Optimal. Leaf size=393 \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{13/2}-\frac{5}{72} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{13/2}+\frac{5}{168} a^6 c^4 x^7 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{13/2}-\frac{5 a^5 c^4 x^6 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{11/2}}{1008}-\frac{11 a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}{1008}-\frac{11}{448} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{55}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{55}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{55 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]
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Rubi [A] time = 0.332593, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6191, 6195, 94, 92, 208} \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{13/2}-\frac{5}{72} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{13/2}+\frac{5}{168} a^6 c^4 x^7 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{13/2}-\frac{5 a^5 c^4 x^6 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{11/2}}{1008}-\frac{11 a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}{1008}-\frac{11}{448} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{55}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{55}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{55 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]
Antiderivative was successfully verified.
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Rule 6191
Rule 6195
Rule 94
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=\left (a^8 c^4\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^4 x^8 \, dx\\ &=-\left (\left (a^8 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{11/2}}{x^{10}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{1}{9} \left (5 a^7 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{11/2}}{x^9} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9-\frac{1}{24} \left (5 a^6 c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{11/2}}{x^8} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{1}{168} \left (5 a^5 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{11/2}}{x^7 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{\left (55 a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{9/2}}{x^6 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{1008}\\ &=-\frac{11 a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5}{1008}-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{1}{112} \left (11 a^3 c^4\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{11}{448} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{11 a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5}{1008}-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{1}{64} \left (11 a^2 c^4\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{11}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{11}{448} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{11 a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5}{1008}-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{1}{192} \left (55 a c^4\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{55}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{11}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{11}{448} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{11 a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5}{1008}-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{1}{128} \left (55 c^4\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{55}{128} c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{55}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{11}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{11}{448} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{11 a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5}{1008}-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9+\frac{\left (55 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{128 a}\\ &=-\frac{55}{128} c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{55}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{11}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{11}{448} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{11 a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5}{1008}-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9-\frac{\left (55 c^4\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 )}{128 a^2}\\ &=-\frac{55}{128} c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{55}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{11}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{11}{448} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4-\frac{11 a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5}{1008}-\frac{5 a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6}{1008}+\frac{5}{168} a^6 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{13/2} x^7-\frac{5}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{13/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{13/2} x^9-\frac{55 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{128 a}\\ \end{align*}
Mathematica [A] time = 0.191668, size = 111, normalized size = 0.28 \[ \frac{c^4 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (896 a^8 x^8+3024 a^7 x^7+1024 a^6 x^6-7224 a^5 x^5-8448 a^4 x^4+3066 a^3 x^3+10240 a^2 x^2+4599 a x-3712\right )-3465 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{8064 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.207, size = 288, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2}{c}^{4}}{8064\,a \left ( ax+1 \right ) } \left ( 896\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{6}{a}^{6}+3024\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{5}{a}^{5}+1920\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}{x}^{4}{a}^{4}-4200\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-6528\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-1134\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-4352\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}+3465\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+8064\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-3465\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\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.
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Maxima [A] time = 1.04752, size = 560, normalized size = 1.42 \begin{align*} -\frac{1}{8064} \,{\left (\frac{3465 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3465 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (3465 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{17}{2}} - 30030 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{2}} + 115038 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{2}} - 255222 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} + 360448 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - 334602 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 115038 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 30030 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 3465 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{9 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{36 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{84 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{126 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac{126 \,{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac{84 \,{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac{36 \,{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac{9 \,{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac{{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62773, size = 419, normalized size = 1.07 \begin{align*} -\frac{3465 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3465 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (896 \, a^{9} c^{4} x^{9} + 3920 \, a^{8} c^{4} x^{8} + 4048 \, a^{7} c^{4} x^{7} - 6200 \, a^{6} c^{4} x^{6} - 15672 \, a^{5} c^{4} x^{5} - 5382 \, a^{4} c^{4} x^{4} + 13306 \, a^{3} c^{4} x^{3} + 14839 \, a^{2} c^{4} x^{2} + 887 \, a c^{4} x - 3712 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{8064 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.20879, size = 500, normalized size = 1.27 \begin{align*} -\frac{1}{8064} \,{\left (\frac{3465 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3465 \, c^{4} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (\frac{30030 \,{\left (a x - 1\right )} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{115038 \,{\left (a x - 1\right )}^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - \frac{334602 \,{\left (a x - 1\right )}^{3} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac{360448 \,{\left (a x - 1\right )}^{4} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} - \frac{255222 \,{\left (a x - 1\right )}^{5} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac{115038 \,{\left (a x - 1\right )}^{6} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} - \frac{30030 \,{\left (a x - 1\right )}^{7} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{7}} + \frac{3465 \,{\left (a x - 1\right )}^{8} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{8}} - 3465 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{9}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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