Optimal. Leaf size=50 \[ x \tanh ^2(c+d x) \sqrt {b \coth ^4(c+d x)}-\frac {\tanh (c+d x) \sqrt {b \coth ^4(c+d x)}}{d} \]
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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 = {3658, 3473, 8} \[ x \tanh ^2(c+d x) \sqrt {b \coth ^4(c+d x)}-\frac {\tanh (c+d x) \sqrt {b \coth ^4(c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
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] time = 0.03, size = 41, normalized size = 0.82 \[ -\frac {\tanh (c+d x) \sqrt {b \coth ^4(c+d x)} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.10, size = 415, normalized size = 8.30 \[ \frac {{\left (d x \cosh \left (d x + c\right )^{2} + {\left (d x e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x e^{\left (2 \, d x + 2 \, c\right )} + d x\right )} \sinh \left (d x + c\right )^{2} - d x + {\left (d x \cosh \left (d x + c\right )^{2} - d x - 2\right )} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, {\left (d x \cosh \left (d x + c\right )^{2} - d x - 2\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d x \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 2\right )} \sqrt {\frac {b e^{\left (8 \, d x + 8 \, c\right )} + 4 \, b e^{\left (6 \, d x + 6 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (8 \, d x + 8 \, c\right )} - 4 \, e^{\left (6 \, d x + 6 \, c\right )} + 6 \, e^{\left (4 \, d x + 4 \, c\right )} - 4 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{d \cosh \left (d x + c\right )^{2} + {\left (d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d e^{\left (2 \, d x + 2 \, c\right )} + d\right )} \sinh \left (d x + c\right )^{2} + {\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 27, normalized size = 0.54 \[ \frac {{\left (d x + c - \frac {2}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )} \sqrt {b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 55, normalized size = 1.10 \[ -\frac {\sqrt {b \left (\coth ^{4}\left (d x +c \right )\right )}\, \left (2 \coth \left (d x +c \right )+\ln \left (\coth \left (d x +c \right )-1\right )-\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 34, normalized size = 0.68 \[ \frac {{\left (d x + c\right )} \sqrt {b}}{d} + \frac {2 \, \sqrt {b}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
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 \[ \int \sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^4} \,d x \]
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 \[ \int \sqrt {b \coth ^{4}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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