Optimal. Leaf size=104 \[ -\frac {1}{5 a c^2 (c-a c x)^{5/2}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {1}{4 a c^4 \sqrt {c-a c x}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6302, 6265, 21,
53, 65, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}-\frac {1}{4 a c^4 \sqrt {c-a c x}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {1}{5 a c^2 (c-a c x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 53
Rule 65
Rule 212
Rule 6265
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{9/2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{9/2}} \, dx\\ &=-\int \frac {1-a x}{(1+a x) (c-a c x)^{9/2}} \, dx\\ &=-\frac {\int \frac {1}{(1+a x) (c-a c x)^{7/2}} \, dx}{c}\\ &=-\frac {1}{5 a c^2 (c-a c x)^{5/2}}-\frac {\int \frac {1}{(1+a x) (c-a c x)^{5/2}} \, dx}{2 c^2}\\ &=-\frac {1}{5 a c^2 (c-a c x)^{5/2}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {\int \frac {1}{(1+a x) (c-a c x)^{3/2}} \, dx}{4 c^3}\\ &=-\frac {1}{5 a c^2 (c-a c x)^{5/2}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {1}{4 a c^4 \sqrt {c-a c x}}-\frac {\int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{8 c^4}\\ &=-\frac {1}{5 a c^2 (c-a c x)^{5/2}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {1}{4 a c^4 \sqrt {c-a c x}}+\frac {\text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{4 a c^5}\\ &=-\frac {1}{5 a c^2 (c-a c x)^{5/2}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {1}{4 a c^4 \sqrt {c-a c x}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 39, normalized size = 0.38 \begin {gather*} -\frac {\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {1}{2} (1-a x)\right )}{5 a c^2 (c-a c x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 78, normalized size = 0.75
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 c^{\frac {7}{2}}}+\frac {1}{8 c^{3} \sqrt {-a c x +c}}+\frac {1}{12 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}+\frac {1}{10 c \left (-a c x +c \right )^{\frac {5}{2}}}\right )}{c a}\) | \(78\) |
default | \(-\frac {2 \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 c^{\frac {7}{2}}}+\frac {1}{8 c^{3} \sqrt {-a c x +c}}+\frac {1}{12 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}+\frac {1}{10 c \left (-a c x +c \right )^{\frac {5}{2}}}\right )}{c a}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 101, normalized size = 0.97 \begin {gather*} -\frac {\frac {15 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} + \frac {4 \, {\left (15 \, {\left (a c x - c\right )}^{2} - 10 \, {\left (a c x - c\right )} c + 12 \, c^{2}\right )}}{{\left (-a c x + c\right )}^{\frac {5}{2}} c^{3}}}{240 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 252, normalized size = 2.42 \begin {gather*} \left [\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 4 \, {\left (15 \, a^{2} x^{2} - 40 \, a x + 37\right )} \sqrt {-a c x + c}}{240 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}, -\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 2 \, {\left (15 \, a^{2} x^{2} - 40 \, a x + 37\right )} \sqrt {-a c x + c}}{120 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.74, size = 100, normalized size = 0.96 \begin {gather*} - \frac {1}{5 a c^{2} \left (- a c x + c\right )^{\frac {5}{2}}} - \frac {1}{6 a c^{3} \left (- a c x + c\right )^{\frac {3}{2}}} - \frac {1}{4 a c^{4} \sqrt {- a c x + c}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{8 a c^{4} \sqrt {- c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 93, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{8 \, a \sqrt {-c} c^{4}} - \frac {15 \, {\left (a c x - c\right )}^{2} - 10 \, {\left (a c x - c\right )} c + 12 \, c^{2}}{60 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 79, normalized size = 0.76 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{8\,a\,c^{9/2}}-\frac {\frac {c-a\,c\,x}{6\,c^2}+\frac {1}{5\,c}+\frac {{\left (c-a\,c\,x\right )}^2}{4\,c^3}}{a\,c\,{\left (c-a\,c\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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