Optimal. Leaf size=121 \[ -\frac{c^2 (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac{7 c^2 (a x+1) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{7 c^2 \sin ^{-1}(a x)}{8 a} \]
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Rubi [A] time = 0.0715503, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6138, 671, 641, 195, 216} \[ -\frac{c^2 (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac{7 c^2 (a x+1) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}+\frac{7 c^2 \sin ^{-1}(a x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 6138
Rule 671
Rule 641
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1+a x)^3 \sqrt{1-a^2 x^2} \, dx\\ &=-\frac{c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{5} \left (7 c^2\right ) \int (1+a x)^2 \sqrt{1-a^2 x^2} \, dx\\ &=-\frac{7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac{c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^2\right ) \int (1+a x) \sqrt{1-a^2 x^2} \, dx\\ &=-\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}-\frac{7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac{c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^2\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}-\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}-\frac{7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac{c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{8} \left (7 c^2\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{7}{8} c^2 x \sqrt{1-a^2 x^2}-\frac{7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}-\frac{7 c^2 (1+a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac{c^2 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^2 \sin ^{-1}(a x)}{8 a}\\ \end{align*}
Mathematica [A] time = 0.0990525, size = 75, normalized size = 0.62 \[ \frac{c^2 \left (\sqrt{1-a^2 x^2} \left (24 a^4 x^4+90 a^3 x^3+112 a^2 x^2+15 a x-136\right )-210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{120 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.066, size = 183, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}{a}^{5}{x}^{6}}{5}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{11\,{c}^{2}{a}^{3}{x}^{4}}{15}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{31\,a{c}^{2}{x}^{2}}{15}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{17\,{c}^{2}}{15\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{3\,{a}^{4}{c}^{2}{x}^{5}}{4}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{5\,{a}^{2}{c}^{2}{x}^{3}}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{x{c}^{2}}{8}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{7\,{c}^{2}}{8}\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 = 1.46709, size = 234, normalized size = 1.93 \begin{align*} -\frac{a^{5} c^{2} x^{6}}{5 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{3 \, a^{4} c^{2} x^{5}}{4 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{11 \, a^{3} c^{2} x^{4}}{15 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{5 \, a^{2} c^{2} x^{3}}{8 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{31 \, a c^{2} x^{2}}{15 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{c^{2} x}{8 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{7 \, c^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}}} - \frac{17 \, c^{2}}{15 \, \sqrt{-a^{2} x^{2} + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62267, size = 209, normalized size = 1.73 \begin{align*} -\frac{210 \, c^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (24 \, a^{4} c^{2} x^{4} + 90 \, a^{3} c^{2} x^{3} + 112 \, a^{2} c^{2} x^{2} + 15 \, a c^{2} x - 136 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{120 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 16.9266, size = 340, normalized size = 2.81 \begin{align*} a^{3} c^{2} \left (\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 3 a^{2} c^{2} \left (\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{8 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left (a x \right )}}{8 a^{3}} & \text{otherwise} \end{cases}\right ) + 3 a c^{2} \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right ) + c^{2} \left (\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14445, size = 105, normalized size = 0.87 \begin{align*} \frac{7 \, c^{2} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} + \frac{1}{120} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (15 \, c^{2} + 2 \,{\left (56 \, a c^{2} + 3 \,{\left (4 \, a^{3} c^{2} x + 15 \, a^{2} c^{2}\right )} x\right )} x\right )} x - \frac{136 \, c^{2}}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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