Optimal. Leaf size=75 \[ \frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c \csc ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.224189, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6177, 1805, 1807, 844, 216, 266, 63, 208} \[ \frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c \csc ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 1805
Rule 1807
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 ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^4}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-c^4+\frac{4 c^4 x}{a}+\frac{c^4 x^2}{a^2}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\operatorname{Subst}\left (\int \frac{-\frac{4 c^4}{a}-\frac{c^4 x}{a^2}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{c \csc ^{-1}(a x)}{a}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{c \csc ^{-1}(a x)}{a}-(4 a c) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{c \csc ^{-1}(a x)}{a}-\frac{4 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [C] time = 0.461815, size = 234, normalized size = 3.12 \[ \frac{\sqrt{2} c (a x+1) (a x-1)^3 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+5 a^2 c x^2 \left ((a x+1) \left (\sqrt{\frac{1}{a x}+1} \left (a^2 x^2-3 a x+2\right )+6 a x \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-2 a x \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )\right )-4 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\right )}{5 a^4 x^3 \sqrt{1-\frac{1}{a x}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.127, size = 376, normalized size = 5. \begin{align*} -{\frac{c}{ \left ( ax-1 \right ) a} \left ( 4\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) -4\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+8\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\,ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +4\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-8\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+4\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}-\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \sqrt{{a}^{2}}-4\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \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 = 1.55299, size = 182, normalized size = 2.43 \begin{align*} -2 \, a{\left (\frac{c \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac{c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{2 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{2 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{4 \, c \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98663, size = 227, normalized size = 3.03 \begin{align*} -\frac{2 \, c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 4 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 4 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (a c x + 9 \, c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c \left (\int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x^{2} + x}\, dx + \int - \frac{2 a \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}\, dx + \int \frac{a^{2} x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}\, dx\right )}{a} \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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