Optimal. Leaf size=127 \[ \frac{8 a (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\sqrt{1-a^2 x^2}}{c^3 x}+\frac{a (79 a x+60)}{15 c^3 \sqrt{1-a^2 x^2}}+\frac{4 a (8 a x+5)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{4 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^3} \]
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Rubi [A] time = 0.351343, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 852, 1805, 807, 266, 63, 208} \[ \frac{8 a (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\sqrt{1-a^2 x^2}}{c^3 x}+\frac{a (79 a x+60)}{15 c^3 \sqrt{1-a^2 x^2}}+\frac{4 a (8 a x+5)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{4 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^3} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)^3} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^2 (c-a c x)^4} \, dx\\ &=\frac{\int \frac{(c+a c x)^4}{x^2 \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^7}\\ &=\frac{8 a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 c^4-20 a c^4 x-27 a^2 c^4 x^2}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^7}\\ &=\frac{8 a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a (5+8 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{15 c^4+60 a c^4 x+64 a^2 c^4 x^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^7}\\ &=\frac{8 a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a (5+8 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (60+79 a x)}{15 c^3 \sqrt{1-a^2 x^2}}-\frac{\int \frac{-15 c^4-60 a c^4 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{15 c^7}\\ &=\frac{8 a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a (5+8 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (60+79 a x)}{15 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^3 x}+\frac{(4 a) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=\frac{8 a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a (5+8 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (60+79 a x)}{15 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^3 x}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{c^3}\\ &=\frac{8 a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a (5+8 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (60+79 a x)}{15 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^3 x}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a c^3}\\ &=\frac{8 a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a (5+8 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (60+79 a x)}{15 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^3 x}-\frac{4 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^3}\\ \end{align*}
Mathematica [A] time = 0.0546973, size = 101, normalized size = 0.8 \[ \frac{94 a^4 x^4-128 a^3 x^3-73 a^2 x^2-60 a x (a x-1)^2 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+134 a x-15}{15 c^3 x (a x-1)^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 248, normalized size = 2. \begin{align*} -{\frac{1}{{c}^{3}} \left ( 2\,{\frac{1}{a} \left ( 1/5\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-2/5\,a \left ( 1/3\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-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 ) \right ) }+{\frac{1}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+4\,a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+5\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) } \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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (a c x - c\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48822, size = 338, normalized size = 2.66 \begin{align*} \frac{104 \, a^{4} x^{4} - 312 \, a^{3} x^{3} + 312 \, a^{2} x^{2} - 104 \, a x + 60 \,{\left (a^{4} x^{4} - 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (94 \, a^{3} x^{3} - 222 \, a^{2} x^{2} + 149 \, a x - 15\right )} \sqrt{-a^{2} x^{2} + 1}}{15 \,{\left (a^{3} c^{3} x^{4} - 3 \, a^{2} c^{3} x^{3} + 3 \, a c^{3} x^{2} - c^{3} x\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 x}{a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23041, size = 363, normalized size = 2.86 \begin{align*} -\frac{4 \, a^{2} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{c^{3}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{2 \, c^{3} x{\left | a \right |}} - \frac{{\left (15 \, a^{2} - \frac{491 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{x} + \frac{1690 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{2} x^{2}} - \frac{2570 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{4} x^{3}} + \frac{1815 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{6} x^{4}} - \frac{555 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{8} x^{5}}\right )} a^{2} x}{30 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{5}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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