Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]
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Rubi [A] time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3176, 3205, 63, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \frac {\coth (e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx &=\int \frac {\coth (e+f x)}{\sqrt {a \cosh ^2(e+f x)}} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cosh ^2(e+f x)}\right )}{a f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 1.58 \[ \frac {\cosh (e+f x) \left (\log \left (\sinh \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cosh \left (\frac {1}{2} (e+f x)\right )\right )\right )}{f \sqrt {a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 174, normalized size = 5.61 \[ \left [\frac {\sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} \log \left (\frac {\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1}{\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1}\right )}{a f e^{\left (2 \, f x + 2 \, e\right )} + a f}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} \sqrt {-a}}{a \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a \cosh \left (f x + e\right ) + {\left (a e^{\left (2 \, f x + 2 \, e\right )} + a\right )} \sinh \left (f x + e\right )}\right )}{a f}\right ] \]
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 [C] time = 0.16, size = 33, normalized size = 1.06 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {1}{\sinh \left (f x +e \right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 40, normalized size = 1.29 \[ -\frac {\log \left (e^{\left (-f x - e\right )} + 1\right )}{\sqrt {a} f} + \frac {\log \left (e^{\left (-f x - e\right )} - 1\right )}{\sqrt {a} f} \]
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 \[ \int \frac {\mathrm {coth}\left (e+f\,x\right )}{\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\coth {\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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