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 = 0.04, antiderivative size = 60, 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) \, _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 {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3740
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) \text {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*}
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Mathematica [A]
time = 0.04, size = 58, normalized size = 0.97 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 3.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {b \left (\coth ^{m}\left (d x +c \right )\right )}}\, dx\]
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 \frac {1}{\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 \frac {1}{\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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