Optimal. Leaf size=267 \[ -\frac{2^{\frac{n-1}{2}} n \left (1-a^2 x^2\right )^{3/2} (1-a x)^{\frac{3-n}{2}} \text{Hypergeometric2F1}\left (\frac{3-n}{2},\frac{3-n}{2},\frac{5-n}{2},\frac{1}{2} (1-a x)\right )}{a^4 (3-n) x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac{n-1}{2}} \left (-a (2 n+3) n x+n^2+2 n+2\right ) (1-a x)^{\frac{1}{2} (-n-1)}}{a^4 \left (1-n^2\right ) x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}-\frac{\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.291179, antiderivative size = 267, normalized size of antiderivative = 1., 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 = {6160, 6150, 100, 145, 69} \[ -\frac{2^{\frac{n-1}{2}} n \left (1-a^2 x^2\right )^{3/2} (1-a x)^{\frac{3-n}{2}} \, _2F_1\left (\frac{3-n}{2},\frac{3-n}{2};\frac{5-n}{2};\frac{1}{2} (1-a x)\right )}{a^4 (3-n) x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac{n-1}{2}} \left (-a (2 n+3) n x+n^2+2 n+2\right ) (1-a x)^{\frac{1}{2} (-n-1)}}{a^4 \left (1-n^2\right ) x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}-\frac{\left (1-a^2 x^2\right )^{3/2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6160
Rule 6150
Rule 100
Rule 145
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac{\left (1-a^2 x^2\right )^{3/2} \int \frac{e^{n \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2} \int x^3 (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x}-\frac{\left (1-a^2 x^2\right )^{3/2} \int x (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} (-2-a n x) \, dx}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x}+\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (2+2 n+n^2-a n (3+2 n) x\right ) \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}+\frac{\left (n \left (1-a^2 x^2\right )^{3/2}\right ) \int (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}} \, dx}{a^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x}+\frac{(1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-1+n)} \left (2+2 n+n^2-a n (3+2 n) x\right ) \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (1-n^2\right ) \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}-\frac{2^{\frac{1}{2} (-1+n)} n (1-a x)^{\frac{3-n}{2}} \left (1-a^2 x^2\right )^{3/2} \, _2F_1\left (\frac{3-n}{2},\frac{3-n}{2};\frac{5-n}{2};\frac{1}{2} (1-a x)\right )}{a^4 (3-n) \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ \end{align*}
Mathematica [A] time = 0.244913, size = 186, normalized size = 0.7 \[ \frac{\left (1-a^2 x^2\right )^{3/2} (1-a x)^{\frac{1}{2} (-n-1)} \left (\frac{a^2 2^{\frac{n+3}{2}} n (a x-1)^2 \text{Hypergeometric2F1}\left (\frac{3}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},\frac{5}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )}{n-3}+\frac{4 a^2 \left (n^2 (2 a x-1)+n (3 a x-2)-2\right ) (a x+1)^{\frac{n-1}{2}}}{n^2-1}-4 a^4 x^2 (a x+1)^{\frac{n-1}{2}}\right )}{4 a^6 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.118, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{3}{2}}}}\, 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 (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{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{a^{4} x^{4} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]