Optimal. Leaf size=135 \[ \frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{6 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{33 c^3 \csc ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.429622, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6177, 1805, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{6 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{33 c^3 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 1805
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^6}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-c^6+\frac{6 c^6 x}{a}+\frac{16 c^6 x^2}{a^2}-\frac{6 c^6 x^3}{a^3}+\frac{c^6 x^4}{a^4}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\operatorname{Subst}\left (\int \frac{-\frac{6 c^6}{a}-\frac{16 c^6 x}{a^2}+\frac{6 c^6 x^2}{a^3}-\frac{c^6 x^3}{a^4}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{a^2 \operatorname{Subst}\left (\int \frac{\frac{12 c^6}{a^3}+\frac{33 c^6 x}{a^4}-\frac{12 c^6 x^2}{a^5}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c^3}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{12 c^6}{a^5}-\frac{33 c^6 x}{a^6}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c^3}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\left (33 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{\left (6 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}-\left (6 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}-\frac{6 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [C] time = 0.462017, size = 663, normalized size = 4.91 \[ \frac{c^3 \left (70 \sqrt{2} a^6 x^6 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )-280 \sqrt{2} a^5 x^5 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+350 \sqrt{2} a^4 x^4 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+126 \sqrt{2} a^2 x^2 (a x-1)^3 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )-350 \sqrt{2} a^2 x^2 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+90 \sqrt{2} a x (a x-1)^4 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{7}{2},\frac{9}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+280 \sqrt{2} a x \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )-70 \sqrt{2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+630 a^7 x^7 \sqrt{\frac{1}{a x}+1}-32340 a^6 x^6 \sqrt{\frac{1}{a x}+1}+17955 a^5 x^5 \sqrt{\frac{1}{a x}+1}+16800 a^4 x^4 \sqrt{\frac{1}{a x}+1}-3465 a^3 x^3 \sqrt{\frac{1}{a x}+1}+420 a^2 x^2 \sqrt{\frac{1}{a x}+1}+44730 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-2520 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )+44730 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-2520 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )-3780 a^6 x^6 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\right )}{630 a^6 x^5 \sqrt{1-\frac{1}{a x}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.134, size = 450, normalized size = 3.3 \begin{align*} -{\frac{{c}^{3}}{2\,{x}^{2}{a}^{3} \left ( ax-1 \right ) } \left ( -12\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{5}{a}^{5}+12\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-57\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+12\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}-33\,{a}^{4}{x}^{4}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +23\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-78\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-66\,{a}^{3}{x}^{3}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +32\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+10\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-33\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+12\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-33\,{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) - \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\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 = 1.58562, size = 304, normalized size = 2.25 \begin{align*} -{\left (\frac{33 \, c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{6 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{6 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{32 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}} + \frac{11 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 6 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 13 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95096, size = 339, normalized size = 2.51 \begin{align*} -\frac{66 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (2 \, a^{3} c^{3} x^{3} + 78 \, a^{2} c^{3} x^{2} + 11 \, a c^{3} x - c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \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*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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