Optimal. Leaf size=74 \[ -\frac {b \sqrt [3]{b \coth ^3(c+d x)}}{d}-\frac {b \coth ^2(c+d x) \sqrt [3]{b \coth ^3(c+d x)}}{3 d}+b x \sqrt [3]{b \coth ^3(c+d x)} \tanh (c+d x) \]
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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 = {3739, 3554, 8}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 43, normalized size = 0.58 \begin {gather*} -\frac {\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 ) \tanh (c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs.
\(2(66)=132\).
time = 2.65, size = 145, normalized size = 1.96
method | result | size |
risch | \(\frac {b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {1}{3}} x}{1+{\mathrm e}^{2 d x +2 c}}-\frac {4 b \left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {1}{3}} \left (3 \,{\mathrm e}^{4 d x +4 c}-3 \,{\mathrm e}^{2 d x +2 c}+2\right )}{3 \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 87, normalized size = 1.18 \begin {gather*} \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 )}} \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. 1046 vs.
\(2 (66) = 132\).
time = 0.38, size = 1046, normalized size = 14.14 \begin {gather*} -\frac {{\left (3 \, b d x \cosh \left (d x + c\right )^{6} - 3 \, {\left (b d x e^{\left (2 \, d x + 2 \, c\right )} - b d x\right )} \sinh \left (d x + c\right )^{6} - 18 \, {\left (b d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - b d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} - 3 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (15 \, b d x \cosh \left (d x + c\right )^{2} - 3 \, b d x - {\left (15 \, b d x \cosh \left (d x + c\right )^{2} - 3 \, b d x - 4 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b\right )} \sinh \left (d x + c\right )^{4} + 12 \, {\left (5 \, b d x \cosh \left (d x + c\right )^{3} - {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right ) - {\left (5 \, b d x \cosh \left (d x + c\right )^{3} - {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )^{3} - 3 \, b d x + 3 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (15 \, b d x \cosh \left (d x + c\right )^{4} + 3 \, b d x - 6 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{2} - {\left (15 \, b d x \cosh \left (d x + c\right )^{4} + 3 \, b d x - 6 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b\right )} \sinh \left (d x + c\right )^{2} - {\left (3 \, b d x \cosh \left (d x + c\right )^{6} - 3 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{4} - 3 \, b d x + 3 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{2} - 8 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, {\left (3 \, b d x \cosh \left (d x + c\right )^{5} - 2 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right ) - {\left (3 \, b d x \cosh \left (d x + c\right )^{5} - 2 \, {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, b d x + 4 \, b\right )} \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right ) - 8 \, b\right )} \left (\frac {b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{3 \, {\left (d \cosh \left (d x + c\right )^{6} + {\left (d e^{\left (2 \, d x + 2 \, c\right )} + d\right )} \sinh \left (d x + c\right )^{6} + 6 \, {\left (d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} - 3 \, d \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} + {\left (5 \, d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right ) + {\left (5 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} - 6 \, d \cosh \left (d x + c\right )^{2} + {\left (5 \, d \cosh \left (d x + c\right )^{4} - 6 \, d \cosh \left (d x + c\right )^{2} + d\right )} e^{\left (2 \, d x + 2 \, c\right )} + d\right )} \sinh \left (d x + c\right )^{2} + {\left (d \cosh \left (d x + c\right )^{6} - 3 \, d \cosh \left (d x + c\right )^{4} + 3 \, d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, {\left (d \cosh \left (d x + c\right )^{5} - 2 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right ) + {\left (d \cosh \left (d x + c\right )^{5} - 2 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right ) - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (66) = 132\).
time = 106.75, size = 162, normalized size = 2.19 \begin {gather*} \begin {cases} x \left (b \coth ^{3}{\left (c \right )}\right )^{\frac {4}{3}} & \text {for}\: d = 0 \\- \frac {\left (b \coth ^{3}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}\right )^{\frac {4}{3}} \log {\left (- e^{- d x} \right )}}{d} & \text {for}\: c = \log {\left (- e^{- d x} \right )} \\- \frac {\left (b \coth ^{3}{\left (d x + \log {\left (e^{- d x} \right )} \right )}\right )^{\frac {4}{3}} \log {\left (e^{- d x} \right )}}{d} & \text {for}\: c = \log {\left (e^{- d x} \right )} \\x \left (\frac {b}{\tanh ^{3}{\left (c + d x \right )}}\right )^{\frac {4}{3}} \tanh ^{4}{\left (c + d x \right )} - \frac {\left (\frac {b}{\tanh ^{3}{\left (c + d x \right )}}\right )^{\frac {4}{3}} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {\left (\frac {b}{\tanh ^{3}{\left (c + d x \right )}}\right )^{\frac {4}{3}} \tanh {\left (c + d x \right )}}{3 d} & \text {otherwise} \end {cases} \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 = -0.01 \begin {gather*} \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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