Optimal. Leaf size=102 \[ -\frac{6 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.112237, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6185, 6184} \[ -\frac{6 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6185
Rule 6184
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{6 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c \left (9-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{6 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.581169, size = 110, normalized size = 1.08 \[ \frac{e^{n \coth ^{-1}(a x)} \left (3 a \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )-3 a n^2 x-2 \left (n^2-1\right ) n \cosh \left (2 \coth ^{-1}(a x)\right )+27 a x+2 n^3-26 n\right )}{4 a c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 84, normalized size = 0.8 \begin{align*}{\frac{ \left ( 6\,{x}^{3}{a}^{3}-6\,n{a}^{2}{x}^{2}+3\,a{n}^{2}x-{n}^{3}-9\,ax+7\,n \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{a \left ({n}^{4}-10\,{n}^{2}+9 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \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 (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56895, size = 335, normalized size = 3.28 \begin{align*} -\frac{{\left (6 \, a^{3} x^{3} + 6 \, a^{2} n x^{2} + n^{3} + 3 \,{\left (a n^{2} - 3 \, a\right )} x - 7 \, n\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{3} n^{4} - 10 \, a c^{3} n^{2} +{\left (a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right )} x^{4} + 9 \, a c^{3} - 2 \,{\left (a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3}\right )} x^{2}} \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 (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]