Optimal. Leaf size=114 \[ -\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.107458, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6124, 835, 807, 266, 63, 208} \[ -\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6124
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^5} \, dx &=\int \frac{1+a x}{x^5 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{1}{4} \int \frac{-4 a-3 a^2 x}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}+\frac{1}{12} \int \frac{9 a^2+8 a^3 x}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{1}{24} \int \frac{-16 a^3-9 a^4 x}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}+\frac{1}{8} \left (3 a^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}+\frac{1}{16} \left (3 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{1}{8} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{4 x^4}-\frac{a \sqrt{1-a^2 x^2}}{3 x^3}-\frac{3 a^2 \sqrt{1-a^2 x^2}}{8 x^2}-\frac{2 a^3 \sqrt{1-a^2 x^2}}{3 x}-\frac{3}{8} a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0600546, size = 75, normalized size = 0.66 \[ \frac{1}{24} \left (-\frac{\sqrt{1-a^2 x^2} \left (16 a^3 x^3+9 a^2 x^2+8 a x+6\right )}{x^4}-9 a^4 \log \left (\sqrt{1-a^2 x^2}+1\right )+9 a^4 \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, size = 100, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{x}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\,{a}^{2}}{4} \left ( -{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) } \right ) }+a \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42476, size = 144, normalized size = 1.26 \begin{align*} -\frac{3}{8} \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{3}}{3 \, x} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{8 \, x^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} a}{3 \, x^{3}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72512, size = 151, normalized size = 1.32 \begin{align*} \frac{9 \, a^{4} x^{4} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (16 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 8 \, a x + 6\right )} \sqrt{-a^{2} x^{2} + 1}}{24 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.57736, size = 258, normalized size = 2.26 \begin{align*} a \left (\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19903, size = 369, normalized size = 3.24 \begin{align*} \frac{{\left (3 \, a^{5} + \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{3}}{x} + \frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a}{x^{2}} + \frac{72 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a x^{3}}\right )} a^{8} x^{4}}{192 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}{\left | a \right |}} - \frac{3 \, a^{5} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{8 \,{\left | a \right |}} - \frac{\frac{72 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{5}{\left | a \right |}}{x} + \frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{3}{\left | a \right |}}{x^{2}} + \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a{\left | a \right |}}{x^{3}} + \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}{\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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