Optimal. Leaf size=50 \[ \frac{\sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{f} \]
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Rubi [A] time = 0.0977005, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3176, 3205, 50, 63, 206} \[ \frac{\sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \coth (e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt{a \cosh ^2(e+f x)} \coth (e+f x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a x}}{1-x} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{\sqrt{a \cosh ^2(e+f x)}}{f}-\frac{a \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{\sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a \cosh ^2(e+f x)}\right )}{f}\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{f}+\frac{\sqrt{a \cosh ^2(e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.0588644, size = 42, normalized size = 0.84 \[ \frac{\text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \left (\cosh (e+f x)+\log \left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.089, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{\sinh \left ( fx+e \right ) }{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.781, size = 92, normalized size = 1.84 \begin{align*} \frac{{\left (\sqrt{a} e^{\left (-2 \, f x - 2 \, e\right )} + \sqrt{a}\right )} e^{\left (f x + e\right )}}{2 \, f} - \frac{\sqrt{a} \log \left (e^{\left (-f x - e\right )} + 1\right )}{f} + \frac{\sqrt{a} \log \left (e^{\left (-f x - e\right )} - 1\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8532, size = 549, normalized size = 10.98 \begin{align*} \frac{{\left (2 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} +{\left (\cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} + 2 \,{\left (\cosh \left (f x + e\right ) e^{\left (f x + e\right )} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )\right )} \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 )\right )} \sqrt{a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{2 \,{\left (f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + f \cosh \left (f x + e\right ) +{\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \coth{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20647, size = 69, normalized size = 1.38 \begin{align*} \frac{\sqrt{a}{\left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )} - 2 \, \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, \log \left ({\left | e^{\left (f x + e\right )} - 1 \right |}\right )\right )}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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