Optimal. Leaf size=116 \[ -\frac{4 a^3 \sqrt{1-a^2 x^2}}{a x+1}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{11}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.747871, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6124, 6742, 271, 264, 266, 51, 63, 208, 651} \[ -\frac{4 a^3 \sqrt{1-a^2 x^2}}{a x+1}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}+\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 x^3}+\frac{11}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6124
Rule 6742
Rule 271
Rule 264
Rule 266
Rule 51
Rule 63
Rule 208
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1-a x)^2}{x^4 (1+a x) \sqrt{1-a^2 x^2}} \, dx\\ &=\int \left (\frac{1}{x^4 \sqrt{1-a^2 x^2}}-\frac{3 a}{x^3 \sqrt{1-a^2 x^2}}+\frac{4 a^2}{x^2 \sqrt{1-a^2 x^2}}-\frac{4 a^3}{x \sqrt{1-a^2 x^2}}+\frac{4 a^4}{(1+a x) \sqrt{1-a^2 x^2}}\right ) \, dx\\ &=-\left ((3 a) \int \frac{1}{x^3 \sqrt{1-a^2 x^2}} \, dx\right )+\left (4 a^2\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\left (4 a^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+\left (4 a^4\right ) \int \frac{1}{(1+a x) \sqrt{1-a^2 x^2}} \, dx+\int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 x^3}-\frac{4 a^2 \sqrt{1-a^2 x^2}}{x}-\frac{4 a^3 \sqrt{1-a^2 x^2}}{1+a x}-\frac{1}{2} (3 a) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^2\right )+\frac{1}{3} \left (2 a^2\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\left (2 a^3\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}}{3 x^3}+\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{4 a^3 \sqrt{1-a^2 x^2}}{1+a x}+(4 a) \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{1}{4} \left (3 a^3\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}}{3 x^3}+\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{4 a^3 \sqrt{1-a^2 x^2}}{1+a x}+4 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\frac{1}{2} (3 a) \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}}{3 x^3}+\frac{3 a \sqrt{1-a^2 x^2}}{2 x^2}-\frac{14 a^2 \sqrt{1-a^2 x^2}}{3 x}-\frac{4 a^3 \sqrt{1-a^2 x^2}}{1+a x}+\frac{11}{2} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.075407, size = 82, normalized size = 0.71 \[ \frac{1}{6} \left (-\frac{\sqrt{1-a^2 x^2} \left (52 a^3 x^3+19 a^2 x^2-7 a x+2\right )}{x^3 (a x+1)}+33 a^3 \log \left (\sqrt{1-a^2 x^2}+1\right )-33 a^3 \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.057, size = 338, normalized size = 2.9 \begin{align*} -{\frac{16\,{a}^{2}}{3\,x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{16\,{a}^{4}x}{3} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-8\,{a}^{4}x\sqrt{-{a}^{2}{x}^{2}+1}-8\,{\frac{{a}^{4}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+2\,{\frac{a \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{ \left ( x+{a}^{-1} \right ) ^{2}}}+{\frac{16\,{a}^{3}}{3} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+8\,{a}^{4}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }x+8\,{\frac{{a}^{4}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}} \right ) }-{\frac{11\,{a}^{3}}{6} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{11\,{a}^{3}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{11\,{a}^{3}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{3\,a}{2\,{x}^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{3\,{x}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{ \left ( x+{a}^{-1} \right ) ^{3}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86297, size = 219, normalized size = 1.89 \begin{align*} -\frac{24 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 33 \,{\left (a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (52 \, a^{3} x^{3} + 19 \, a^{2} x^{2} - 7 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \,{\left (a x^{4} + x^{3}\right )}} \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*} \int \frac{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{x^{4} \left (a x + 1\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21526, size = 358, normalized size = 3.09 \begin{align*} \frac{{\left (a^{4} - \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2}}{x} + \frac{48 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}} + \frac{249 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} + \frac{11 \, a^{4} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} - \frac{\frac{57 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4}}{x} - \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2}}{x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{x^{3}}}{24 \, a^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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