Optimal. Leaf size=255 \[ \frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{\frac{1}{a x}+1}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^3} \]
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Rubi [A] time = 0.172283, antiderivative size = 255, 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} \[ \frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{\frac{1}{a x}+1}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^3} \]
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 )^{9/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{a}-\frac{5 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{9/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{a \operatorname{Subst}\left (\int \frac{\frac{21}{a^2}+\frac{32 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{7 c^3}\\ &=-\frac{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{105}{a^3}-\frac{159 x}{a^4}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{35 c^3}\\ &=-\frac{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{\frac{315}{a^4}+\frac{528 x}{a^5}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{105 c^3}\\ &=-\frac{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{315}{a^5}-\frac{843 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{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\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{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\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{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\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{8}{7 a c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}-\frac{53}{35 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \sqrt{1+\frac{1}{a x}}}-\frac{88}{35 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}+\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 = 0.208061, size = 101, normalized size = 0.4 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (35 a^5 x^5-286 a^4 x^4+368 a^3 x^3+125 a^2 x^2-423 a x+176\right )}{35 (a x-1)^4 (a x+1)}+3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.206, 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 = 1.02702, size = 259, normalized size = 1.02 \begin{align*} \frac{1}{560} \, a{\left (\frac{\frac{51 \,{\left (a x - 1\right )}}{a x + 1} + \frac{294 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{2170 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{3640 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 5}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}} + \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}} + \frac{35 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65872, size = 468, normalized size = 1.84 \begin{align*} \frac{105 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, 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} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, 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 [A] time = 1.27798, size = 277, normalized size = 1.09 \begin{align*} \frac{1}{560} \, a{\left (\frac{1680 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{1680 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac{{\left (a x + 1\right )}^{3}{\left (\frac{56 \,{\left (a x - 1\right )}}{a x + 1} + \frac{350 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{2520 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 5\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{35 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac{1120 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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