Optimal. Leaf size=50 \[ \frac {x \coth ^2(c+d x)}{\sqrt {b \coth ^4(c+d x)}}-\frac {\coth (c+d x)}{d \sqrt {b \coth ^4(c+d x)}} \]
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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 = {3658, 3473, 8} \[ \frac {x \coth ^2(c+d x)}{\sqrt {b \coth ^4(c+d x)}}-\frac {\coth (c+d x)}{d \sqrt {b \coth ^4(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \coth ^4(c+d x)}} \, dx &=\frac {\coth ^2(c+d x) \int \tanh ^2(c+d x) \, dx}{\sqrt {b \coth ^4(c+d x)}}\\ &=-\frac {\coth (c+d x)}{d \sqrt {b \coth ^4(c+d x)}}+\frac {\coth ^2(c+d x) \int 1 \, dx}{\sqrt {b \coth ^4(c+d x)}}\\ &=-\frac {\coth (c+d x)}{d \sqrt {b \coth ^4(c+d x)}}+\frac {x \coth ^2(c+d x)}{\sqrt {b \coth ^4(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 40, normalized size = 0.80 \[ \frac {\coth (c+d x) \left (\tanh ^{-1}(\tanh (c+d x)) \coth (c+d x)-1\right )}{d \sqrt {b \coth ^4(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.30, size = 422, normalized size = 8.44 \[ \frac {{\left (d x \cosh \left (d x + c\right )^{2} + {\left (d x e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x e^{\left (2 \, d x + 2 \, c\right )} + d x\right )} \sinh \left (d x + c\right )^{2} + d x + {\left (d x \cosh \left (d x + c\right )^{2} + d x + 2\right )} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, {\left (d x \cosh \left (d x + c\right )^{2} + d x + 2\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d x \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 2\right )} \sqrt {\frac {b e^{\left (8 \, d x + 8 \, c\right )} + 4 \, b e^{\left (6 \, d x + 6 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (8 \, d x + 8 \, c\right )} - 4 \, e^{\left (6 \, d x + 6 \, c\right )} + 6 \, e^{\left (4 \, d x + 4 \, c\right )} - 4 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{b d \cosh \left (d x + c\right )^{2} + {\left (b d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b d e^{\left (2 \, d x + 2 \, c\right )} + b d\right )} \sinh \left (d x + c\right )^{2} + b d + {\left (b d \cosh \left (d x + c\right )^{2} + b d\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (b d \cosh \left (d x + c\right )^{2} + b d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (b d \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + b d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 32, normalized size = 0.64 \[ \frac {\frac {d x + c}{\sqrt {b}} + \frac {2}{\sqrt {b} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 59, normalized size = 1.18 \[ -\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )-1\right ) \coth \left (d x +c \right )-\ln \left (\coth \left (d x +c \right )+1\right ) \coth \left (d x +c \right )+2\right )}{2 d \sqrt {b \left (\coth ^{4}\left (d x +c \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 36, normalized size = 0.72 \[ \frac {d x + c}{\sqrt {b} d} - \frac {2 \, \sqrt {b}}{{\left (b e^{\left (-2 \, d x - 2 \, c\right )} + b\right )} d} \]
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 \[ \int \frac {1}{\sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^4}} \,d x \]
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 \[ \int \frac {1}{\sqrt {b \coth ^{4}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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