Optimal. Leaf size=189 \[ \frac{(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{2 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{2 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.202343, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6192, 6193, 43} \[ \frac{(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{2 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{2 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{9/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{\left (c-a^2 c x^2\right )^{9/2} \int (-1+a x)^6 (1+a x)^3 \, dx}{a^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{\left (c-a^2 c x^2\right )^{9/2} \int \left (8 (-1+a x)^6+12 (-1+a x)^7+6 (-1+a x)^8+(-1+a x)^9\right ) \, dx}{a^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ &=-\frac{8 (1-a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}+\frac{3 (1-a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{2 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}-\frac{2 (1-a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}+\frac{(1-a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac{1}{a^2 x^2}\right )^{9/2} x^9}\\ \end{align*}
Mathematica [A] time = 0.0564303, size = 71, normalized size = 0.38 \[ \frac{c^4 (a x-1)^7 \left (21 a^3 x^3+77 a^2 x^2+98 a x+44\right ) \sqrt{c-a^2 c x^2}}{210 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 100, normalized size = 0.5 \begin{align*}{\frac{x \left ( 21\,{a}^{9}{x}^{9}-70\,{x}^{8}{a}^{8}+240\,{x}^{6}{a}^{6}-210\,{x}^{5}{a}^{5}-252\,{x}^{4}{a}^{4}+420\,{x}^{3}{a}^{3}-315\,ax+210 \right ) }{210\, \left ( ax+1 \right ) ^{3} \left ( ax-1 \right ) ^{6}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09494, size = 275, normalized size = 1.46 \begin{align*} \frac{{\left (21 \, a^{11} \sqrt{-c} c^{4} x^{11} - 49 \, a^{10} \sqrt{-c} c^{4} x^{10} - 70 \, a^{9} \sqrt{-c} c^{4} x^{9} + 240 \, a^{8} \sqrt{-c} c^{4} x^{8} + 30 \, a^{7} \sqrt{-c} c^{4} x^{7} - 462 \, a^{6} \sqrt{-c} c^{4} x^{6} + 168 \, a^{5} \sqrt{-c} c^{4} x^{5} + 420 \, a^{4} \sqrt{-c} c^{4} x^{4} - 315 \, a^{3} \sqrt{-c} c^{4} x^{3} - 105 \, a^{2} \sqrt{-c} c^{4} x^{2} - 210 \, \sqrt{-c} c^{4}\right )}{\left (a x - 1\right )}^{2}}{210 \,{\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )}{\left (a x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.66939, size = 212, normalized size = 1.12 \begin{align*} \frac{{\left (21 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} + 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} - 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} + 210 \, c^{4} x\right )} \sqrt{-a^{2} c}}{210 \, a} \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{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]