Optimal. Leaf size=291 \[ -\frac {3 \tanh (c+d x) \left (b \coth ^4(c+d x)\right )^{2/3}}{5 d}-\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\sqrt {3} \left (b \coth ^4(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\sqrt {3} \left (b \coth ^4(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {8}{3}}(c+d x)} \]
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Rubi [A] time = 0.22, antiderivative size = 291, 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, 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 204
Rule 206
Rule 296
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3473
Rule 3476
Rule 3658
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*}
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Mathematica [A] time = 0.42, 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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fricas [B] time = 1.79, size = 618, normalized size = 2.12 \[ -\frac {10 \, {\left (\sqrt {3} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} \sinh \left (d x + c\right )^{2} - \sqrt {3}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 10 \, {\left (\sqrt {3} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} \sinh \left (d x + c\right )^{2} - \sqrt {3}\right )} {\left (b^{2}\right )}^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 5 \, \left (-b^{2}\right )^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + 5 \, {\left (b^{2}\right )}^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 10 \, \left (-b^{2}\right )^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) - 10 \, {\left (b^{2}\right )}^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right ) + 12 \, {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1\right )} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}}{20 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} - d\right )}} \]
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 \[ \int \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \left (b \left (\coth ^{4}\left (d x +c \right )\right )\right )^{\frac {2}{3}}\, dx \]
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 \[ \int \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {2}{3}}\,{d x} \]
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 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^{2/3} \,d x \]
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 \[ \int \left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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