Optimal. Leaf size=105 \[ \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {5 c^2 \csc ^{-1}(a x)}{a}-\frac {5 c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Rubi [A]
time = 0.22, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6312, 1819,
1821, 1823, 858, 222, 272, 65, 214} \begin {gather*} \frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {5 c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {5 c^2 \csc ^{-1}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1819
Rule 1821
Rule 1823
Rule 6312
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^5}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-c^5+\frac {5 c^5 x}{a}+\frac {5 c^5 x^2}{a^2}-\frac {c^5 x^3}{a^3}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\text {Subst}\left (\int \frac {-\frac {5 c^5}{a}-\frac {5 c^5 x}{a^2}+\frac {c^5 x^2}{a^3}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {a^2 \text {Subst}\left (\int \frac {\frac {5 c^5}{a^3}+\frac {5 c^5 x}{a^4}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {5 c^2 \csc ^{-1}(a x)}{a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {5 c^2 \csc ^{-1}(a x)}{a}-\left (5 a c^2\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {5 c^2 \csc ^{-1}(a x)}{a}-\frac {5 c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.26, size = 424, normalized size = 4.04 \begin {gather*} \frac {c^2 \left (-35 a^2 \sqrt {1+\frac {1}{a x}} x^2+315 a^3 \sqrt {1+\frac {1}{a x}} x^3+280 a^4 \sqrt {1+\frac {1}{a x}} x^4-595 a^5 \sqrt {1+\frac {1}{a x}} x^5+35 a^6 \sqrt {1+\frac {1}{a x}} x^6+910 a^4 \sqrt {1-\frac {1}{a x}} x^4 \text {ArcSin}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+910 a^5 \sqrt {1-\frac {1}{a x}} x^5 \text {ArcSin}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-105 a^4 \sqrt {1-\frac {1}{a x}} x^4 \text {ArcSin}\left (\frac {1}{a x}\right )-105 a^5 \sqrt {1-\frac {1}{a x}} x^5 \text {ArcSin}\left (\frac {1}{a x}\right )-175 a^5 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {1+\frac {1}{a x}} x^5 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+7 \sqrt {2} a x (-1+a x)^3 (1+a x) \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+5 \sqrt {2} (-1+a x)^4 (1+a x) \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )\right )}{35 a^5 \sqrt {1-\frac {1}{a x}} x^4 (1+a x)} \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. \(599\) vs.
\(2(97)=194\).
time = 0.12, size = 600, normalized size = 5.71
method | result | size |
risch | \(\frac {\left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{x \,a^{2}}+\frac {\left (a \sqrt {\left (a x +1\right ) \left (a x -1\right )}-\frac {5 a^{2} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {16 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}+5 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{2} \left (a x -1\right )}\) | \(169\) |
default | \(-\frac {\left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}-4 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-7 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-5 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+8 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -8 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+8 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -11 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-10 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-4 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-5 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -5 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right ) c^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a^{2} x \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )}\) | \(600\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 149, normalized size = 1.42 \begin {gather*} -{\left (\frac {4 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {10 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {5 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {5 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {16 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 120, normalized size = 1.14 \begin {gather*} -\frac {10 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 5 \, a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 5 \, a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c^{2} x^{2} + 18 \, a c^{2} x + c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \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 {c^{2} \left (\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\right )\, dx + \int \frac {3 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\, dx + \int \left (- \frac {3 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{3} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{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 = 0.10, size = 117, normalized size = 1.11 \begin {gather*} \frac {16\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}+\frac {4\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {10\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,10{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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