Optimal. Leaf size=183 \[ \frac {\sqrt {1-a^2 x^2}}{8 a c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^2 (1+a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{4 a c^2 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a c^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6278, 6275, 46,
213} \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{8 a c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{4 a c^2 (a x+1) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^2 (a x+1)^2 \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a c^2 \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{(1-a x)^2 (1+a x)^3} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{8 (-1+a x)^2}+\frac {1}{4 (1+a x)^3}+\frac {1}{4 (1+a x)^2}-\frac {3}{8 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{8 a c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^2 (1+a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{4 a c^2 (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (3 \sqrt {1-a^2 x^2}\right ) \int \frac {1}{-1+a^2 x^2} \, dx}{8 c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{8 a c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^2 (1+a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{4 a c^2 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 83, normalized size = 0.45 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (2-3 a x-3 a^2 x^2+3 (-1+a x) (1+a x)^2 \tanh ^{-1}(a x)\right )}{8 a (-1+a x) (c+a c x)^2 \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 166, normalized size = 0.91
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (3 \ln \left (a x -1\right ) a^{3} x^{3}-3 \ln \left (a x +1\right ) a^{3} x^{3}+3 \ln \left (a x -1\right ) a^{2} x^{2}-3 \ln \left (a x +1\right ) a^{2} x^{2}+6 a^{2} x^{2}-3 \ln \left (a x -1\right ) a x +3 \ln \left (a x +1\right ) a x +6 a x -3 \ln \left (a x -1\right )+3 \ln \left (a x +1\right )-4\right )}{16 \left (a^{2} x^{2}-1\right ) c^{3} a \left (a x +1\right )^{2} \left (a x -1\right )}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 83, normalized size = 0.45 \begin {gather*} -\frac {3 \, a^{2} x^{2} + 3 \, a x - 2}{8 \, {\left (a^{4} c^{\frac {5}{2}} x^{3} + a^{3} c^{\frac {5}{2}} x^{2} - a^{2} c^{\frac {5}{2}} x - a c^{\frac {5}{2}}\right )}} + \frac {3 \, \log \left (a x + 1\right )}{16 \, a c^{\frac {5}{2}}} - \frac {3 \, \log \left (a x - 1\right )}{16 \, a c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 451, normalized size = 2.46 \begin {gather*} \left [\frac {3 \, {\left (a^{5} x^{5} + a^{4} x^{4} - 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + a x + 1\right )} \sqrt {c} \log \left (-\frac {a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) - 4 \, {\left (2 \, a^{3} x^{3} - a^{2} x^{2} - 5 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{32 \, {\left (a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\right )}}, \frac {3 \, {\left (a^{5} x^{5} + a^{4} x^{4} - 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a \sqrt {-c} x}{a^{4} c x^{4} - c}\right ) - 2 \, {\left (2 \, a^{3} x^{3} - a^{2} x^{2} - 5 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{16 \, {\left (a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{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 {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}\, 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 {\sqrt {1-a^2\,x^2}}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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