Optimal. Leaf size=157 \[ \frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{c^3 (a x+8) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}-\frac{3 c^3 (8-a x) \sqrt{1-a^2 x^2}}{8 a^2 x}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{8 a}-\frac{3 c^3 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.315477, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6157, 6148, 1807, 811, 813, 844, 216, 266, 63, 208} \[ \frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{c^3 (a x+8) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}-\frac{3 c^3 (8-a x) \sqrt{1-a^2 x^2}}{8 a^2 x}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{8 a}-\frac{3 c^3 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6148
Rule 1807
Rule 811
Rule 813
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx &=-\frac{c^3 \int \frac{e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac{c^3 \int \frac{(1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}{x^6} \, dx}{a^6}\\ &=\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2} \left (-15 a-15 a^2 x-5 a^3 x^2\right )}{x^5} \, dx}{5 a^6}\\ &=\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac{c^3 \int \frac{\left (60 a^2+5 a^3 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{20 a^6}\\ &=\frac{c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac{c^3 \int \frac{\left (240 a^4+30 a^5 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx}{80 a^6}\\ &=-\frac{3 c^3 (8-a x) \sqrt{1-a^2 x^2}}{8 a^2 x}+\frac{c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac{c^3 \int \frac{-60 a^5+480 a^6 x}{x \sqrt{1-a^2 x^2}} \, dx}{160 a^6}\\ &=-\frac{3 c^3 (8-a x) \sqrt{1-a^2 x^2}}{8 a^2 x}+\frac{c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\left (3 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx+\frac{\left (3 c^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac{3 c^3 (8-a x) \sqrt{1-a^2 x^2}}{8 a^2 x}+\frac{c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac{3 c^3 \sin ^{-1}(a x)}{a}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac{3 c^3 (8-a x) \sqrt{1-a^2 x^2}}{8 a^2 x}+\frac{c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac{3 c^3 \sin ^{-1}(a x)}{a}-\frac{\left (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 )}{8 a^3}\\ &=-\frac{3 c^3 (8-a x) \sqrt{1-a^2 x^2}}{8 a^2 x}+\frac{c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac{c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac{3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac{3 c^3 \sin ^{-1}(a x)}{a}-\frac{3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{8 a}\\ \end{align*}
Mathematica [C] time = 0.100065, size = 186, normalized size = 1.18 \[ \frac{c^3 \left (-8 a^5 x^5 \left (a^2 x^2-1\right )^3 \text{Hypergeometric2F1}\left (2,\frac{5}{2},\frac{7}{2},1-a^2 x^2\right )+40 a^2 x^2 \sqrt{1-a^2 x^2} \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},a^2 x^2\right )-8 a^6 x^6+75 a^5 x^5+24 a^4 x^4-105 a^3 x^3-24 a^2 x^2+45 a^5 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+30 a x+8\right )}{40 a^6 x^5 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.072, size = 227, normalized size = 1.5 \begin{align*} -{{c}^{3}a{x}^{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{19\,{c}^{3}}{8\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{19\,{c}^{3}x}{5}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-3\,{\frac{{c}^{3}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{3\,{c}^{3}}{4\,{a}^{5}{x}^{4}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{17\,{c}^{3}}{8\,{x}^{2}{a}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{3\,{c}^{3}}{8\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{22\,{c}^{3}}{5\,{a}^{2}x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{c}^{3}}{5\,{a}^{6}{x}^{5}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{2\,{c}^{3}}{5\,{a}^{4}{x}^{3}}{\frac{1}{\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 = 1.46734, size = 614, normalized size = 3.91 \begin{align*} -a^{3} c^{3}{\left (\frac{x^{2}}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{2}{\sqrt{-a^{2} x^{2} + 1} a^{4}}\right )} + 3 \, a^{2} c^{3}{\left (\frac{x}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} - \frac{8 \, c^{3} x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{6 \, c^{3}{\left (\frac{1}{\sqrt{-a^{2} x^{2} + 1}} - \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )\right )}}{a} + \frac{6 \,{\left (\frac{2 \, a^{2} x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} x}\right )} c^{3}}{a^{2}} - \frac{4 \,{\left (3 \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{3 \, a^{2}}{\sqrt{-a^{2} x^{2} + 1}} + \frac{1}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\right )} c^{3}}{a^{3}} + \frac{3 \,{\left (15 \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{15 \, a^{4}}{\sqrt{-a^{2} x^{2} + 1}} + \frac{5 \, a^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{2}} + \frac{2}{\sqrt{-a^{2} x^{2} + 1} x^{4}}\right )} c^{3}}{8 \, a^{5}} - \frac{{\left (\frac{16 \, a^{6} x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{8 \, a^{4}}{\sqrt{-a^{2} x^{2} + 1} x} - \frac{2 \, a^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{3}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} x^{5}}\right )} c^{3}}{5 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27402, size = 338, normalized size = 2.15 \begin{align*} \frac{240 \, a^{5} c^{3} x^{5} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 15 \, a^{5} c^{3} x^{5} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + 40 \, a^{5} c^{3} x^{5} +{\left (40 \, a^{5} c^{3} x^{5} - 152 \, a^{4} c^{3} x^{4} - 55 \, a^{3} c^{3} x^{3} + 24 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x + 8 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{40 \, a^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.3562, size = 687, normalized size = 4.38 \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 = 1.26753, size = 520, normalized size = 3.31 \begin{align*} -\frac{{\left (2 \, c^{3} + \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{3}}{a^{2} x} + \frac{30 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} - \frac{80 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} - \frac{580 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{320 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}{\left | a \right |}} - \frac{3 \, c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{3 \, 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 )}{8 \,{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{a} - \frac{\frac{580 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac{80 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac{30 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} - \frac{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{320 \, a^{4}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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