Optimal. Leaf size=184 \[ -\frac {4 x \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}+\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {a x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6288, 6285, 90}
\begin {gather*} \frac {2 x^2 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}+\frac {a x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a^3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{-3 \tanh ^{-1}(a x)} x^2 \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {x^2 (1-a x)^2}{1+a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (-\frac {4}{a^2}+\frac {4 x}{a}-3 x^2+a x^3+\frac {4}{a^2 (1+a x)}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {4 x \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}+\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {a x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 69, normalized size = 0.38 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left (-\frac {4 x}{a^2}+\frac {2 x^2}{a}-x^3+\frac {a x^4}{4}+\frac {4 \log (1+a x)}{a^3}\right )}{\sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 79, normalized size = 0.43
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (a^{4} x^{4}-4 a^{3} x^{3}+8 a^{2} x^{2}-16 a x +16 \ln \left (a x +1\right )\right )}{4 \left (a^{2} x^{2}-1\right ) a^{3}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 383, normalized size = 2.08 \begin {gather*} \left [\frac {8 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left (a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) - {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 16 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{4 \, {\left (a^{5} x^{2} - a^{3}\right )}}, \frac {16 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right ) - {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 16 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{4 \, {\left (a^{5} x^{2} - a^{3}\right )}}\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 \frac {x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\left (a x + 1\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^2\,\sqrt {c-a^2\,c\,x^2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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