Optimal. Leaf size=147 \[ -\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{2 a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6302, 6268,
25, 528, 382, 105, 157, 162, 65, 214} \begin {gather*} \frac {3 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}}-\frac {7}{2 a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 65
Rule 105
Rule 157
Rule 162
Rule 214
Rule 382
Rule 528
Rule 6268
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\\ &=-\int \frac {1-a x}{\left (c-\frac {c}{a x}\right )^{7/2} (1+a x)} \, dx\\ &=\frac {a \int \frac {x}{\left (c-\frac {c}{a x}\right )^{5/2} (1+a x)} \, dx}{c}\\ &=\frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}} \, dx}{c}\\ &=-\frac {a \text {Subst}\left (\int \frac {1}{x^2 (a+x) \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-\frac {3 c}{2}-\frac {5 c x}{2 a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=-\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {\frac {9 c^2}{2}+\frac {6 c^2 x}{a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^4}\\ &=-\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{2 a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-\frac {9 c^3}{2}-\frac {21 c^3 x}{4 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c^6}\\ &=-\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{2 a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 a c^3}-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^3}\\ &=-\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{2 a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{2 c^4}+\frac {3 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^4}\\ &=-\frac {4}{3 a c^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{2 a c^3 \sqrt {c-\frac {c}{a x}}}+\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{7/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.07, size = 79, normalized size = 0.54 \begin {gather*} \frac {x \left (3 a x-\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a-\frac {1}{x}}{2 a}\right )-3 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {1}{a x}\right )\right )}{3 c^3 \sqrt {c-\frac {c}{a x}} (-1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(496\) vs.
\(2(120)=240\).
time = 0.24, size = 497, normalized size = 3.38
method | result | size |
risch | \(\frac {a x -1}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {3 \ln \left (\frac {-\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {17 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{6 a^{6} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{8 a^{5} \sqrt {c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{3 a^{7} c \left (x -\frac {1}{a}\right )^{2}}\right ) a^{3} \sqrt {c \left (a x -1\right ) a x}}{c^{3} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(271\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-84 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{3}+60 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x -36 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{3}+3 a^{\frac {7}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{3}+252 \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}-52 \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}+108 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{2}-9 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{2}-252 \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x -108 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x +9 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x +84 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+36 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-3 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{24 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, c^{4} \left (a x -1\right )^{3} \sqrt {\frac {1}{a}}}\) | \(497\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 359, normalized size = 2.44 \begin {gather*} \left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 36 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 4 \, {\left (6 \, a^{3} x^{3} - 29 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{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 {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + 36 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - 2 \, {\left (6 \, a^{3} x^{3} - 29 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\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 {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {7}{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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {a\,x-1}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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