Optimal. Leaf size=156 \[ -\frac{a^4 c^2 \sqrt{1-a^2 x^2}}{16 x^2}+\frac{2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{1}{16} a^6 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.172594, antiderivative size = 156, 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, 835, 807, 266, 47, 63, 208} \[ -\frac{a^4 c^2 \sqrt{1-a^2 x^2}}{16 x^2}+\frac{2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{1}{16} a^6 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6128
Rule 835
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^7} \, dx &=c \int \frac{(c-a c x) \sqrt{1-a^2 x^2}}{x^7} \, dx\\ &=-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}-\frac{1}{6} c \int \frac{\left (6 a c-3 a^2 c x\right ) \sqrt{1-a^2 x^2}}{x^6} \, dx\\ &=-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{1}{30} c \int \frac{\left (15 a^2 c-12 a^3 c x\right ) \sqrt{1-a^2 x^2}}{x^5} \, dx\\ &=-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}-\frac{1}{120} c \int \frac{\left (48 a^3 c-15 a^4 c x\right ) \sqrt{1-a^2 x^2}}{x^4} \, dx\\ &=-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{8} \left (a^4 c^2\right ) \int \frac{\sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{16} \left (a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{a^4 c^2 \sqrt{1-a^2 x^2}}{16 x^2}-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{1}{32} \left (a^6 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a^4 c^2 \sqrt{1-a^2 x^2}}{16 x^2}-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{16} \left (a^4 c^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{a^4 c^2 \sqrt{1-a^2 x^2}}{16 x^2}-\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{a c^2 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{2 a^3 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{16} a^6 c^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0388684, size = 115, normalized size = 0.74 \[ \frac{c^2 \left (32 a^7 x^7-15 a^6 x^6-16 a^5 x^5+5 a^4 x^4-64 a^3 x^3+50 a^2 x^2+15 a^6 x^6 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+48 a x-40\right )}{240 x^6 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 193, normalized size = 1.2 \begin{align*}{c}^{2} \left ( -{\frac{{a}^{2}}{6} \left ( -{\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 ) } \right ) }-a \left ( -{\frac{1}{5\,{x}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{4\,{a}^{2}}{5} \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 ) } \right ) -{\frac{1}{6\,{x}^{6}}\sqrt{-{a}^{2}{x}^{2}+1}}+{a}^{3} \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 ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4252, size = 227, normalized size = 1.46 \begin{align*} \frac{1}{16} \, a^{6} c^{2} \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^{5} c^{2}}{15 \, x} + \frac{\sqrt{-a^{2} x^{2} + 1} a^{4} c^{2}}{16 \, x^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} a^{3} c^{2}}{15 \, x^{3}} + \frac{\sqrt{-a^{2} x^{2} + 1} a^{2} c^{2}}{24 \, x^{4}} + \frac{\sqrt{-a^{2} x^{2} + 1} a c^{2}}{5 \, x^{5}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70295, size = 232, normalized size = 1.49 \begin{align*} -\frac{15 \, a^{6} c^{2} x^{6} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (32 \, a^{5} c^{2} x^{5} - 15 \, a^{4} c^{2} x^{4} + 16 \, a^{3} c^{2} x^{3} - 10 \, a^{2} c^{2} x^{2} - 48 \, a c^{2} x + 40 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{240 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 15.382, size = 644, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25303, size = 572, normalized size = 3.67 \begin{align*} \frac{{\left (5 \, a^{7} c^{2} - \frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{5} c^{2}}{x} + \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{3} c^{2}}{x^{2}} - \frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a c^{2}}{x^{3}} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{2}}{a x^{4}} + \frac{120 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{2}}{a^{3} x^{5}}\right )} a^{12} x^{6}}{1920 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}{\left | a \right |}} + \frac{a^{7} c^{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 )}{16 \,{\left | a \right |}} - \frac{\frac{120 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{9} c^{2}{\left | a \right |}}{x} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{7} c^{2}{\left | a \right |}}{x^{2}} - \frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a^{5} c^{2}{\left | a \right |}}{x^{3}} + \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} a^{3} c^{2}{\left | a \right |}}{x^{4}} - \frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} a c^{2}{\left | a \right |}}{x^{5}} + \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{2}{\left | a \right |}}{a x^{6}}}{1920 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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