Optimal. Leaf size=138 \[ -\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^5}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^5} \]
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Rubi [A] time = 0.389577, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^5}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^5} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^5} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (c-\frac{c x}{a}\right )^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^2}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-5 c^2-\frac{10 c^2 x}{a}-\frac{8 c^2 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{15 c^2+\frac{30 c^2 x}{a}+\frac{26 c^2 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-15 c^2-\frac{30 c^2 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^5}\\ \end{align*}
Mathematica [A] time = 0.0808256, size = 104, normalized size = 0.75 \[ \frac{15 a^4 x^4-76 a^3 x^3+32 a^2 x^2+30 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+82 a x-56}{15 a^2 c^5 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.144, size = 615, normalized size = 4.5 \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 = 1.05843, size = 238, normalized size = 1.72 \begin{align*} \frac{1}{120} \, a{\left (\frac{\frac{32 \,{\left (a x - 1\right )}}{a x + 1} + \frac{310 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac{585 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - a^{2} c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} + \frac{240 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{5}} - \frac{240 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{5}} + \frac{15 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84735, size = 387, normalized size = 2.8 \begin{align*} \frac{30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\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 x}\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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