Optimal. Leaf size=264 \[ -\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt {3} \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt {3} \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)} \]
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Rubi [A] time = 0.21, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3658, 3476, 329, 296, 634, 618, 204, 628, 206} \[ -\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt {3} \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt {3} \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 296
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \sqrt [3]{b \coth ^2(c+d x)} \, dx &=\frac {\sqrt [3]{b \coth ^2(c+d x)} \int \coth ^{\frac {2}{3}}(c+d x) \, dx}{\coth ^{\frac {2}{3}}(c+d x)}\\ &=-\frac {\sqrt [3]{b \coth ^2(c+d x)} \operatorname {Subst}\left (\int \frac {x^{2/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac {2}{3}}(c+d x)}\\ &=-\frac {\left (3 \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^4}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)}\\ &=\frac {\sqrt [3]{b \coth ^2(c+d x)} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \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 {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \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 {2}{3}}(c+d x)}\\ &=\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt [3]{b \coth ^2(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 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(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 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\left (3 \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\left (3 \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}\\ &=\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\left (3 \sqrt [3]{b \coth ^2(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 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\left (3 \sqrt [3]{b \coth ^2(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 d \coth ^{\frac {2}{3}}(c+d x)}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{2 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{2 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt [3]{b \coth ^2(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 151, normalized size = 0.57 \[ \frac {\sqrt [3]{b \coth ^2(c+d x)} \left (-\log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )+\log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )+4 \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 1618, normalized size = 6.13 \[ \text {result too large to display} \]
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 \[ \int \left (b \coth \left (d x + c\right )^{2}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {1}{3}}\, dx \]
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 \[ \int \left (b \coth \left (d x + c\right )^{2}\right )^{\frac {1}{3}}\,{d x} \]
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 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{1/3} \,d x \]
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 \[ \int \sqrt [3]{b \coth ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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