Optimal. Leaf size=104 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}-\frac {1}{4 a c^4 \sqrt {c-a c x}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {1}{5 a c^2 (c-a c x)^{5/2}} \]
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Rubi [A] time = 0.13, 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 = {6167, 6130, 21, 51, 63, 206} \[ -\frac {1}{4 a c^4 \sqrt {c-a c x}}-\frac {1}{6 a c^3 (c-a c x)^{3/2}}-\frac {1}{5 a c^2 (c-a c x)^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 51
Rule 63
Rule 206
Rule 6130
Rule 6167
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 {\operatorname {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] time = 0.04, size = 39, normalized size = 0.38 \[ -\frac {\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {1}{2} (1-a x)\right )}{5 a c^2 (c-a c x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 252, normalized size = 2.42 \[ \left [\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 4 \, {\left (15 \, a^{2} x^{2} - 40 \, a x + 37\right )} \sqrt {-a c x + c}}{240 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}, -\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 2 \, {\left (15 \, a^{2} x^{2} - 40 \, a x + 37\right )} \sqrt {-a c x + c}}{120 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 93, normalized size = 0.89 \[ -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{8 \, a \sqrt {-c} c^{4}} - \frac {15 \, {\left (a c x - c\right )}^{2} - 10 \, {\left (a c x - c\right )} c + 12 \, c^{2}}{60 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 0.75 \[ -\frac {2 \left (\frac {1}{8 c^{3} \sqrt {-a c x +c}}+\frac {1}{12 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}+\frac {1}{10 c \left (-a c x +c \right )^{\frac {5}{2}}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 c^{\frac {7}{2}}}\right )}{c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 101, normalized size = 0.97 \[ -\frac {\frac {15 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} + \frac {4 \, {\left (15 \, {\left (a c x - c\right )}^{2} - 10 \, {\left (a c x - c\right )} c + 12 \, c^{2}\right )}}{{\left (-a c x + c\right )}^{\frac {5}{2}} c^{3}}}{240 \, a c} \]
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 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{8\,a\,c^{9/2}}-\frac {\frac {c-a\,c\,x}{6\,c^2}+\frac {1}{5\,c}+\frac {{\left (c-a\,c\,x\right )}^2}{4\,c^3}}{a\,c\,{\left (c-a\,c\,x\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.75, size = 100, normalized size = 0.96 \[ - \frac {1}{5 a c^{2} \left (- a c x + c\right )^{\frac {5}{2}}} - \frac {1}{6 a c^{3} \left (- a c x + c\right )^{\frac {3}{2}}} - \frac {1}{4 a c^{4} \sqrt {- a c x + c}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{8 a c^{4} \sqrt {- c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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