Optimal. Leaf size=142 \[ \frac{(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{3 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.191245, antiderivative size = 142, 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} \[ \frac{(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{3 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]
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 )^{7/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int (-1+a x)^5 (1+a x)^2 \, dx}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int \left (4 (-1+a x)^5+4 (-1+a x)^6+(-1+a x)^7\right ) \, dx}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{3 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}-\frac{4 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}+\frac{(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ \end{align*}
Mathematica [A] time = 0.0493986, size = 63, normalized size = 0.44 \[ -\frac{c^3 (a x-1)^6 \left (21 a^2 x^2+54 a x+37\right ) \sqrt{c-a^2 c x^2}}{168 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 100, normalized size = 0.7 \begin{align*}{\frac{x \left ( 21\,{a}^{7}{x}^{7}-72\,{x}^{6}{a}^{6}+28\,{x}^{5}{a}^{5}+168\,{x}^{4}{a}^{4}-210\,{x}^{3}{a}^{3}-56\,{a}^{2}{x}^{2}+252\,ax-168 \right ) }{168\, \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) ^{5}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{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 = 1.12915, size = 232, normalized size = 1.63 \begin{align*} -\frac{{\left (21 \, a^{9} \sqrt{-c} c^{3} x^{9} - 51 \, a^{8} \sqrt{-c} c^{3} x^{8} - 44 \, a^{7} \sqrt{-c} c^{3} x^{7} + 196 \, a^{6} \sqrt{-c} c^{3} x^{6} - 42 \, a^{5} \sqrt{-c} c^{3} x^{5} - 266 \, a^{4} \sqrt{-c} c^{3} x^{4} + 196 \, a^{3} \sqrt{-c} c^{3} x^{3} + 84 \, a^{2} \sqrt{-c} c^{3} x^{2} + 168 \, \sqrt{-c} c^{3}\right )}{\left (a x - 1\right )}^{2}}{168 \,{\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 = 1.61501, size = 209, normalized size = 1.47 \begin{align*} -\frac{{\left (21 \, a^{7} c^{3} x^{8} - 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} - 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} - 168 \, c^{3} x\right )} \sqrt{-a^{2} c}}{168 \, 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{7}{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]