Optimal. Leaf size=70 \[ -\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a}-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 \text {ArcSin}(a x)}{8 a^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6263, 794, 201,
222} \begin {gather*} -\frac {c^2 \text {ArcSin}(a x)}{8 a^2}-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 794
Rule 6263
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x (c-a c x)^2 \, dx &=c \int x (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 \int \sqrt {1-a^2 x^2} \, dx}{4 a}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a}-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a}-\frac {c^2 (4-3 a x) \left (1-a^2 x^2\right )^{3/2}}{12 a^2}-\frac {c^2 \sin ^{-1}(a x)}{8 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 67, normalized size = 0.96 \begin {gather*} -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (8-3 a x-8 a^2 x^2+6 a^3 x^3\right )-6 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{24 a^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs.
\(2(60)=120\).
time = 0.77, size = 197, normalized size = 2.81
method | result | size |
risch | \(\frac {\left (6 a^{3} x^{3}-8 a^{2} x^{2}-3 a x +8\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{24 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8 a \sqrt {a^{2}}}\) | \(86\) |
default | \(c^{2} \left (a^{3} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-a^{2} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-a \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{2}}\right )\) | \(197\) |
meijerg | \(\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 a \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{2} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a^{2} \sqrt {\pi }}+\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 a \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{2} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {\pi }}\) | \(221\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 95, normalized size = 1.36 \begin {gather*} -\frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{3} + \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{2} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x}{8 \, a} - \frac {c^{2} \arcsin \left (a x\right )}{8 \, a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{3 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 82, normalized size = 1.17 \begin {gather*} \frac {6 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (6 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.35, size = 330, normalized size = 4.71 \begin {gather*} a^{3} c^{2} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \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 = 69, normalized size = 0.99 \begin {gather*} -\frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, a {\left | a \right |}} - \frac {1}{24} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (3 \, a c^{2} x - 4 \, c^{2}\right )} x - \frac {3 \, c^{2}}{a}\right )} x + \frac {8 \, c^{2}}{a^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 108, normalized size = 1.54 \begin {gather*} \frac {c^2\,x^2\,\sqrt {1-a^2\,x^2}}{3}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2}+\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8\,a}-\frac {a\,c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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