Optimal. Leaf size=393 \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{11/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{72} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{11}{56} a^6 c^4 x^7 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{48} a^5 c^4 x^6 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{11}{48} a^4 c^4 x^5 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{11}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{55}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{55}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{55 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]
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Rubi [A] time = 0.337411, 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} \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{11/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{72} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{11}{56} a^6 c^4 x^7 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{48} a^5 c^4 x^6 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{11}{48} a^4 c^4 x^5 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{11}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{11}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{55}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{55}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{55 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]
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 )^{11/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^{10}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} x^9+\frac{1}{9} \left (11 a^7 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{9/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^9} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} x^9-\frac{1}{8} \left (11 a^6 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^8} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} x^9+\frac{1}{8} \left (11 a^5 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^7} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} x^9-\frac{1}{48} \left (55 a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{5/2}}{x^6} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11}{48} a^4 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} x^9+\frac{1}{16} \left (11 a^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{11}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{11}{48} a^4 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{11}{48} a^4 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{11}{48} a^4 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{11}{48} a^4 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{11}{48} a^4 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{11}{48} a^4 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2} x^5-\frac{11}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2} x^6+\frac{11}{56} a^6 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2} x^7-\frac{11}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{11/2} \left (1+\frac{1}{a x}\right )^{7/2} 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 = 0.18618, 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 = 0.15, size = 288, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ) ^{2}{c}^{4}}{8064\, \left ( ax-1 \right ) a} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}} \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 ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06843, 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}} + 334602 \, 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}} + 255222 \, 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 = 1.68322, size = 416, normalized size = 1.06 \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} - 2128 \, a^{8} c^{4} x^{8} - 2000 \, a^{7} c^{4} x^{7} + 8248 \, a^{6} c^{4} x^{6} - 1224 \, a^{5} c^{4} x^{5} - 11514 \, a^{4} c^{4} x^{4} + 7174 \, a^{3} c^{4} x^{3} + 5641 \, a^{2} c^{4} x^{2} - 8311 \, 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 = 1.1457, size = 267, normalized size = 0.68 \begin{align*} -\frac{55 \, c^{4} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{128 \,{\left | a \right |}} - \frac{1}{8064} \, \sqrt{a^{2} x^{2} - 1}{\left (\frac{3712 \, c^{4} \mathrm{sgn}\left (a x + 1\right )}{a} +{\left (4599 \, c^{4} \mathrm{sgn}\left (a x + 1\right ) - 2 \,{\left (5120 \, a c^{4} \mathrm{sgn}\left (a x + 1\right ) -{\left (1533 \, a^{2} c^{4} \mathrm{sgn}\left (a x + 1\right ) + 4 \,{\left (1056 \, a^{3} c^{4} \mathrm{sgn}\left (a x + 1\right ) -{\left (903 \, a^{4} c^{4} \mathrm{sgn}\left (a x + 1\right ) + 2 \,{\left (64 \, a^{5} c^{4} \mathrm{sgn}\left (a x + 1\right ) + 7 \,{\left (8 \, a^{7} c^{4} x \mathrm{sgn}\left (a x + 1\right ) - 27 \, a^{6} c^{4} \mathrm{sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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