Optimal. Leaf size=132 \[ -\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{2/3} (b \coth (c+d x))^{2/3}+b^{4/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d} \]
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Rubi [A] time = 0.11, 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 = {3476, 329, 275, 292, 31, 634, 617, 204, 628} \[ -\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{2/3} (b \coth (c+d x))^{2/3}+b^{4/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 617
Rule 628
Rule 634
Rule 3476
Rubi steps
\begin {align*} \int \sqrt [3]{b \coth (c+d x)} \, dx &=-\frac {b \operatorname {Subst}\left (\int \frac {\sqrt [3]{x}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {(3 b) \operatorname {Subst}\left (\int \frac {x^3}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}\\ &=-\frac {(3 b) \operatorname {Subst}\left (\int \frac {x}{-b^2+x^3} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}\\ &=-\frac {\sqrt [3]{b} \operatorname {Subst}\left (\int \frac {1}{-b^{2/3}+x} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \operatorname {Subst}\left (\int \frac {-b^{2/3}+x}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}\\ &=-\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \operatorname {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 d}-\frac {(3 b) \operatorname {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 {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}+\frac {\left (3 \sqrt [3]{b}\right ) \operatorname {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 d}\\ &=-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1+\frac {2 (b \coth (c+d x))^{2/3}}{b^{2/3}}}{\sqrt {3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 38, normalized size = 0.29 \[ \frac {3 (b \coth (c+d x))^{4/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\coth ^2(c+d x)\right )}{4 b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 291, normalized size = 2.20 \[ -\frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}}{3 \, b}\right ) - 2 \, \left (-b\right )^{\frac {1}{3}} \log \left (-\left (-b\right )^{\frac {2}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) + \left (-b\right )^{\frac {1}{3}} \log \left (\frac {{\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \left (-b\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \left (-b\right )^{\frac {1}{3}} + {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b\right )} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 217, normalized size = 1.64 \[ -\frac {b {\left (\frac {2 \, \sqrt {3} {\left | b \right |}^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}}{3 \, {\left | b \right |}^{\frac {2}{3}}}\right )}{b^{2}} - \frac {{\left | b \right |}^{\frac {4}{3}} \log \left (\left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} {\left | b \right |}^{\frac {2}{3}} + {\left | b \right |}^{\frac {4}{3}} + \frac {{\left (b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )}{b^{2}} + \frac {2 \, {\left | b \right |}^{\frac {4}{3}} \log \left ({\left | \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} - {\left | b \right |}^{\frac {2}{3}} \right |}\right )}{b^{2}}\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 115, normalized size = 0.87 \[ -\frac {b \ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{2 d \left (b^{2}\right )^{\frac {1}{3}}}+\frac {b \ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {4}{3}}+\left (b^{2}\right )^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{4 d \left (b^{2}\right )^{\frac {1}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 d \left (b^{2}\right )^{\frac {1}{3}}} \]
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 \[ \int \left (b \coth \left (d x + c\right )\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 146, normalized size = 1.11 \[ \frac {{\left (-b\right )}^{1/3}\,\ln \left (81\,{\left (-b\right )}^{16/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}-81\,b^6\right )}{2\,d}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (-\frac {81\,b^6}{d^4}-\frac {81\,{\left (-b\right )}^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (-\frac {81\,b^6}{d^4}+\frac {162\,{\left (-b\right )}^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \]
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 \[ \int \sqrt [3]{b \coth {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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