Optimal. Leaf size=61 \[ c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+\frac{3 c^3 \csc ^{-1}(a x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0557759, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6177, 277, 195, 216} \[ c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+\frac{3 c^3 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6177
Rule 277
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \sqrt{1-\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}\\ &=\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x+\frac{3 c^3 \csc ^{-1}(a x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0697137, size = 51, normalized size = 0.84 \[ \frac{c^3 \left (\sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2+1\right )+3 a x \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{2 a^2 x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.164, size = 105, normalized size = 1.7 \begin{align*} -{\frac{{c}^{3} \left ( ax-1 \right ) ^{2}}{ \left ( 2\,ax+2 \right ){a}^{3}{x}^{2}} \left ( -3\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}-1}-3\,{a}^{2}{x}^{2}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) + \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.52451, size = 204, normalized size = 3.34 \begin{align*} -{\left (\frac{3 \, c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{3 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 2 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 3 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6236, size = 192, normalized size = 3.15 \begin{align*} -\frac{6 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x + c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c^{3} \left (\int \frac{3 a}{\frac{a x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{3 a^{2}}{\frac{a x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{a^{3}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{1}{\frac{a x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.20123, size = 232, normalized size = 3.8 \begin{align*} -\frac{1}{4} \,{\left (\frac{3 \,{\left (\pi + 2 \, \arctan \left (\frac{\frac{a x - 1}{a x + 1} - 1}{2 \, \sqrt{\frac{a x - 1}{a x + 1}}}\right )\right )} c^{3}}{a^{2}} + \frac{4 \,{\left (3 \, c^{3}{\left (\sqrt{\frac{a x - 1}{a x + 1}} - \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{2} + 8 \, c^{3}\right )}}{{\left ({\left (\sqrt{\frac{a x - 1}{a x + 1}} - \frac{1}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )}^{3} + 4 \, \sqrt{\frac{a x - 1}{a x + 1}} - \frac{4}{\sqrt{\frac{a x - 1}{a x + 1}}}\right )} a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]