Optimal. Leaf size=156 \[ -\frac{7 a^4 c^4 \sqrt{1-a^2 x^2}}{16 x^2}+\frac{11 a^3 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{7}{16} a^6 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.263734, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{7 a^4 c^4 \sqrt{1-a^2 x^2}}{16 x^2}+\frac{11 a^3 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{7}{16} a^6 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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)^4}{x^7} \, dx &=c \int \frac{(c-a c x)^3 \sqrt{1-a^2 x^2}}{x^7} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}-\frac{1}{6} c \int \frac{\sqrt{1-a^2 x^2} \left (18 a c^3-21 a^2 c^3 x+6 a^3 c^3 x^2\right )}{x^6} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}+\frac{1}{30} c \int \frac{\left (105 a^2 c^3-66 a^3 c^3 x\right ) \sqrt{1-a^2 x^2}}{x^5} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}-\frac{1}{120} c \int \frac{\left (264 a^3 c^3-105 a^4 c^3 x\right ) \sqrt{1-a^2 x^2}}{x^4} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{11 a^3 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{8} \left (7 a^4 c^4\right ) \int \frac{\sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{11 a^3 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{16} \left (7 a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{7 a^4 c^4 \sqrt{1-a^2 x^2}}{16 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{11 a^3 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}-\frac{1}{32} \left (7 a^6 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{7 a^4 c^4 \sqrt{1-a^2 x^2}}{16 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{11 a^3 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{1}{16} \left (7 a^4 c^4\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{7 a^4 c^4 \sqrt{1-a^2 x^2}}{16 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{6 x^6}+\frac{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}{5 x^5}-\frac{7 a^2 c^4 \left (1-a^2 x^2\right )^{3/2}}{8 x^4}+\frac{11 a^3 c^4 \left (1-a^2 x^2\right )^{3/2}}{15 x^3}+\frac{7}{16} a^6 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0367067, size = 115, normalized size = 0.74 \[ \frac{c^4 \left (176 a^7 x^7-105 a^6 x^6-208 a^5 x^5+275 a^4 x^4-112 a^3 x^3-130 a^2 x^2+105 a^6 x^6 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+144 a x-40\right )}{240 x^6 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 255, normalized size = 1.6 \begin{align*}{c}^{4} \left ({\frac{17\,{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 ) }-{\frac{{a}^{5}}{x}\sqrt{-{a}^{2}{x}^{2}+1}}-3\,a \left ( -1/5\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{5}}}+4/5\,{a}^{2} \left ( -1/3\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{3}}}-2/3\,{\frac{{a}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}{x}} \right ) \right ) -{\frac{1}{6\,{x}^{6}}\sqrt{-{a}^{2}{x}^{2}+1}}-3\,{a}^{4} \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 ) +2\,{a}^{3} \left ( -1/3\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{3}}}-2/3\,{\frac{{a}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}{x}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43569, size = 227, normalized size = 1.46 \begin{align*} \frac{7}{16} \, a^{6} c^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{11 \, \sqrt{-a^{2} x^{2} + 1} a^{5} c^{4}}{15 \, x} + \frac{7 \, \sqrt{-a^{2} x^{2} + 1} a^{4} c^{4}}{16 \, x^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4}}{15 \, x^{3}} - \frac{17 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4}}{24 \, x^{4}} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} a c^{4}}{5 \, x^{5}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6049, size = 239, normalized size = 1.53 \begin{align*} -\frac{105 \, a^{6} c^{4} x^{6} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (176 \, a^{5} c^{4} x^{5} - 105 \, a^{4} c^{4} x^{4} - 32 \, a^{3} c^{4} x^{3} + 170 \, a^{2} c^{4} x^{2} - 144 \, a c^{4} x + 40 \, c^{4}\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 = 17.6997, size = 801, normalized size = 5.13 \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 = 1.17995, size = 572, normalized size = 3.67 \begin{align*} \frac{{\left (5 \, a^{7} c^{4} - \frac{36 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{5} c^{4}}{x} + \frac{105 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{3} c^{4}}{x^{2}} - \frac{140 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a c^{4}}{x^{3}} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}}{a x^{4}} + \frac{600 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{4}}{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{7 \, a^{7} c^{4} \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{600 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{9} c^{4}{\left | a \right |}}{x} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{7} c^{4}{\left | a \right |}}{x^{2}} - \frac{140 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a^{5} c^{4}{\left | a \right |}}{x^{3}} + \frac{105 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} a^{3} c^{4}{\left | a \right |}}{x^{4}} - \frac{36 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} a c^{4}{\left | a \right |}}{x^{5}} + \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{4}{\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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