Optimal. Leaf size=82 \[ -\frac {2 \sqrt {1-a^2 x^2}}{a (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{a \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6278, 6275, 45}
\begin {gather*} -\frac {2 \sqrt {1-a^2 x^2}}{a (a x+1) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (a x+1)}{a \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1-a x}{(1+a x)^2} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{-1-a x}+\frac {2}{(1+a x)^2}\right ) \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{a \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 0.66 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (-\frac {2}{a (1+a x)}-\frac {\log (1+a x)}{a}\right )}{\sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 69, normalized size = 0.84
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x +1\right ) a x +\ln \left (a x +1\right )+2\right )}{\left (a^{2} x^{2}-1\right ) c a \left (a x +1\right )}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 33, normalized size = 0.40 \begin {gather*} -\frac {\log \left (a x + 1\right )}{a \sqrt {c}} - \frac {2}{a^{2} \sqrt {c} x + a \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (74) = 148\).
time = 0.35, size = 381, normalized size = 4.65 \begin {gather*} \left [-\frac {4 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a x - {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 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 )}{2 \, {\left (a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c\right )}}, -\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a x + {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 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 )}{a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c}\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 {\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 {{\left (1-a^2\,x^2\right )}^{3/2}}{\sqrt {c-a^2\,c\,x^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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