Optimal. Leaf size=132 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d} \]
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Rubi [A]
time = 0.07, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3557, 335, 281,
206, 31, 648, 631, 210, 642} \begin {gather*} \frac {\sqrt {3} \text {ArcTan}\left (\frac {b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{2/3} (b \coth (c+d x))^{2/3}+b^{4/3}+(b \coth (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 281
Rule 335
Rule 631
Rule 642
Rule 648
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx &=-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (-b^2+x^2\right )} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {(3 b) \text {Subst}\left (\int \frac {x}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}\\ &=-\frac {(3 b) \text {Subst}\left (\int \frac {1}{-b^2+x^3} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{-b^{2/3}+x} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}-\frac {\text {Subst}\left (\int \frac {-2 b^{2/3}-x}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}\\ &=-\frac {\log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac {\text {Subst}\left (\int \frac {b^{2/3}+2 x}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{4 \sqrt [3]{b} d}+\frac {\left (3 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{4 d}\\ &=-\frac {\log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 (b \coth (c+d x))^{2/3}}{b^{2/3}}\right )}{2 \sqrt [3]{b} d}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (b \coth (c+d x))^{2/3}}{b^{2/3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 98, normalized size = 0.74 \begin {gather*} \frac {\sqrt [3]{\coth (c+d x)} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {1+2 \coth ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )-2 \log \left (1-\coth ^{\frac {2}{3}}(c+d x)\right )+\log \left (1+\coth ^{\frac {2}{3}}(c+d x)+\coth ^{\frac {4}{3}}(c+d x)\right )\right )}{4 d \sqrt [3]{b \coth (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.23, size = 109, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {4}{3}}+\left (b^{2}\right )^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}\right )}{d}\) | \(109\) |
default | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {4}{3}}+\left (b^{2}\right )^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}\right )}{d}\) | \(109\) |
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. 742 vs.
\(2 (99) = 198\).
time = 0.55, size = 1598, normalized size = 12.11 \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}{\sqrt [3]{b \coth {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs.
\(2 (99) = 198\).
time = 0.49, size = 216, normalized size = 1.64 \begin {gather*} \frac {b {\left (\frac {2 \, \sqrt {3} {\left | b \right |}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}}{3 \, {\left | b \right |}^{\frac {2}{3}}}\right )}{b^{2}} + \frac {{\left | b \right |}^{\frac {2}{3}} \log \left (\left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} {\left | b \right |}^{\frac {2}{3}} + {\left | b \right |}^{\frac {4}{3}} + \frac {{\left (b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )}{b^{2}} - \frac {2 \, {\left | b \right |}^{\frac {2}{3}} \log \left ({\left | \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} - {\left | b \right |}^{\frac {2}{3}} \right |}\right )}{b^{2}}\right )}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.65, size = 147, normalized size = 1.11 \begin {gather*} \frac {\ln \left (162\,{\left (-b\right )}^{11/3}+162\,b^3\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}\right )}{2\,{\left (-b\right )}^{1/3}\,d}+\frac {\ln \left (\frac {81\,{\left (-b\right )}^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{d^3}+\frac {162\,b^3\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,{\left (-b\right )}^{1/3}\,d}-\frac {\ln \left (\frac {81\,{\left (-b\right )}^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{d^3}-\frac {162\,b^3\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,{\left (-b\right )}^{1/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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