Optimal. Leaf size=45 \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a^4}+\frac {c \left (1-a^2 x^2\right )^{5/2}}{5 a^4} \]
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Rubi [A]
time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6263, 272, 45}
\begin {gather*} \frac {c \left (1-a^2 x^2\right )^{5/2}}{5 a^4}-\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 6263
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^3 (c-a c x) \, dx &=c \int x^3 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {1}{2} c \text {Subst}\left (\int x \sqrt {1-a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{2} c \text {Subst}\left (\int \left (\frac {\sqrt {1-a^2 x}}{a^2}-\frac {\left (1-a^2 x\right )^{3/2}}{a^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a^4}+\frac {c \left (1-a^2 x^2\right )^{5/2}}{5 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 0.71 \begin {gather*} -\frac {c \left (1-a^2 x^2\right )^{3/2} \left (2+3 a^2 x^2\right )}{15 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs.
\(2(37)=74\).
time = 0.40, size = 110, normalized size = 2.44
method | result | size |
trager | \(\frac {c \left (3 a^{4} x^{4}-a^{2} x^{2}-2\right ) \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\) | \(37\) |
gosper | \(-\frac {\left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (3 a^{2} x^{2}+2\right ) c}{15 a^{4} \sqrt {-a^{2} x^{2}+1}}\) | \(43\) |
risch | \(-\frac {c \left (3 a^{4} x^{4}-a^{2} x^{2}-2\right ) \left (a^{2} x^{2}-1\right )}{15 a^{4} \sqrt {-a^{2} x^{2}+1}}\) | \(46\) |
meijerg | \(\frac {c \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a^{4} \sqrt {\pi }}+\frac {c \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a^{4} \sqrt {\pi }}\) | \(94\) |
default | \(-c \left (a^{2} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )+\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}+\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 58, normalized size = 1.29 \begin {gather*} \frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} c x^{4} - \frac {\sqrt {-a^{2} x^{2} + 1} c x^{2}}{15 \, a^{2}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c}{15 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 39, normalized size = 0.87 \begin {gather*} \frac {{\left (3 \, a^{4} c x^{4} - a^{2} c x^{2} - 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{15 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 66, normalized size = 1.47 \begin {gather*} \begin {cases} \frac {c x^{4} \sqrt {- a^{2} x^{2} + 1}}{5} - \frac {c x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{2}} - \frac {2 c \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {c x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 47, normalized size = 1.04 \begin {gather*} \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c - 5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{15 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 36, normalized size = 0.80 \begin {gather*} -\frac {5\,c\,{\left (1-a^2\,x^2\right )}^{3/2}-3\,c\,{\left (1-a^2\,x^2\right )}^{5/2}}{15\,a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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