Optimal. Leaf size=343 \[ c^4 x \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{9/2}+\frac{8 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{7/2}}{7 a}+\frac{7 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{5/2}}{6 a}+\frac{29 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{3/2}}{30 a}+\frac{19 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{1-\frac{1}{a x}}}{40 a}+\frac{7 c^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{1-\frac{1}{a x}}}{40 a}-\frac{c^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{1-\frac{1}{a x}}}{16 a}-\frac{19 c^4 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{16 a}+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}-\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.256663, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ c^4 x \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{9/2}+\frac{8 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{7/2}}{7 a}+\frac{7 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{5/2}}{6 a}+\frac{29 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \left (1-\frac{1}{a x}\right )^{3/2}}{30 a}+\frac{19 c^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{1-\frac{1}{a x}}}{40 a}+\frac{7 c^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{1-\frac{1}{a x}}}{40 a}-\frac{c^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{1-\frac{1}{a x}}}{16 a}-\frac{19 c^4 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{16 a}+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}-\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 97
Rule 154
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx &=-\left (c^4 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{9/2} \left (1+\frac{x}{a}\right )^{7/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x-c^4 \operatorname{Subst}\left (\int \frac{\left (-\frac{1}{a}-\frac{8 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{5/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{1}{7} \left (a c^4\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{7}{a^2}-\frac{49 x}{a^3}\right ) \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{5/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{1}{42} \left (a^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{42}{a^3}-\frac{203 x}{a^4}\right ) \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{5/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{1}{210} \left (a^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{210}{a^4}-\frac{399 x}{a^5}\right ) \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{40 a}+\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{1}{840} \left (a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{840}{a^5}+\frac{441 x}{a^6}\right ) \left (1+\frac{x}{a}\right )^{5/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{7 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{40 a}+\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{40 a}+\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x+\frac{\left (a^5 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{2520}{a^6}+\frac{315 x}{a^7}\right ) \left (1+\frac{x}{a}\right )^{3/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2520}\\ &=-\frac{c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{16 a}+\frac{7 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{40 a}+\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{40 a}+\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{\left (a^6 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{5040}{a^7}-\frac{5985 x}{a^8}\right ) \sqrt{1+\frac{x}{a}}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5040}\\ &=-\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{16 a}+\frac{7 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{40 a}+\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{40 a}+\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x+\frac{\left (a^7 c^4\right ) \operatorname{Subst}\left (\int \frac{\frac{5040}{a^8}+\frac{11025 x}{a^9}}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5040}\\ &=-\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{16 a}+\frac{7 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{40 a}+\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{40 a}+\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x+\frac{\left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^2}+\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{16 a}+\frac{7 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{40 a}+\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{40 a}+\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{c^4 \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 )}{a^2}+\frac{\left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{16 a^2}\\ &=-\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{16 a}+\frac{7 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{40 a}+\frac{19 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{40 a}+\frac{29 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}{30 a}+\frac{7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}{6 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{7/2} x+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}-\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.276369, size = 120, normalized size = 0.35 \[ \frac{c^4 \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (1680 a^7 x^7+2816 a^6 x^6+3045 a^5 x^5-1952 a^4 x^4-1330 a^3 x^3+1056 a^2 x^2+280 a x-240\right )}{x^6}-1680 a^6 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+3675 a^6 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{1680 a^7} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.152, size = 320, normalized size = 0.9 \begin{align*} -{\frac{ \left ( ax+1 \right ){c}^{4}}{1680\,{a}^{8}{x}^{7}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -1680\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{8}{a}^{8}+1680\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{6}{a}^{6}-3675\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{7}{a}^{7}-3675\,{a}^{7}{x}^{7}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +1680\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{7}{a}^{8}+1995\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{5}{a}^{5}-1136\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}{x}^{4}{a}^{4}-1050\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+816\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+280\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-240\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \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 = 1.62895, size = 513, normalized size = 1.5 \begin{align*} -\frac{1}{840} \,{\left (\frac{3675 \, c^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{840 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{840 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{1995 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{2}} + 10185 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{2}} + 17619 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} + 4569 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} + 71801 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 72051 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 31465 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 5355 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{6 \,{\left (a x - 1\right )} a^{2}}{a x + 1} + \frac{14 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{14 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{14 \,{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac{14 \,{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac{6 \,{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac{{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45438, size = 490, normalized size = 1.43 \begin{align*} -\frac{7350 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (1680 \, a^{8} c^{4} x^{8} + 4496 \, a^{7} c^{4} x^{7} + 5861 \, a^{6} c^{4} x^{6} + 1093 \, a^{5} c^{4} x^{5} - 3282 \, a^{4} c^{4} x^{4} - 274 \, a^{3} c^{4} x^{3} + 1336 \, a^{2} c^{4} x^{2} + 40 \, a c^{4} x - 240 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{1680 \, a^{8} x^{7}} \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 = 1.22717, size = 707, normalized size = 2.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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