Optimal. Leaf size=86 \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a^3 c (2-n)}+\frac{e^{n \tanh ^{-1}(a x)}}{a^3 c n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.135161, antiderivative size = 127, normalized size of antiderivative = 1.48, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6150, 90, 79, 69} \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^3 c}-\frac{x (a x+1)^{n/2} (1-a x)^{-n/2}}{a^2 c}+\frac{(1-n) (a x+1)^{n/2} (1-a x)^{-n/2}}{a^3 c n} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6150
Rule 90
Rule 79
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx &=\frac{\int x^2 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} \, dx}{c}\\ &=-\frac{x (1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c}-\frac{\int (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} (-1-a n x) \, dx}{a^2 c}\\ &=\frac{(1-n) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^3 c n}-\frac{x (1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c}+\frac{n \int (1-a x)^{-1-\frac{n}{2}} (1+a x)^{n/2} \, dx}{a^2 c}\\ &=\frac{(1-n) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^3 c n}-\frac{x (1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c}+\frac{2^{1+\frac{n}{2}} (1-a x)^{-n/2} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^3 c}\\ \end{align*}
Mathematica [A] time = 0.0631519, size = 82, normalized size = 0.95 \[ \frac{(1-a x)^{-n/2} \left (2^{\frac{n}{2}+1} n \text{Hypergeometric2F1}\left (-\frac{n}{2},-\frac{n}{2},1-\frac{n}{2},\frac{1}{2} (1-a x)\right )-(a x+1)^{n/2} (a n x+n-1)\right )}{a^3 c n} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{2}}{-{a}^{2}c{x}^{2}+c}}\, 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{x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{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{x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, 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*} - \frac{\int \frac{x^{2} e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \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{x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]