Optimal. Leaf size=289 \[ -\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}} \]
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Rubi [A]
time = 0.16, antiderivative size = 289, 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 = {3739, 3555,
3557, 335, 302, 648, 632, 210, 642, 212} \begin {gather*} \frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \text {ArcTan}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \text {ArcTan}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 302
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3555
Rule 3557
Rule 3739
Rubi steps
\begin {align*} \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx &=\frac {\coth ^{\frac {4}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {4}{3}}(c+d x)} \, dx}{\left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \int \coth ^{\frac {2}{3}}(c+d x) \, dx}{\left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {x^{2/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 41, normalized size = 0.14 \begin {gather*} -\frac {3 \coth (c+d x) \, _2F_1\left (-\frac {1}{6},1;\frac {5}{6};\coth ^2(c+d x)\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.83, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {2}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2066 vs.
\(2 (239) = 478\).
time = 0.42, size = 2066, normalized size = 7.15 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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