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 = 0.10, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 206
Rule 3176
Rule 3205
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 2.39, size = 200, normalized size = 4.00 \[ \frac {{\left (2 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + {\left (\cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} + 2 \, {\left (\cosh \left (f x + e\right ) e^{\left (f x + e\right )} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )\right )} \log \left (\frac {\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1}{\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1}\right )\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{2 \, {\left (f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + f \cosh \left (f x + e\right ) + {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 51, normalized size = 1.02 \[ \frac {\sqrt {a} {\left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )} - 2 \, \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, \log \left ({\left | e^{\left (f x + e\right )} - 1 \right |}\right )\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 42, normalized size = 0.84 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {a \left (\cosh ^{2}\left (f x +e \right )\right )}{\sinh \left (f x +e \right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.34, size = 68, normalized size = 1.36 \[ \frac {{\left (\sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + \sqrt {a}\right )} e^{\left (f x + e\right )}}{2 \, f} - \frac {\sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right )}{f} + \frac {\sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {coth}\left (e+f\,x\right )\,\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]
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 \[ \int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \coth {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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