Optimal. Leaf size=60 \[ \frac{2 \coth (c+d x) \, _2F_1\left (1,\frac{2-m}{4};\frac{6-m}{4};\coth ^2(c+d x)\right )}{d (2-m) \sqrt{b \coth ^m(c+d x)}} \]
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Rubi [A] time = 0.047278, antiderivative size = 60, normalized size of antiderivative = 1., 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 = {3659, 3476, 364} \[ \frac{2 \coth (c+d x) \, _2F_1\left (1,\frac{2-m}{4};\frac{6-m}{4};\coth ^2(c+d x)\right )}{d (2-m) \sqrt{b \coth ^m(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \coth ^m(c+d x)}} \, dx &=\frac{\coth ^{\frac{m}{2}}(c+d x) \int \coth ^{-\frac{m}{2}}(c+d x) \, dx}{\sqrt{b \coth ^m(c+d x)}}\\ &=-\frac{\coth ^{\frac{m}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{x^{-m/2}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \sqrt{b \coth ^m(c+d x)}}\\ &=\frac{2 \coth (c+d x) \, _2F_1\left (1,\frac{2-m}{4};\frac{6-m}{4};\coth ^2(c+d x)\right )}{d (2-m) \sqrt{b \coth ^m(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0473323, size = 58, normalized size = 0.97 \[ -\frac{2 \coth (c+d x) \, _2F_1\left (1,\frac{2-m}{4};\frac{6-m}{4};\coth ^2(c+d x)\right )}{d (m-2) \sqrt{b \coth ^m(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.195, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{b \left ({\rm coth} \left (dx+c\right ) \right ) ^{m}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \coth \left (d x + c\right )^{m}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \coth ^{m}{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \coth \left (d x + c\right )^{m}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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