Optimal. Leaf size=127 \[ \frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35 c^4 \sin ^{-1}(a x)}{128 a} \]
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Rubi [A] time = 0.0682698, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6139, 641, 195, 216} \[ \frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35 c^4 \sin ^{-1}(a x)}{128 a} \]
Antiderivative was successfully verified.
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Rule 6139
Rule 641
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=c^4 \int (1-a x) \left (1-a^2 x^2\right )^{7/2} \, dx\\ &=\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+c^4 \int \left (1-a^2 x^2\right )^{7/2} \, dx\\ &=\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{8} \left (7 c^4\right ) \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{48} \left (35 c^4\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{64} \left (35 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{1}{128} \left (35 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{35}{128} c^4 x \sqrt{1-a^2 x^2}+\frac{35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac{7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac{1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac{c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac{35 c^4 \sin ^{-1}(a x)}{128 a}\\ \end{align*}
Mathematica [A] time = 0.144339, size = 107, normalized size = 0.84 \[ \frac{c^4 \left (\sqrt{1-a^2 x^2} \left (128 a^8 x^8-144 a^7 x^7-512 a^6 x^6+600 a^5 x^5+768 a^4 x^4-978 a^3 x^3-512 a^2 x^2+837 a x+128\right )-630 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{1152 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.07, size = 201, normalized size = 1.6 \begin{align*} -{\frac{{c}^{4}{a}^{5}{x}^{6}}{9} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{4}{a}^{3}{x}^{4}}{3} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{{c}^{4}a{x}^{2}}{3} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{4}}{9\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{4}{c}^{4}{x}^{5}}{8} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{19\,{a}^{2}{c}^{4}{x}^{3}}{48} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{29\,{c}^{4}x}{64} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{c}^{4}x}{128}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{35\,{c}^{4}}{128}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48355, size = 246, normalized size = 1.94 \begin{align*} -\frac{1}{9} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{5} c^{4} x^{6} + \frac{1}{8} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{4} c^{4} x^{5} + \frac{1}{3} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{3} c^{4} x^{4} - \frac{19}{48} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} c^{4} x^{3} - \frac{1}{3} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a c^{4} x^{2} + \frac{29}{64} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{4} x + \frac{35}{128} \, \sqrt{-a^{2} x^{2} + 1} c^{4} x + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{4}}{9 \, a} + \frac{35 \, c^{4} \arcsin \left (a x\right )}{128 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22347, size = 312, normalized size = 2.46 \begin{align*} -\frac{630 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (128 \, a^{8} c^{4} x^{8} - 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} + 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} - 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} + 837 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{1152 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 20.0465, size = 996, normalized size = 7.84 \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 [A] time = 1.19541, size = 167, normalized size = 1.31 \begin{align*} \frac{35 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{128 \,{\left | a \right |}} + \frac{1}{1152} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{128 \, c^{4}}{a} +{\left (837 \, c^{4} - 2 \,{\left (256 \, a c^{4} +{\left (489 \, a^{2} c^{4} - 4 \,{\left (96 \, a^{3} c^{4} +{\left (75 \, a^{4} c^{4} - 2 \,{\left (32 \, a^{5} c^{4} -{\left (8 \, a^{7} c^{4} x - 9 \, a^{6} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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