Optimal. Leaf size=144 \[ -\frac{38 a^4 c \sqrt{1-a^2 x^2}}{15 x}-\frac{13 a^3 c \sqrt{1-a^2 x^2}}{8 x^2}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{13}{8} a^5 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.25023, antiderivative size = 144, 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, 1807, 835, 807, 266, 63, 208} \[ -\frac{38 a^4 c \sqrt{1-a^2 x^2}}{15 x}-\frac{13 a^3 c \sqrt{1-a^2 x^2}}{8 x^2}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{13}{8} a^5 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6148
Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx &=c \int \frac{(1+a x)^3}{x^6 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{1}{5} c \int \frac{-15 a-19 a^2 x-5 a^3 x^2}{x^5 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}+\frac{1}{20} c \int \frac{76 a^2+65 a^3 x}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{1}{60} c \int \frac{-195 a^3-152 a^4 x}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{13 a^3 c \sqrt{1-a^2 x^2}}{8 x^2}+\frac{1}{120} c \int \frac{304 a^4+195 a^5 x}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{13 a^3 c \sqrt{1-a^2 x^2}}{8 x^2}-\frac{38 a^4 c \sqrt{1-a^2 x^2}}{15 x}+\frac{1}{8} \left (13 a^5 c\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{13 a^3 c \sqrt{1-a^2 x^2}}{8 x^2}-\frac{38 a^4 c \sqrt{1-a^2 x^2}}{15 x}+\frac{1}{16} \left (13 a^5 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{13 a^3 c \sqrt{1-a^2 x^2}}{8 x^2}-\frac{38 a^4 c \sqrt{1-a^2 x^2}}{15 x}-\frac{1}{8} \left (13 a^3 c\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{c \sqrt{1-a^2 x^2}}{5 x^5}-\frac{3 a c \sqrt{1-a^2 x^2}}{4 x^4}-\frac{19 a^2 c \sqrt{1-a^2 x^2}}{15 x^3}-\frac{13 a^3 c \sqrt{1-a^2 x^2}}{8 x^2}-\frac{38 a^4 c \sqrt{1-a^2 x^2}}{15 x}-\frac{13}{8} a^5 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [C] time = 0.121111, size = 110, normalized size = 0.76 \[ -3 a^5 c \sqrt{1-a^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},3,\frac{3}{2},1-a^2 x^2\right )-\frac{c \sqrt{1-a^2 x^2} \left (76 a^4 x^4+15 a^3 x^3+38 a^2 x^2+6\right )}{30 x^5}-\frac{1}{2} a^5 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.049, size = 292, normalized size = 2. \begin{align*} -c \left ( -3\,a \left ( -1/4\,{\frac{1}{{x}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}}+5/4\,{a}^{2} \left ( -1/2\,{\frac{1}{{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}}+3/2\,{a}^{2} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) \right ) \right ) +3\,{a}^{4} \left ( -{\frac{1}{x\sqrt{-{a}^{2}{x}^{2}+1}}}+2\,{\frac{{a}^{2}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{1}{5\,{x}^{5}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{16\,{a}^{2}}{5} \left ( -{\frac{1}{3\,{x}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{4\,{a}^{2}}{3} \left ( -{\frac{1}{x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+2\,{\frac{{a}^{2}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) } \right ) }+{a}^{5} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) +2\,{a}^{3} \left ( -1/2\,{\frac{1}{{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}}+3/2\,{a}^{2} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97974, size = 230, normalized size = 1.6 \begin{align*} \frac{38 \, a^{6} c x}{15 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{13}{8} \, a^{5} c \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{13 \, a^{5} c}{8 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{19 \, a^{4} c}{15 \, \sqrt{-a^{2} x^{2} + 1} x} - \frac{7 \, a^{3} c}{8 \, \sqrt{-a^{2} x^{2} + 1} x^{2}} - \frac{16 \, a^{2} c}{15 \, \sqrt{-a^{2} x^{2} + 1} x^{3}} - \frac{3 \, a c}{4 \, \sqrt{-a^{2} x^{2} + 1} x^{4}} - \frac{c}{5 \, \sqrt{-a^{2} x^{2} + 1} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61866, size = 197, normalized size = 1.37 \begin{align*} \frac{195 \, a^{5} c x^{5} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (304 \, a^{4} c x^{4} + 195 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 90 \, a c x + 24 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{120 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 31.3626, size = 518, normalized size = 3.6 \begin{align*} a^{3} c \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 ) + 3 a^{2} c \left (\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right ) + 3 a c \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 \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 = 1.16176, size = 448, normalized size = 3.11 \begin{align*} \frac{{\left (6 \, a^{6} c + \frac{45 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c}{x} + \frac{170 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c}{x^{2}} + \frac{480 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c}{x^{3}} + \frac{1380 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c}{a^{2} x^{4}}\right )} a^{10} x^{5}}{960 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}{\left | a \right |}} - \frac{13 \, a^{6} c \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{8 \,{\left | a \right |}} - \frac{\frac{1380 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{8} c}{x} + \frac{480 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{6} c}{x^{2}} + \frac{170 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a^{4} c}{x^{3}} + \frac{45 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} a^{2} c}{x^{4}} + \frac{6 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c}{x^{5}}}{960 \, a^{4}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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