Optimal. Leaf size=83 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}}-\frac {1}{2 a c^3 \sqrt {c-a c x}}-\frac {1}{3 a c^2 (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {1}{2 a c^3 \sqrt {c-a c x}}-\frac {1}{3 a c^2 (c-a c x)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}} \]
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)^{7/2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\\ &=-\int \frac {1-a x}{(1+a x) (c-a c x)^{7/2}} \, dx\\ &=-\frac {\int \frac {1}{(1+a x) (c-a c x)^{5/2}} \, dx}{c}\\ &=-\frac {1}{3 a c^2 (c-a c x)^{3/2}}-\frac {\int \frac {1}{(1+a x) (c-a c x)^{3/2}} \, dx}{2 c^2}\\ &=-\frac {1}{3 a c^2 (c-a c x)^{3/2}}-\frac {1}{2 a c^3 \sqrt {c-a c x}}-\frac {\int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{4 c^3}\\ &=-\frac {1}{3 a c^2 (c-a c x)^{3/2}}-\frac {1}{2 a c^3 \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 )}{2 a c^4}\\ &=-\frac {1}{3 a c^2 (c-a c x)^{3/2}}-\frac {1}{2 a c^3 \sqrt {c-a c x}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 39, normalized size = 0.47 \[ -\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {1}{2} (1-a x)\right )}{3 a c^2 (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.64, size = 196, normalized size = 2.36 \[ \left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 4 \, \sqrt {-a c x + c} {\left (3 \, a x - 5\right )}}{24 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 2 \, \sqrt {-a c x + c} {\left (3 \, a x - 5\right )}}{12 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 73, normalized size = 0.88 \[ -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{4 \, a \sqrt {-c} c^{3}} - \frac {3 \, a c x - 5 \, c}{6 \, {\left (a c x - c\right )} \sqrt {-a c x + c} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 64, normalized size = 0.77 \[ -\frac {2 \left (\frac {1}{4 c^{2} \sqrt {-a c x +c}}+\frac {1}{6 c \left (-a c x +c \right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 c^{\frac {5}{2}}}\right )}{c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 81, normalized size = 0.98 \[ -\frac {\frac {3 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {5}{2}}} - \frac {4 \, {\left (3 \, a c x - 5 \, c\right )}}{{\left (-a c x + c\right )}^{\frac {3}{2}} c^{2}}}{24 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 65, normalized size = 0.78 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{4\,a\,c^{7/2}}-\frac {\frac {c-a\,c\,x}{2\,c^2}+\frac {1}{3\,c}}{a\,c\,{\left (c-a\,c\,x\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.54, size = 82, normalized size = 0.99 \[ - \frac {1}{3 a c^{2} \left (- a c x + c\right )^{\frac {3}{2}}} - \frac {1}{2 a c^{3} \sqrt {- a c x + c}} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{4 a c^{3} \sqrt {- c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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