Optimal. Leaf size=291 \[ -\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac{2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\sqrt{3} \left (b \coth ^4(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt{3}}\right )}{2 d \coth ^{\frac{8}{3}}(c+d x)}-\frac{\sqrt{3} \left (b \coth ^4(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt{3}}\right )}{2 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac{8}{3}}(c+d x)}-\frac{3 \tanh (c+d x) \left (b \coth ^4(c+d x)\right )^{2/3}}{5 d} \]
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Rubi [A] time = 0.215087, antiderivative size = 291, normalized size of antiderivative = 1., 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, 3473, 3476, 329, 296, 634, 618, 204, 628, 206} \[ -\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac{2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\sqrt{3} \left (b \coth ^4(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt{3}}\right )}{2 d \coth ^{\frac{8}{3}}(c+d x)}-\frac{\sqrt{3} \left (b \coth ^4(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt{3}}\right )}{2 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac{8}{3}}(c+d x)}-\frac{3 \tanh (c+d x) \left (b \coth ^4(c+d x)\right )^{2/3}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3476
Rule 329
Rule 296
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx &=\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \int \coth ^{\frac{8}{3}}(c+d x) \, dx}{\coth ^{\frac{8}{3}}(c+d x)}\\ &=-\frac{3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \int \coth ^{\frac{2}{3}}(c+d x) \, dx}{\coth ^{\frac{8}{3}}(c+d x)}\\ &=-\frac{3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}-\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \operatorname{Subst}\left (\int \frac{x^{2/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac{8}{3}}(c+d x)}\\ &=-\frac{3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}-\frac{\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{x^4}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac{8}{3}}(c+d x)}\\ &=-\frac{3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac{8}{3}}(c+d x)}\\ &=\frac{\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac{8}{3}}(c+d x)}-\frac{3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}-\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}-\frac{\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}-\frac{\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}\\ &=\frac{\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac{8}{3}}(c+d x)}-\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac{2}{3}}(c+d x)\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac{2}{3}}(c+d x)\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}-\frac{3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}+\frac{\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac{8}{3}}(c+d x)}\\ &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt{3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac{8}{3}}(c+d x)}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt{3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac{8}{3}}(c+d x)}-\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac{2}{3}}(c+d x)\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}+\frac{\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac{2}{3}}(c+d x)\right )}{4 d \coth ^{\frac{8}{3}}(c+d x)}-\frac{3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.408708, size = 166, normalized size = 0.57 \[ \frac{\left (b \coth ^4(c+d x)\right )^{2/3} \left (-12 \coth ^{\frac{5}{3}}(c+d x)+20 \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )+5 \left (-\log \left (\coth ^{\frac{2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )+\log \left (\coth ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt{3}}\right )\right )\right )}{20 d \coth ^{\frac{8}{3}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.094, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ({\rm coth} \left (dx+c\right ) \right ) ^{4} \right ) ^{{\frac{2}{3}}}\, 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 \left (b \coth \left (d x + c\right )^{4}\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26442, size = 1783, normalized size = 6.13 \begin{align*} \text{result too large to display} \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 \left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac{2}{3}}\, 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 \left (b \coth \left (d x + c\right )^{4}\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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