Optimal. Leaf size=123 \[ \frac {7}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {7 c^2 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^2 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^2 \text {ArcSin}(a x)}{8 a} \]
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Rubi [A]
time = 0.05, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 = {6274, 685, 655,
201, 222} \begin {gather*} \frac {c^2 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^2 (1-a x) \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 \text {ArcSin}(a x)}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 655
Rule 685
Rule 6274
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*}
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Mathematica [A]
time = 0.07, size = 75, normalized size = 0.61 \begin {gather*} -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (-136-15 a x+112 a^2 x^2-90 a^3 x^3+24 a^4 x^4\right )+210 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.07, size = 194, normalized size = 1.58
method | result | size |
risch | \(\frac {\left (24 a^{4} x^{4}-90 a^{3} x^{3}+112 a^{2} x^{2}-15 a x -136\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{120 a \sqrt {-a^{2} x^{2}+1}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8 \sqrt {a^{2}}}\) | \(91\) |
default | \(c^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{5 a}-\frac {3 x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}-\frac {9 x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}+\frac {\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+4 a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{a}\right )\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 147, normalized size = 1.20 \begin {gather*} -\frac {3}{4} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} x - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{2}}{5 \, a} + 2 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2} x - \frac {9}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3 \, a} - \frac {2 i \, c^{2} \arcsin \left (a x + 2\right )}{a} - \frac {9 \, c^{2} \arcsin \left (a x\right )}{8 \, a} + \frac {4 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 92, normalized size = 0.75 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.67, size = 338, normalized size = 2.75 \begin {gather*} - 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 x \sqrt {a^{2} x^{2} - 1}}{2} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {x}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 78, normalized size = 0.63 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 128, normalized size = 1.04 \begin {gather*} \frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {7\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {17\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a}-\frac {14\,a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15}+\frac {3\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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