Optimal. Leaf size=50 \[ -\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a \cosh ^2(e+f x)}}{f} \]
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Rubi [A]
time = 0.07, 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 = {3255, 3284, 52,
65, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 3255
Rule 3284
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 {\text {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 \text {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 {\text {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 = 0.05, size = 42, normalized size = 0.84 \begin {gather*} \frac {\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 ) \text {sech}(e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.99, size = 42, normalized size = 0.84
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {a \left (\cosh ^{2}\left (f x +e \right )\right )}{\sinh \left (f x +e \right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(42\) |
risch | \(\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}+\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {\ln \left ({\mathrm e}^{f x}+{\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{f \left ({\mathrm e}^{2 f x +2 e}+1\right )}+\frac {\ln \left ({\mathrm e}^{f x}-{\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{f \left ({\mathrm e}^{2 f x +2 e}+1\right )}\) | \(220\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 72, normalized size = 1.44 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 200 vs.
\(2 (42) = 84\).
time = 0.49, size = 200, normalized size = 4.00 \begin {gather*} \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 {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*} \int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \coth {\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 47, normalized size = 0.94 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \mathrm {coth}\left (e+f\,x\right )\,\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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