Optimal. Leaf size=74 \[ -\frac {b \sqrt [3]{b \coth ^3(c+d x)}}{d}-\frac {b \coth ^2(c+d x) \sqrt [3]{b \coth ^3(c+d x)}}{3 d}+b x \tanh (c+d x) \sqrt [3]{b \coth ^3(c+d x)} \]
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Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ -\frac {b \coth ^2(c+d x) \sqrt [3]{b \coth ^3(c+d x)}}{3 d}-\frac {b \sqrt [3]{b \coth ^3(c+d x)}}{d}+b x \tanh (c+d x) \sqrt [3]{b \coth ^3(c+d x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \left (b \coth ^3(c+d x)\right )^{4/3} \, dx &=\left (b \sqrt [3]{b \coth ^3(c+d x)} \tanh (c+d x)\right ) \int \coth ^4(c+d x) \, dx\\ &=-\frac {b \coth ^2(c+d x) \sqrt [3]{b \coth ^3(c+d x)}}{3 d}+\left (b \sqrt [3]{b \coth ^3(c+d x)} \tanh (c+d x)\right ) \int \coth ^2(c+d x) \, dx\\ &=-\frac {b \sqrt [3]{b \coth ^3(c+d x)}}{d}-\frac {b \coth ^2(c+d x) \sqrt [3]{b \coth ^3(c+d x)}}{3 d}+\left (b \sqrt [3]{b \coth ^3(c+d x)} \tanh (c+d x)\right ) \int 1 \, dx\\ &=-\frac {b \sqrt [3]{b \coth ^3(c+d x)}}{d}-\frac {b \coth ^2(c+d x) \sqrt [3]{b \coth ^3(c+d x)}}{3 d}+b x \sqrt [3]{b \coth ^3(c+d x)} \tanh (c+d x)\\ \end {align*}
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Mathematica [C] time = 0.07, size = 43, normalized size = 0.58 \[ -\frac {\tanh (c+d x) \left (b \coth ^3(c+d x)\right )^{4/3} \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.45, size = 1046, normalized size = 14.14 \[ \text {result too large to display} \]
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 \[ \int \left (b \coth \left (d x + c\right )^{3}\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 145, normalized size = 1.96 \[ \frac {b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {1}{3}} x}{1+{\mathrm e}^{2 d x +2 c}}-\frac {4 b \left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {1}{3}} \left (3 \,{\mathrm e}^{4 d x +4 c}-3 \,{\mathrm e}^{2 d x +2 c}+2\right )}{3 \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 87, normalized size = 1.18 \[ \frac {{\left (d x + c\right )} b^{\frac {4}{3}}}{d} - \frac {4 \, {\left (3 \, b^{\frac {4}{3}} e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, b^{\frac {4}{3}} e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, b^{\frac {4}{3}}\right )}}{3 \, d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{4/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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