Optimal. Leaf size=179 \[ -\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6329, 105, 157,
12, 94, 214} \begin {gather*} \frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {\frac {1}{a x}+1}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 94
Rule 105
Rule 157
Rule 214
Rule 6329
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{a}-\frac {3 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a \text {Subst}\left (\int \frac {-\frac {1}{a^2}+\frac {4 x}{a^3}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^2 \text {Subst}\left (\int \frac {-\frac {3}{a^3}+\frac {5 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^2}\\ &=-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^3 \text {Subst}\left (\int -\frac {3}{a^4 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c^2}\\ &=-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c^2}\\ &=-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c^2}\\ &=-\frac {2}{a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 85, normalized size = 0.47 \begin {gather*} \frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-8-5 a x+7 a^2 x^2+3 a^3 x^3\right )}{3 (-1+a x) (1+a x)^2}-\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(529\) vs.
\(2(155)=310\).
time = 0.16, size = 530, normalized size = 2.96
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2}}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {\sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{4 a^{6} \left (x -\frac {1}{a}\right )}+\frac {19 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{12 a^{6} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{6 a^{7} \left (x +\frac {1}{a}\right )^{2}}\right ) a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{2} \left (a x -1\right )}\) | \(213\) |
default | \(-\frac {\left (-45 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+21 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-45 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-11 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+90 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-5 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x +90 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+19 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -45 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}+24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right ) \sqrt {\frac {a x -1}{a x +1}}}{24 a \left (a x -1\right )^{2} \sqrt {a^{2}}\, \left (a x +1\right )^{2} c^{2} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}\) | \(530\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 163, normalized size = 0.91 \begin {gather*} -\frac {1}{12} \, a {\left (\frac {3 \, {\left (\frac {9 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 18 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 119, normalized size = 0.66 \begin {gather*} -\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (3 \, a^{3} x^{3} + 7 \, a^{2} x^{2} - 5 \, a x - 8\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\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*} \frac {a^{4} \int \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 137, normalized size = 0.77 \begin {gather*} \frac {\frac {9\,\left (a\,x-1\right )}{a\,x+1}-1}{4\,a\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}+\frac {3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a\,c^2}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12\,a\,c^2}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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