Optimal. Leaf size=309 \[ -\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\sqrt {3} \coth ^{\frac {2}{3}}(c+d x) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\sqrt {3} \coth ^{\frac {2}{3}}(c+d x) \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^2(c+d x)}} \]
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Rubi [A] time = 0.18, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3658, 3474, 3476, 329, 210, 634, 618, 204, 628, 206} \[ -\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\sqrt {3} \coth ^{\frac {2}{3}}(c+d x) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\sqrt {3} \coth ^{\frac {2}{3}}(c+d x) \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 210
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3474
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{4/3}} \, dx &=\frac {\coth ^{\frac {2}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {8}{3}}(c+d x)} \, dx}{b \sqrt [3]{b \coth ^2(c+d x)}}\\ &=-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)} \, dx}{b \sqrt [3]{b \coth ^2(c+d x)}}\\ &=-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {1}{x^{2/3} \left (-1+x^2\right )} \, dx,x,\coth (c+d x)\right )}{b d \sqrt [3]{b \coth ^2(c+d x)}}\\ &=-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\left (3 \coth ^{\frac {2}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^2(c+d x)}}\\ &=-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^2(c+d x)}}\\ &=\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {2}{3}}(c+d x)}{b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\left (3 \coth ^{\frac {2}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\left (3 \coth ^{\frac {2}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}\\ &=\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {2}{3}}(c+d x)}{b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\left (3 \coth ^{\frac {2}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\left (3 \coth ^{\frac {2}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {2}{3}}(c+d x)}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {2}{3}}(c+d x)}{2 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {2}{3}}(c+d x)}{b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {3 \tanh (c+d x)}{5 b d \sqrt [3]{b \coth ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 43, normalized size = 0.14 \[ -\frac {3 \coth (c+d x) \, _2F_1\left (-\frac {5}{6},1;\frac {1}{6};\coth ^2(c+d x)\right )}{5 d \left (b \coth ^2(c+d x)\right )^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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