Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3205, 63, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 3205
Rubi steps
\begin {align*} \int \frac {\coth (c+d x)}{\sqrt {a \cosh ^2(c+d x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cosh ^2(c+d x)}\right )}{a d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 1.58 \[ \frac {\cosh (c+d x) \left (\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d \sqrt {a \cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 174, normalized size = 5.61 \[ \left [\frac {\sqrt {a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} \log \left (\frac {\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1}\right )}{a d e^{\left (2 \, d x + 2 \, c\right )} + a d}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} \sqrt {-a}}{a \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + a \cosh \left (d x + c\right ) + {\left (a e^{\left (2 \, d x + 2 \, c\right )} + a\right )} \sinh \left (d x + c\right )}\right )}{a d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 31, normalized size = 1.00 \[ -\frac {\cosh \left (d x +c \right ) \arctanh \left (\cosh \left (d x +c \right )\right )}{\sqrt {a \left (\cosh ^{2}\left (d x +c \right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 40, normalized size = 1.29 \[ -\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{\sqrt {a} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{\sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {coth}\left (c+d\,x\right )}{\sqrt {a\,{\mathrm {cosh}\left (c+d\,x\right )}^2}} \,d x \]
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 \[ \int \frac {\coth {\left (c + d x \right )}}{\sqrt {a \cosh ^{2}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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