Optimal. Leaf size=83 \[ -\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac {3}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {3 c^2 \sin ^{-1}(a x)}{8 a} \]
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Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6138, 641, 195, 216} \[ -\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac {3}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {3 c^2 \sin ^{-1}(a x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 6138
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1+a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+c^2 \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} \left (3 c^2\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {3}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{8} \left (3 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {3 c^2 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 75, normalized size = 0.90 \[ -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (8 a^4 x^4+10 a^3 x^3-16 a^2 x^2-25 a x+8\right )+30 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{40 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 92, normalized size = 1.11 \[ -\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (8 \, a^{4} c^{2} x^{4} + 10 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} - 25 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 78, normalized size = 0.94 \[ \frac {3 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, {\left | a \right |}} + \frac {1}{40} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (25 \, c^{2} + 2 \, {\left (8 \, a c^{2} - {\left (4 \, a^{3} c^{2} x + 5 \, a^{2} c^{2}\right )} x\right )} x\right )} x - \frac {8 \, c^{2}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 137, normalized size = 1.65 \[ -\frac {c^{2} a^{3} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}+\frac {2 c^{2} a \,x^{2} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{5 a}-\frac {c^{2} a^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{4}+\frac {5 c^{2} x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 118, normalized size = 1.42 \[ -\frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2} x^{4} - \frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x^{3} + \frac {2}{5} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{2} + \frac {5}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {3 \, c^{2} \arcsin \left (a x\right )}{8 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{5 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 82, normalized size = 0.99 \[ \frac {3\,c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {c^2\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{4}-\frac {c^2\,{\left (1-a^2\,x^2\right )}^{5/2}}{5\,a}-\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{8\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.15, size = 143, normalized size = 1.72 \[ \begin {cases} \frac {- \frac {c^{2} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} + c^{2} \left (\begin {cases} \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - c^{2} \left (\begin {cases} - \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} + \frac {\operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - c^{2} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\c^{2} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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