Optimal. Leaf size=253 \[ \frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{\frac{1}{a x}+1}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^3} \]
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Rubi [A] time = 0.16675, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{\frac{1}{a x}+1}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^3} \]
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 )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{a}-\frac{5 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a \operatorname{Subst}\left (\int \frac{-\frac{3}{a^2}+\frac{8 x}{a^3}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{21}{a^3}+\frac{33 x}{a^4}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 c^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{-\frac{105}{a^4}+\frac{108 x}{a^5}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{35 c^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{315}{a^5}+\frac{213 x}{a^6}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{105 c^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^5 \operatorname{Subst}\left (\int -\frac{315}{a^6 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{105 c^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\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^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}-\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^3}\\ &=-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^3}\\ \end{align*}
Mathematica [A] time = 0.207562, size = 101, normalized size = 0.4 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (35 a^5 x^5+286 a^4 x^4+368 a^3 x^3-125 a^2 x^2-423 a x-176\right )}{35 (a x-1) (a x+1)^4}-3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.161, size = 714, normalized size = 2.8 \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 = 1.1127, size = 269, normalized size = 1.06 \begin{align*} -\frac{1}{560} \, a{\left (\frac{35 \,{\left (\frac{33 \,{\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{5 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 56 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 350 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 2520 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac{1680 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{1680 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36585, size = 416, normalized size = 1.64 \begin{align*} -\frac{105 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (35 \, a^{5} x^{5} + 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} - 125 \, a^{2} x^{2} - 423 \, a x - 176\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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