Optimal. Leaf size=313 \[ -\frac{1}{7} a^6 c^3 x^7 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{5}{42} a^5 c^3 x^6 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{14} a^4 c^3 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{56} a^3 c^3 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{1}{24} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{5}{48} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{5}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{16 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.250656, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6191, 6195, 94, 92, 208} \[ -\frac{1}{7} a^6 c^3 x^7 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{5}{42} a^5 c^3 x^6 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{14} a^4 c^3 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{56} a^3 c^3 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{1}{24} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{5}{48} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{5}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{16 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6191
Rule 6195
Rule 94
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=-\left (\left (a^6 c^3\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^3 x^6 \, dx\right )\\ &=\left (a^6 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{7/2}}{x^8} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{7} \left (5 a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{7/2}}{x^7} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7+\frac{1}{14} \left (5 a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}}{x^6} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{14} a^4 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{14} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{7/2}}{x^5 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{56} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{14} a^4 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/2}}{x^4 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{56} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{14} a^4 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{24} \left (5 a c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^3 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{56} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{14} a^4 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{16} \left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{16} c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{56} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{14} a^4 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a}\\ &=\frac{5}{16} c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{56} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{14} a^4 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7+\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a}} \, dx,x,\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{16 a^2}\\ &=\frac{5}{16} c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{5}{48} a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{24} a^2 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{56} a^3 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{1}{14} a^4 c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{5}{42} a^5 c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6-\frac{1}{7} a^6 c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7+\frac{5 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.187864, size = 95, normalized size = 0.3 \[ \frac{c^3 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (-48 a^6 x^6-56 a^5 x^5+144 a^4 x^4+182 a^3 x^3-144 a^2 x^2-231 a x+48\right )+105 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{336 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.138, size = 231, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{3}}{336\,a} \left ( 48\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}{x}^{4}{a}^{4}+56\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-96\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-126\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-64\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}+105\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+112\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-105\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05407, size = 455, normalized size = 1.45 \begin{align*} \frac{1}{336} \,{\left (\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (105 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{2}} - 700 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} + 1981 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - 3072 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 1981 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 700 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 105 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{7 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{21 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{35 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{35 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac{21 \,{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac{7 \,{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac{{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.3204, size = 344, normalized size = 1.1 \begin{align*} \frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (48 \, a^{7} c^{3} x^{7} + 104 \, a^{6} c^{3} x^{6} - 88 \, a^{5} c^{3} x^{5} - 326 \, a^{4} c^{3} x^{4} - 38 \, a^{3} c^{3} x^{3} + 375 \, a^{2} c^{3} x^{2} + 183 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{336 \, a} \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*} - c^{3} \left (\int \frac{3 a^{2} x^{2}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{3 a^{4} x^{4}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{a^{6} x^{6}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{1}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19224, size = 408, normalized size = 1.3 \begin{align*} \frac{1}{336} \,{\left (\frac{105 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{3} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (\frac{700 \,{\left (a x - 1\right )} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{1981 \,{\left (a x - 1\right )}^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - \frac{3072 \,{\left (a x - 1\right )}^{3} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac{1981 \,{\left (a x - 1\right )}^{4} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} - \frac{700 \,{\left (a x - 1\right )}^{5} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac{105 \,{\left (a x - 1\right )}^{6} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} - 105 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{7}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]