Optimal. Leaf size=92 \[ \frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+4 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {15 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
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Rubi [A]
time = 0.16, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6310, 6313,
1819, 1821, 821, 272, 65, 214} \begin {gather*} -\frac {1}{2} a c x^2 \sqrt {1-\frac {1}{a^2 x^2}}+4 c x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {15 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1819
Rule 1821
Rule 6310
Rule 6313
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx &=-\left ((a c) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x \, dx\right )\\ &=(a c) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4}{x^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-(a c) \text {Subst}\left (\int \frac {-1+\frac {4 x}{a}-\frac {7 x^2}{a^2}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{2} (a c) \text {Subst}\left (\int \frac {-\frac {8}{a}+\frac {15 x}{a^2}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+4 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {(15 c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+4 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {(15 c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a}\\ &=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+4 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {1}{2} (15 a c) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=\frac {8 c \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+4 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {15 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 68, normalized size = 0.74 \begin {gather*} \frac {1}{2} c \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (24+7 a x-a^2 x^2\right )}{1+a x}-\frac {15 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}\right ) \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. \(421\) vs.
\(2(80)=160\).
time = 0.10, size = 422, normalized size = 4.59
method | result | size |
risch | \(-\frac {\left (a x -8\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{2 a}-\frac {\left (\frac {15 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{2 \sqrt {a^{2}}}-\frac {8 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a x -1}\) | \(136\) |
default | \(\frac {\left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+16 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-16 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -8 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+32 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -32 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +16 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-16 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right ) c \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )}\) | \(422\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 156, normalized size = 1.70 \begin {gather*} \frac {1}{2} \, a {\left (\frac {2 \, {\left (9 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 7 \, c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\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}} - a^{2}} - \frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {16 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 81, normalized size = 0.88 \begin {gather*} -\frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c x^{2} - 7 \, a c x - 24 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a} \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*} - c \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \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 = 0.07, size = 117, normalized size = 1.27 \begin {gather*} \frac {7\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}-9\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {15\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {8\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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