Optimal. Leaf size=106 \[ -\frac {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {13 c^3 \csc ^{-1}(a x)}{2 a}-\frac {4 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Rubi [A]
time = 0.22, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6312, 1821,
1823, 858, 222, 272, 65, 214} \begin {gather*} c^3 x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}-\frac {4 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}-\frac {13 c^3 \csc ^{-1}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1821
Rule 1823
Rule 6312
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^4}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\text {Subst}\left (\int \frac {\frac {4 c^4}{a}-\frac {6 c^4 x}{a^2}+\frac {4 c^4 x^2}{a^3}-\frac {c^4 x^3}{a^4}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {a^2 \text {Subst}\left (\int \frac {-\frac {8 c^4}{a^3}+\frac {13 c^4 x}{a^4}-\frac {8 c^4 x^2}{a^5}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {a^4 \text {Subst}\left (\int \frac {\frac {8 c^4}{a^5}-\frac {13 c^4 x}{a^6}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\left (13 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}+\frac {\left (4 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {13 c^3 \csc ^{-1}(a x)}{2 a}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=-\frac {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {13 c^3 \csc ^{-1}(a x)}{2 a}-\left (4 a c^3\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 {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {13 c^3 \csc ^{-1}(a x)}{2 a}-\frac {4 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 167, normalized size = 1.58 \begin {gather*} \frac {c^3 \left (-1+8 a x-a^2 x^2-8 a^3 x^3+2 a^4 x^4+10 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \text {ArcSin}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-8 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \text {ArcSin}\left (\frac {1}{a x}\right )-8 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{2 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3} \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. \(265\) vs.
\(2(94)=188\).
time = 0.10, size = 266, normalized size = 2.51
method | result | size |
risch | \(-\frac {\left (a x +1\right ) \left (8 a x -1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{2 x^{2} a^{3}}+\frac {\left (a^{2} \sqrt {\left (a x +1\right ) \left (a x -1\right )}-\frac {4 a^{3} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}-\frac {13 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{3} \left (a x -1\right )}\) | \(149\) |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{3} \left (-8 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+16 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-16 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+8 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -13 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-13 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}+8 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{2 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} \sqrt {a^{2}}\, x^{2}}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs.
\(2 (94) = 188\).
time = 0.46, size = 201, normalized size = 1.90 \begin {gather*} {\left (\frac {13 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {4 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {4 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {11 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 5 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 143, normalized size = 1.35 \begin {gather*} \frac {26 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 8 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 8 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{3} c^{3} x^{3} - 6 \, a^{2} c^{3} x^{2} - 7 \, a c^{3} x + c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \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^{3} \left (\int a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{3}}\right )\, dx + \int \frac {3 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\, dx + \int \left (- \frac {3 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x}\right )\, dx\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs.
\(2 (94) = 188\).
time = 0.42, size = 232, normalized size = 2.19 \begin {gather*} \frac {13 \, c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {4 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 8 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 8 \, a c^{3} \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2} a {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 163, normalized size = 1.54 \begin {gather*} \frac {2\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}+11\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a+\frac {a\,\left (a\,x-1\right )}{a\,x+1}-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {13\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {8\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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