Optimal. Leaf size=58 \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3476, 329, 298, 203, 206} \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 3476
Rubi steps
\begin {align*} \int \sqrt {b \coth (c+d x)} \, dx &=-\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^2}{-b^2+x^4} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}\\ &=-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.88 \[ \frac {\sqrt {b \coth (c+d x)} \left (\tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right )-\tan ^{-1}\left (\sqrt {\coth (c+d x)}\right )\right )}{d \sqrt {\coth (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 594, normalized size = 10.24 \[ \left [-\frac {2 \, \sqrt {-b} \arctan \left (\frac {{\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b}\right ) - \sqrt {-b} \log \left (-\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \sqrt {-b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}} - 2 \, b}{\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4}}\right )}{4 \, d}, -\frac {2 \, \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b}\right ) - \sqrt {b} \log \left (2 \, b \cosh \left (d x + c\right )^{4} + 8 \, b \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 12 \, b \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 8 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, b \sinh \left (d x + c\right )^{4} + 2 \, {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4} + {\left (6 \, \cosh \left (d x + c\right )^{2} - 1\right )} \sinh \left (d x + c\right )^{2} - \cosh \left (d x + c\right )^{2} + 2 \, {\left (2 \, \cosh \left (d x + c\right )^{3} - \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b} \sqrt {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}} - b\right )}{4 \, d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 101, normalized size = 1.74 \[ \frac {{\left (2 \, \sqrt {b} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right ) - \sqrt {b} \log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 47, normalized size = 0.81 \[ -\frac {\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {b}}{d}+\frac {\arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \coth \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 41, normalized size = 0.71 \[ -\frac {\sqrt {b}\,\left (\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )-\mathrm {atanh}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )\right )}{d} \]
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 \[ \int \sqrt {b \coth {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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