Optimal. Leaf size=169 \[ -\frac{c^4 (7 a x+6) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 (35 a x+24) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (35 a x+16) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (16-35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}+\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}+\frac{c^4 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.219308, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6157, 6148, 811, 813, 844, 216, 266, 63, 208} \[ -\frac{c^4 (7 a x+6) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 (35 a x+24) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (35 a x+16) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (16-35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}+\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}+\frac{c^4 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6148
Rule 811
Rule 813
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx &=\frac{c^4 \int \frac{e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac{c^4 \int \frac{(1+a x) \left (1-a^2 x^2\right )^{7/2}}{x^8} \, dx}{a^8}\\ &=-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac{c^4 \int \frac{\left (12 a^2+14 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{12 a^8}\\ &=\frac{c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \int \frac{\left (96 a^4+140 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{96 a^8}\\ &=-\frac{c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac{c^4 \int \frac{\left (384 a^6+840 a^7 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx}{384 a^8}\\ &=\frac{c^4 (16-35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \int \frac{-1680 a^7+768 a^8 x}{x \sqrt{1-a^2 x^2}} \, dx}{768 a^8}\\ &=\frac{c^4 (16-35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+c^4 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{\left (35 c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{16 a}\\ &=\frac{c^4 (16-35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \sin ^{-1}(a x)}{a}-\frac{\left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac{c^4 (16-35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \sin ^{-1}(a x)}{a}+\frac{\left (35 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 )}{16 a^3}\\ &=\frac{c^4 (16-35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16+35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24+35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6+7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \sin ^{-1}(a x)}{a}+\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}\\ \end{align*}
Mathematica [C] time = 0.0318889, size = 70, normalized size = 0.41 \[ \frac{c^4 \left (-7 a^7 \left (1-a^2 x^2\right )^{9/2} \text{Hypergeometric2F1}\left (4,\frac{9}{2},\frac{11}{2},1-a^2 x^2\right )-\frac{9 \text{Hypergeometric2F1}\left (-\frac{7}{2},-\frac{7}{2},-\frac{5}{2},a^2 x^2\right )}{x^7}\right )}{63 a^8} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.061, size = 233, normalized size = 1.4 \begin{align*} -{\frac{{c}^{4}}{a}\sqrt{-{a}^{2}{x}^{2}+1}}+{{c}^{4}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{{c}^{4}}{6\,{a}^{7}{x}^{6}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{19\,{c}^{4}}{24\,{a}^{5}{x}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{29\,{c}^{4}}{16\,{x}^{2}{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{35\,{c}^{4}}{16\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{176\,{c}^{4}}{105\,{a}^{2}x}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{4}}{7\,{a}^{8}{x}^{7}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{22\,{c}^{4}}{35\,{a}^{6}{x}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{122\,{c}^{4}}{105\,{a}^{4}{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.50612, size = 707, normalized size = 4.18 \begin{align*} \frac{c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{4 \, c^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{a} - \frac{3 \,{\left (a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\sqrt{-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{4}}{a^{3}} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} c^{4}}{a^{2} x} - \frac{2 \,{\left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x} + \frac{\sqrt{-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{4}}{a^{4}} + \frac{{\left (3 \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{x^{4}}\right )} c^{4}}{2 \, a^{5}} + \frac{4 \,{\left (\frac{8 \, \sqrt{-a^{2} x^{2} + 1} a^{4}}{x} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{3}} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1}}{x^{5}}\right )} c^{4}}{15 \, a^{6}} - \frac{{\left (15 \, a^{6} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{15 \, \sqrt{-a^{2} x^{2} + 1} a^{4}}{x^{2}} + \frac{10 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{4}} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1}}{x^{6}}\right )} c^{4}}{48 \, a^{7}} - \frac{{\left (\frac{16 \, \sqrt{-a^{2} x^{2} + 1} a^{6}}{x} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1} a^{4}}{x^{3}} + \frac{6 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{5}} + \frac{5 \, \sqrt{-a^{2} x^{2} + 1}}{x^{7}}\right )} c^{4}}{35 \, a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11245, size = 413, normalized size = 2.44 \begin{align*} -\frac{3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3675 \, a^{7} c^{4} x^{7} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + 1680 \, a^{7} c^{4} x^{7} +{\left (1680 \, a^{7} c^{4} x^{7} - 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} + 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} - 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x + 240 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 133.666, size = 1119, normalized size = 6.62 \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 = 1.20922, size = 682, normalized size = 4.04 \begin{align*} \frac{{\left (15 \, c^{4} + \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac{189 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} - \frac{525 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} + \frac{1295 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} + \frac{4935 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} - \frac{9765 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{13440 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}{\left | a \right |}} + \frac{c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{35 \, 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{\sqrt{-a^{2} x^{2} + 1} c^{4}}{a} + \frac{\frac{9765 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{4}}{x} - \frac{4935 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac{1295 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} + \frac{525 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac{189 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} - \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{13440 \, a^{6}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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