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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6139, 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 6139
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 = 0.66, size = 93, normalized size = 1.12 \[ -\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (8 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} + 25 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, 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 = 109, normalized size = 1.31 \[ -\frac {c^{2} a \,x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5}+\frac {c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 a}+\frac {c^{2} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 90, normalized size = 1.08 \[ -\frac {1}{5} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a c^{2} x^{2} + \frac {1}{4} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} x + \frac {3}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{5 \, a} + \frac {3 \, c^{2} \arcsin \left (a x\right )}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 128, normalized size = 1.54 \[ \frac {5\,c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {c^2\,\sqrt {1-a^2\,x^2}}{5\,a}-\frac {2\,a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{5}-\frac {a^2\,c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}+\frac {a^3\,c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.89, size = 337, normalized size = 4.06 \[ a^{3} c^{2} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} \frac {i a^{2} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {i a^{2} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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