Optimal. Leaf size=129 \[ \frac{a^3 c^3 \sqrt{1-a^2 x^2}}{4 x^2}-\frac{7 a^2 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{1}{4} a^5 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.190251, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6128, 1807, 835, 807, 266, 47, 63, 208} \[ \frac{a^3 c^3 \sqrt{1-a^2 x^2}}{4 x^2}-\frac{7 a^2 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{1}{4} a^5 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6128
Rule 1807
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)^3}{x^6} \, dx &=c \int \frac{(c-a c x)^2 \sqrt{1-a^2 x^2}}{x^6} \, dx\\ &=-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{1}{5} c \int \frac{\left (10 a c^2-7 a^2 c^2 x\right ) \sqrt{1-a^2 x^2}}{x^5} \, dx\\ &=-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}+\frac{1}{20} c \int \frac{\left (28 a^2 c^2-10 a^3 c^2 x\right ) \sqrt{1-a^2 x^2}}{x^4} \, dx\\ &=-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}-\frac{7 a^2 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{1}{2} \left (a^3 c^3\right ) \int \frac{\sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}-\frac{7 a^2 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{1}{4} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=\frac{a^3 c^3 \sqrt{1-a^2 x^2}}{4 x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}-\frac{7 a^2 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{8} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a^3 c^3 \sqrt{1-a^2 x^2}}{4 x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}-\frac{7 a^2 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{1}{4} \left (a^3 c^3\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^3 c^3 \sqrt{1-a^2 x^2}}{4 x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{a c^3 \left (1-a^2 x^2\right )^{3/2}}{2 x^4}-\frac{7 a^2 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{1}{4} a^5 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0332235, size = 107, normalized size = 0.83 \[ -\frac{c^3 \left (28 a^6 x^6-15 a^5 x^5-44 a^4 x^4+45 a^3 x^3+4 a^2 x^2+15 a^5 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-30 a x+12\right )}{60 x^5 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 190, normalized size = 1.5 \begin{align*} -{c}^{3} \left ( 2\,a \left ( -1/4\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{4}}}+3/4\,{a}^{2} \left ( -1/2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{2}}}-1/2\,{a}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) \right ) -{\frac{{a}^{4}}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+{\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 ) }-2\,{a}^{3} \left ( -1/2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{2}}}-1/2\,{a}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42988, size = 196, normalized size = 1.52 \begin{align*} -\frac{1}{4} \, a^{5} c^{3} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{7 \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{3}}{15 \, x} - \frac{\sqrt{-a^{2} x^{2} + 1} a^{3} c^{3}}{4 \, x^{2}} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3}}{15 \, x^{3}} + \frac{\sqrt{-a^{2} x^{2} + 1} a c^{3}}{2 \, x^{4}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6267, size = 207, normalized size = 1.6 \begin{align*} \frac{15 \, a^{5} c^{3} x^{5} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (28 \, a^{4} c^{3} x^{4} - 15 \, a^{3} c^{3} x^{3} - 16 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x - 12 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.1219, size = 476, normalized size = 3.69 \begin{align*} - a^{4} c^{3} \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right ) + 2 a^{3} c^{3} \left (\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{2} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right ) - 2 a c^{3} \left (\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}\right ) + c^{3} \left (\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1874, size = 401, normalized size = 3.11 \begin{align*} \frac{{\left (3 \, a^{6} c^{3} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{3}}{x} + \frac{25 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{3}}{x^{2}} - \frac{90 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{3}}{a^{2} x^{4}}\right )} a^{10} x^{5}}{480 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}{\left | a \right |}} - \frac{a^{6} c^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{4 \,{\left | a \right |}} + \frac{\frac{90 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{8} c^{3}}{x} - \frac{25 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a^{4} c^{3}}{x^{3}} + \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} a^{2} c^{3}}{x^{4}} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{3}}{x^{5}}}{480 \, a^{4}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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