Optimal. Leaf size=78 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2}{b d \sqrt {b \coth (c+d x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3555, 3557,
335, 304, 209, 212} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2}{b d \sqrt {b \coth (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 3555
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{(b \coth (c+d x))^{3/2}} \, dx &=-\frac {2}{b d \sqrt {b \coth (c+d x)}}+\frac {\int \sqrt {b \coth (c+d x)} \, dx}{b^2}\\ &=-\frac {2}{b d \sqrt {b \coth (c+d x)}}-\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{b d}\\ &=-\frac {2}{b d \sqrt {b \coth (c+d x)}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{-b^2+x^4} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{b d}\\ &=-\frac {2}{b d \sqrt {b \coth (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{b d}-\frac {\text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{b d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2}{b d \sqrt {b \coth (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 36, normalized size = 0.46 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};\coth ^2(c+d x)\right )}{b d \sqrt {b \coth (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.10, size = 62, normalized size = 0.79
method | result | size |
derivativedivides | \(-\frac {2 b \left (\frac {1}{b^{2} \sqrt {b \coth \left (d x +c \right )}}-\frac {\arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {5}{2}}}+\frac {\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {5}{2}}}\right )}{d}\) | \(62\) |
default | \(-\frac {2 b \left (\frac {1}{b^{2} \sqrt {b \coth \left (d x +c \right )}}-\frac {\arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {5}{2}}}+\frac {\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {5}{2}}}\right )}{d}\) | \(62\) |
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 \begin {gather*} \text {Failed to integrate} \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. 437 vs.
\(2 (64) = 128\).
time = 0.59, size = 923, normalized size = 11.83 \begin {gather*} \left [-\frac {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} \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 ) + {\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} \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 ) + 8 \, {\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 {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{4 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2} + b^{2} d\right )}}, -\frac {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} \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 ) - {\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} \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 ) + 8 \, {\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 {\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}}}{4 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2} + b^{2} d\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 \frac {1}{\left (b \coth {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs.
\(2 (64) = 128\).
time = 0.48, size = 174, normalized size = 2.23 \begin {gather*} \frac {\frac {2 \, \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} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} - \frac {\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 )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} - \frac {8}{{\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} + \sqrt {b}\right )} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}{2 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 64, normalized size = 0.82 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{b^{3/2}\,d}-\frac {\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{b^{3/2}\,d}-\frac {2}{b\,d\,\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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