Optimal. Leaf size=235 \[ \frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.138806, antiderivative size = 235, 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 = {6197, 6193, 75} \[ \frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6197
Rule 6193
Rule 75
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \, dx &=\frac{\left (c^2 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^2 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \frac{(-1+a x)^4 (1+a x)}{x^5} \, dx}{a^5 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c^2 \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \left (a^5+\frac{1}{x^5}-\frac{3 a}{x^4}+\frac{2 a^2}{x^3}+\frac{2 a^3}{x^2}-\frac{3 a^4}{x}\right ) \, dx}{a^5 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 \sqrt{1-\frac{1}{a^2 x^2}} x^4}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}} x^3}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}-\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}} x}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^2 \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0629846, size = 81, normalized size = 0.34 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} \left (-\frac{a^2}{x^2}+a^5 x-\frac{2 a^3}{x}-3 a^4 \log (x)-\frac{5 a^4}{4}+\frac{a}{x^3}-\frac{1}{4 x^4}\right )}{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.227, size = 96, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -4\,{x}^{5}{a}^{5}+12\,{a}^{4}\ln \left ( x \right ){x}^{4}+8\,{x}^{3}{a}^{3}+4\,{a}^{2}{x}^{2}-4\,ax+1 \right ) x}{4\, \left ( ax-1 \right ) ^{3} \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{5}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{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]
________________________________________________________________________________________
Fricas [A] time = 1.6958, size = 158, normalized size = 0.67 \begin{align*} \frac{{\left (4 \, a^{5} c^{2} x^{5} - 12 \, a^{4} c^{2} x^{4} \log \left (x\right ) - 8 \, a^{3} c^{2} x^{3} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right )} \sqrt{a^{2} c}}{4 \, a^{6} x^{4}} \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 (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{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]