x^2\right )^{-1+p} \, dx\\ &=\frac{\left (c-\frac{c}{a^2 x^2}\right )^p x (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac{1}{2} (1-2 p),1-p;\frac{1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac{a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^3 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac{1}{2} (3-2 p),1-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^p x^2 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (1-p,1-p;2-p;a^2 x^2\right )}{1-p}\\ \end{align*}
Mathematica [C] time = 0.101709, size = 142, normalized size = 0.65 \[ \frac{x (1-a x)^{-p} \left (-\left (a^2 x^2-1\right )^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \left ((1-a x)^p \left (a^2 x^2-1\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2}-p,-p,\frac{3}{2}-p,a^2 x^2\right )-2 (a x-1)^p \left (1-a^2 x^2\right )^p F_1(1-2 p;-p,1-p;2-2 p;a x,-a x)\right )}{2 p-1} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.219, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{a}^{2}{x}^{2}+1}{ \left ( ax+1 \right ) ^{2}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \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 \frac{{\left (a^{2} x^{2} - 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{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 (-\frac{{\left (a x - 1\right )} \left (\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}}{a x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 12.5377, size = 695, normalized size = 3.19 \begin{align*} \text{result too large to display} \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 -\frac{{\left (a^{2} x^{2} - 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]