Optimal. Leaf size=63 \[ \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Rubi [A]
time = 0.09, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6312, 864, 827,
858, 222, 272, 65, 214} \begin {gather*} \frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}{a}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {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 827
Rule 858
Rule 864
Rule 6312
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=-\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac {c x}{a}\right )} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\left (c^3 \text {Subst}\left (\int \frac {\left (\frac {1}{c}+\frac {x}{a c}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {1}{2} c^3 \text {Subst}\left (\int \frac {-\frac {2}{a c}+\frac {2 x}{a^2 c}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {c^2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}-\frac {c^2 \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}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}-\frac {c^2 \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}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}+\left (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}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(63)=126\).
time = 0.10, size = 154, normalized size = 2.44 \begin {gather*} -\frac {c^2 \left (-1+a x+a^2 x^2-a^3 x^3+4 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \text {ArcSin}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \text {ArcSin}\left (\frac {1}{a x}\right )-a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^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. \(173\) vs.
\(2(59)=118\).
time = 0.06, size = 174, normalized size = 2.76
method | result | size |
risch | \(-\frac {\left (a x -1\right ) c^{2}}{x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (a \sqrt {\left (a x +1\right ) \left (a x -1\right )}+\frac {a^{2} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c^{2} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{2} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(137\) |
default | \(\frac {\left (a x -1\right )^{2} c^{2} \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +\sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x \right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{2} x \sqrt {a^{2}}}\) | \(174\) |
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. 125 vs.
\(2 (59) = 118\).
time = 0.47, size = 125, normalized size = 1.98 \begin {gather*} -{\left (\frac {4 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{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 = 114, normalized size = 1.81 \begin {gather*} -\frac {2 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c^{2} x^{2} - 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 {2 a}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{2}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs.
\(2 (59) = 118\).
time = 0.42, size = 138, normalized size = 2.19 \begin {gather*} -\frac {2 \, c^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{2}}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {2 \, c^{2}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 90, normalized size = 1.43 \begin {gather*} \frac {4\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {2\,c^2\,\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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