Optimal. Leaf size=85 \[ -\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6287, 1823,
794, 223, 209} \begin {gather*} -\frac {\sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2}-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 794
Rule 1823
Rule 6287
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx &=c \int \frac {x (1-a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x \left (-5 a^2 c+6 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{3 a^2}\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {c \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{a}\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {c \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{a}\\ &=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5-3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 80, normalized size = 0.94 \begin {gather*} \frac {-\left (\left (5-3 a x+a^2 x^2\right ) \sqrt {c-a^2 c x^2}\right )+3 \sqrt {c} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{3 a^2} \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. \(153\) vs.
\(2(71)=142\).
time = 0.06, size = 154, normalized size = 1.81
method | result | size |
risch | \(\frac {\left (a^{2} x^{2}-3 a x +5\right ) \left (a^{2} x^{2}-1\right ) c}{3 a^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {\arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c}{a \sqrt {c \,a^{2}}}\) | \(80\) |
default | \(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,a^{2}}+\frac {x \sqrt {-a^{2} c \,x^{2}+c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {c \,a^{2}}}}{a}-\frac {2 \left (\sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}+\frac {a c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {c \,a^{2}}}\right )}{a^{2}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 70, normalized size = 0.82 \begin {gather*} \frac {\sqrt {-a^{2} c x^{2} + c} x}{a} - \frac {\sqrt {c} \arcsin \left (a x\right )}{a^{2}} - \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2}} + \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 150, normalized size = 1.76 \begin {gather*} \left [-\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 3 \, a x + 5\right )} - 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{6 \, a^{2}}, -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 3 \, a x + 5\right )} - 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{3 \, a^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {x \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\right )\, dx - \int \frac {a x^{2} \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs.
\(2 (71) = 142\).
time = 0.46, size = 174, normalized size = 2.05 \begin {gather*} \frac {{\left (24 \, a^{4} \sqrt {c} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - \frac {{\left (9 \, a^{4} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 3 \, a^{4} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 8 \, a^{4} c^{2} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )\right )} {\left (a x + 1\right )}^{3}}{c^{3}}\right )} {\left | a \right |}}{12 \, a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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