Optimal. Leaf size=161 \[ -\frac{16 a^2 \sqrt{1-a^2 x^2}}{3 c^2 x}-\frac{5 a \sqrt{1-a^2 x^2}}{2 c^2 x^2}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}+\frac{5 a x+6}{3 c^2 x^3 \sqrt{1-a^2 x^2}}+\frac{a x+1}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{5 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^2} \]
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Rubi [A] time = 0.188517, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6148, 823, 835, 807, 266, 63, 208} \[ -\frac{16 a^2 \sqrt{1-a^2 x^2}}{3 c^2 x}-\frac{5 a \sqrt{1-a^2 x^2}}{2 c^2 x^2}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}+\frac{5 a x+6}{3 c^2 x^3 \sqrt{1-a^2 x^2}}+\frac{a x+1}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{5 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 823
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^4 \left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{1+a x}{x^4 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{6 a^2+5 a^3 x}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6+5 a x}{3 c^2 x^3 \sqrt{1-a^2 x^2}}+\frac{\int \frac{24 a^4+15 a^5 x}{x^4 \sqrt{1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6+5 a x}{3 c^2 x^3 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}-\frac{\int \frac{-45 a^5-48 a^6 x}{x^3 \sqrt{1-a^2 x^2}} \, dx}{9 a^4 c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6+5 a x}{3 c^2 x^3 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}-\frac{5 a \sqrt{1-a^2 x^2}}{2 c^2 x^2}+\frac{\int \frac{96 a^6+45 a^7 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{18 a^4 c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6+5 a x}{3 c^2 x^3 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}-\frac{5 a \sqrt{1-a^2 x^2}}{2 c^2 x^2}-\frac{16 a^2 \sqrt{1-a^2 x^2}}{3 c^2 x}+\frac{\left (5 a^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6+5 a x}{3 c^2 x^3 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}-\frac{5 a \sqrt{1-a^2 x^2}}{2 c^2 x^2}-\frac{16 a^2 \sqrt{1-a^2 x^2}}{3 c^2 x}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6+5 a x}{3 c^2 x^3 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}-\frac{5 a \sqrt{1-a^2 x^2}}{2 c^2 x^2}-\frac{16 a^2 \sqrt{1-a^2 x^2}}{3 c^2 x}-\frac{(5 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 )}{2 c^2}\\ &=\frac{1+a x}{3 c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6+5 a x}{3 c^2 x^3 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{1-a^2 x^2}}{3 c^2 x^3}-\frac{5 a \sqrt{1-a^2 x^2}}{2 c^2 x^2}-\frac{16 a^2 \sqrt{1-a^2 x^2}}{3 c^2 x}-\frac{5 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0510961, size = 110, normalized size = 0.68 \[ \frac{32 a^5 x^5-17 a^4 x^4-31 a^3 x^3+11 a^2 x^2-15 a^3 x^3 (a x-1) \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a x+2}{6 c^2 x^3 (a x-1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 260, normalized size = 1.6 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{8\,{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}}-2\,{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -{\frac{{a}^{2}}{4\,x+4\,{a}^{-1}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{{a}^{2}}{2} \left ({\frac{1}{3\,a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-{\frac{1}{3}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) }-{\frac{9\,{a}^{2}}{4}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}+a \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 ) -{\frac{1}{3\,{x}^{3}}\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 = 1.03299, size = 258, normalized size = 1.6 \begin{align*} -\frac{\frac{15 \, a^{4} \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right )}{c^{2}} - \frac{15 \, a^{4} \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right )}{c^{2}} - \frac{2 \,{\left (15 \,{\left (a^{2} x^{2} - 1\right )}^{2} a^{4} + 10 \,{\left (a^{2} x^{2} - 1\right )} a^{4} - 2 \, a^{4}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} c^{2} -{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}}{12 \, a} + \frac{16 \, a^{6} x^{6} - 24 \, a^{4} x^{4} + 6 \, a^{2} x^{2} + 1}{3 \,{\left (a^{2} c^{2} x^{5} - c^{2} x^{3}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53575, size = 360, normalized size = 2.24 \begin{align*} \frac{14 \, a^{6} x^{6} - 14 \, a^{5} x^{5} - 14 \, a^{4} x^{4} + 14 \, a^{3} x^{3} + 15 \,{\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (32 \, a^{5} x^{5} - 17 \, a^{4} x^{4} - 31 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + a x + 2\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \,{\left (a^{3} c^{2} x^{6} - a^{2} c^{2} x^{5} - a c^{2} x^{4} + c^{2} 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*} \frac{\int \frac{a}{a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{8} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}} \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{a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt{-a^{2} x^{2} + 1} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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