Optimal. Leaf size=327 \[ \frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (\frac{1}{a x}+1\right )^{7/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (\frac{1}{a x}+1\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4} \]
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Rubi [A] time = 0.221861, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (\frac{1}{a x}+1\right )^{7/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (\frac{1}{a x}+1\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 103
Rule 152
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{11/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{a}-\frac{7 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{11/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{a \operatorname{Subst}\left (\int \frac{-\frac{9}{a^2}+\frac{24 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{11/2}} \, dx,x,\frac{1}{x}\right )}{3 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\frac{9}{a^3}-\frac{75 x}{a^4}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{11/2}} \, dx,x,\frac{1}{x}\right )}{3 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{a^3 \operatorname{Subst}\left (\int \frac{\frac{81}{a^4}-\frac{336 x}{a^5}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{27 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{a^4 \operatorname{Subst}\left (\int \frac{\frac{567}{a^5}-\frac{1251 x}{a^6}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{189 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{a^5 \operatorname{Subst}\left (\int \frac{\frac{2835}{a^6}-\frac{3636 x}{a^7}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{945 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{a^6 \operatorname{Subst}\left (\int \frac{\frac{8505}{a^7}-\frac{6471 x}{a^8}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2835 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{a^7 \operatorname{Subst}\left (\int \frac{8505}{a^8 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2835 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{3 \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 c^4}\\ &=-\frac{4}{3 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left (1+\frac{1}{a x}\right )^{9/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^4}\\ \end{align*}
Mathematica [A] time = 0.26325, size = 117, normalized size = 0.36 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (315 a^7 x^7+2669 a^6 x^6+2967 a^5 x^5-4029 a^4 x^4-7399 a^3 x^3-339 a^2 x^2+4047 a x+1664\right )}{315 (a x-1)^2 (a x+1)^5}-3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.181, size = 766, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07162, size = 312, normalized size = 0.95 \begin{align*} \frac{1}{20160} \, a{\left (\frac{105 \,{\left (\frac{29 \,{\left (a x - 1\right )}}{a x + 1} - \frac{414 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} + \frac{35 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} + 450 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 2961 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 14700 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 95445 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}} - \frac{60480 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} + \frac{60480 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36328, size = 610, normalized size = 1.87 \begin{align*} -\frac{945 \,{\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 945 \,{\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (315 \, a^{7} x^{7} + 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} - 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} - 339 \, a^{2} x^{2} + 4047 \, a x + 1664\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{315 \,{\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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