Optimal. Leaf size=69 \[ \frac {3 \coth ^{1-m}(c+d x) \, _2F_1\left (1,\frac {1}{6} (3-4 m);\frac {1}{6} (9-4 m);\coth ^2(c+d x)\right )}{b d (3-4 m) \sqrt [3]{b \coth ^m(c+d x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 69, 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 = {3659, 3476, 364} \[ \frac {3 \coth ^{1-m}(c+d x) \, _2F_1\left (1,\frac {1}{6} (3-4 m);\frac {1}{6} (9-4 m);\coth ^2(c+d x)\right )}{b d (3-4 m) \sqrt [3]{b \coth ^m(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{4/3}} \, dx &=\frac {\coth ^{\frac {m}{3}}(c+d x) \int \coth ^{-\frac {4 m}{3}}(c+d x) \, dx}{b \sqrt [3]{b \coth ^m(c+d x)}}\\ &=-\frac {\coth ^{\frac {m}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {x^{-4 m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{b d \sqrt [3]{b \coth ^m(c+d x)}}\\ &=\frac {3 \coth ^{1-m}(c+d x) \, _2F_1\left (1,\frac {1}{6} (3-4 m);\frac {1}{6} (9-4 m);\coth ^2(c+d x)\right )}{b d (3-4 m) \sqrt [3]{b \coth ^m(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 60, normalized size = 0.87 \[ -\frac {3 \coth (c+d x) \, _2F_1\left (1,\frac {1}{6} (3-4 m);\frac {1}{6} (9-4 m);\coth ^2(c+d x)\right )}{d (4 m-3) \left (b \coth ^m(c+d x)\right )^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \left (\coth ^{m}\left (d x +c \right )\right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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