Optimal. Leaf size=50 \[ -\frac {\sqrt {b \coth ^4(c+d x)} \tanh (c+d x)}{d}+x \sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x) \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 8}
\begin {gather*} x \tanh ^2(c+d x) \sqrt {b \coth ^4(c+d x)}-\frac {\tanh (c+d x) \sqrt {b \coth ^4(c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
Rubi steps
\begin {align*} \int \sqrt {b \coth ^4(c+d x)} \, dx &=\left (\sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x)\right ) \int \coth ^2(c+d x) \, dx\\ &=-\frac {\sqrt {b \coth ^4(c+d x)} \tanh (c+d x)}{d}+\left (\sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x)\right ) \int 1 \, dx\\ &=-\frac {\sqrt {b \coth ^4(c+d x)} \tanh (c+d x)}{d}+x \sqrt {b \coth ^4(c+d x)} \tanh ^2(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 = 0.02, size = 41, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {b \coth ^4(c+d x)} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(c+d x)\right ) \tanh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.60, size = 55, normalized size = 1.10
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \left (\coth ^{4}\left (d x +c \right )\right )}\, \left (2 \coth \left (d x +c \right )+\ln \left (\coth \left (d x +c \right )-1\right )-\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )^{2}}\) | \(55\) |
default | \(-\frac {\sqrt {b \left (\coth ^{4}\left (d x +c \right )\right )}\, \left (2 \coth \left (d x +c \right )+\ln \left (\coth \left (d x +c \right )-1\right )-\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )^{2}}\) | \(55\) |
risch | \(\frac {\sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} x}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}-\frac {2 \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} d}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 34, normalized size = 0.68 \begin {gather*} \frac {{\left (d x + c\right )} \sqrt {b}}{d} + \frac {2 \, \sqrt {b}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \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. 415 vs.
\(2 (46) = 92\).
time = 0.35, size = 415, normalized size = 8.30 \begin {gather*} \frac {{\left (d x \cosh \left (d x + c\right )^{2} + {\left (d x e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x e^{\left (2 \, d x + 2 \, c\right )} + d x\right )} \sinh \left (d x + c\right )^{2} - d x + {\left (d x \cosh \left (d x + c\right )^{2} - d x - 2\right )} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, {\left (d x \cosh \left (d x + c\right )^{2} - d x - 2\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d x \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 2\right )} \sqrt {\frac {b e^{\left (8 \, d x + 8 \, c\right )} + 4 \, b e^{\left (6 \, d x + 6 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (8 \, d x + 8 \, c\right )} - 4 \, e^{\left (6 \, d x + 6 \, c\right )} + 6 \, e^{\left (4 \, d x + 4 \, c\right )} - 4 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{d \cosh \left (d x + c\right )^{2} + {\left (d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d e^{\left (2 \, d x + 2 \, c\right )} + d\right )} \sinh \left (d x + c\right )^{2} + {\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - d} \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 {b \coth ^{4}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 27, normalized size = 0.54 \begin {gather*} \frac {{\left (d x + c - \frac {2}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )} \sqrt {b}}{d} \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 \sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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