Optimal. Leaf size=236 \[ -\frac {\sqrt {3} b^{4/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {\sqrt {3} b^{4/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {b^{4/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{4/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3554, 3557,
335, 216, 648, 632, 210, 642, 212} \begin {gather*} -\frac {\sqrt {3} b^{4/3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {\sqrt {3} b^{4/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{2 d}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{4/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{4/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3554
Rule 3557
Rubi steps
\begin {align*} \int (b \coth (c+d x))^{4/3} \, dx &=-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}+b^2 \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx\\ &=-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}-\frac {b^3 \text {Subst}\left (\int \frac {1}{x^{2/3} \left (-b^2+x^2\right )} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {1}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}\\ &=-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}+\frac {b^{4/3} \text {Subst}\left (\int \frac {\sqrt [3]{b}-\frac {x}{2}}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}+\frac {b^{4/3} \text {Subst}\left (\int \frac {\sqrt [3]{b}+\frac {x}{2}}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}+\frac {b^{5/3} \text {Subst}\left (\int \frac {1}{b^{2/3}-x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}\\ &=\frac {b^{4/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}-\frac {b^{4/3} \text {Subst}\left (\int \frac {-\sqrt [3]{b}+2 x}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 d}+\frac {b^{4/3} \text {Subst}\left (\int \frac {\sqrt [3]{b}+2 x}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 d}+\frac {\left (3 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 d}+\frac {\left (3 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 d}\\ &=\frac {b^{4/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{4/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {\left (3 b^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 d}-\frac {\left (3 b^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 d}\\ &=-\frac {\sqrt {3} b^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {\sqrt {3} b^{4/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {b^{4/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{4/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 36, normalized size = 0.15 \begin {gather*} \frac {3 b \sqrt [3]{b \coth (c+d x)} \left (-1+\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};\coth ^2(c+d x)\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.54, size = 199, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {3 b \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\frac {\left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{3 b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{6 b^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right )}{3 b^{\frac {2}{3}}}\right ) b}{2}-\frac {\left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{3 b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{6 b^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b^{\frac {2}{3}}}\right ) b}{2}\right )}{d}\) | \(199\) |
default | \(-\frac {3 b \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\frac {\left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{3 b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{6 b^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right )}{3 b^{\frac {2}{3}}}\right ) b}{2}-\frac {\left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{3 b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{6 b^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b^{\frac {2}{3}}}\right ) b}{2}\right )}{d}\) | \(199\) |
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 [A]
time = 0.37, size = 292, normalized size = 1.24 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} b + 2 \, \sqrt {3} \left (-b\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) - 2 \, \sqrt {3} b^{\frac {4}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} b^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + \left (-b\right )^{\frac {1}{3}} b \log \left (\left (-b\right )^{\frac {2}{3}} - \left (-b\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) + b^{\frac {4}{3}} \log \left (b^{\frac {2}{3}} - b^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-b\right )^{\frac {1}{3}} b \log \left (\left (-b\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, b^{\frac {4}{3}} \log \left (b^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + 12 \, b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{4 \, d} \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 \left (b \coth {\left (c + d x \right )}\right )^{\frac {4}{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.88, size = 249, normalized size = 1.06 \begin {gather*} -\frac {3\,b\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d}-\frac {b^{4/3}\,\mathrm {atan}\left (\frac {{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{b^{1/3}}\right )\,1{}\mathrm {i}}{d}-\frac {b^{4/3}\,\ln \left (\frac {486\,b^{37/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d^4}-\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}-\frac {b^{4/3}\,\ln \left (\frac {486\,b^{37/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d^4}-\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {b^{4/3}\,\ln \left (\frac {972\,b^{37/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d^4}+\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d}+\frac {b^{4/3}\,\ln \left (\frac {972\,b^{37/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d^4}+\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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