Optimal. Leaf size=105 \[ -\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6107, 6105}
\begin {gather*} -\frac {2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6105
Rule 6107
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\tanh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 64, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {c-a^2 c x^2} \left (7-6 a^2 x^2+\left (-9 a x+6 a^3 x^3\right ) \tanh ^{-1}(a x)\right )}{9 a c^3 \left (-1+a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.60, size = 160, normalized size = 1.52
method | result | size |
default | \(\frac {\left (a x +1\right ) \left (3 \arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{72 a \left (a x -1\right )^{2} c^{3}}-\frac {3 \left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{8 a \,c^{3} \left (a x -1\right )}-\frac {3 \left (\arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{8 a \left (a x +1\right ) c^{3}}+\frac {\left (a x -1\right ) \left (1+3 \arctanh \left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{72 a \left (a x +1\right )^{2} c^{3}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 90, normalized size = 0.86 \begin {gather*} -\frac {1}{9} \, a {\left (\frac {6}{\sqrt {-a^{2} c x^{2} + c} a^{2} c^{2}} + \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c}\right )} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{2}} + \frac {x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 84, normalized size = 0.80 \begin {gather*} \frac {\sqrt {-a^{2} c x^{2} + c} {\left (12 \, a^{2} x^{2} - 3 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 14\right )}}{18 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\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 {\operatorname {atanh}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 111, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (\frac {2 \, a^{2} x^{2}}{c} - \frac {3}{c}\right )} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, {\left (a^{2} c x^{2} - c\right )}^{2}} - \frac {6 \, a^{2} c x^{2} - 7 \, c}{9 \, {\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} c x^{2} + c} a c^{2}} \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 {\mathrm {atanh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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