Optimal. Leaf size=54 \[ \frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {2+m}{4};\frac {6+m}{4};\coth ^2(c+d x)\right )}{d (2+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, 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 = {3740, 3557,
371} \begin {gather*} \frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {m+2}{4};\frac {m+6}{4};\coth ^2(c+d x)\right )}{d (m+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int \sqrt {b \coth ^m(c+d x)} \, dx &=\left (\coth ^{-\frac {m}{2}}(c+d x) \sqrt {b \coth ^m(c+d x)}\right ) \int \coth ^{\frac {m}{2}}(c+d x) \, dx\\ &=-\frac {\left (\coth ^{-\frac {m}{2}}(c+d x) \sqrt {b \coth ^m(c+d x)}\right ) \text {Subst}\left (\int \frac {x^{m/2}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {2+m}{4};\frac {6+m}{4};\coth ^2(c+d x)\right )}{d (2+m)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {2+m}{4};\frac {6+m}{4};\coth ^2(c+d x)\right )}{d (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.55, size = 0, normalized size = 0.00 \[\int \sqrt {b \left (\coth ^{m}\left (d x +c \right )\right )}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 ^{m}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 )}^m} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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