Optimal. Leaf size=36 \[ -\frac {(a+b) \coth (c+d x)}{d}+x (a+b)-\frac {a \coth ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3629, 12, 3473, 8} \[ -\frac {(a+b) \coth (c+d x)}{d}+x (a+b)-\frac {a \coth ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 3473
Rule 3629
Rubi steps
\begin {align*} \int \coth ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac {a \coth ^3(c+d x)}{3 d}+\int (a+b) \coth ^2(c+d x) \, dx\\ &=-\frac {a \coth ^3(c+d x)}{3 d}+(a+b) \int \coth ^2(c+d x) \, dx\\ &=-\frac {(a+b) \coth (c+d x)}{d}-\frac {a \coth ^3(c+d x)}{3 d}+(a+b) \int 1 \, dx\\ &=(a+b) x-\frac {(a+b) \coth (c+d x)}{d}-\frac {a \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 61, normalized size = 1.69 \[ -\frac {a \coth ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(c+d x)\right )}{3 d}-\frac {b \coth (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 156, normalized size = 4.33 \[ -\frac {{\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, {\left (a + b\right )} d x + 4 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right ) + 3 \, {\left (3 \, {\left (a + b\right )} d x - {\left (3 \, {\left (a + b\right )} d x + 4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a + 3 \, b\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 86, normalized size = 2.39 \[ \frac {3 \, {\left (d x + c\right )} {\left (a + b\right )} - \frac {2 \, {\left (6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + 3 \, b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 46, normalized size = 1.28 \[ \frac {a \left (d x +c -\coth \left (d x +c \right )-\frac {\left (\coth ^{3}\left (d x +c \right )\right )}{3}\right )+b \left (d x +c -\coth \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 105, normalized size = 2.92 \[ \frac {1}{3} \, a {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 162, normalized size = 4.50 \[ \frac {\frac {2\,b}{3\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a+b\right )}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {2\,\left (2\,a+b\right )}{3\,d}-\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a+b\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+x\,\left (a+b\right )-\frac {2\,\left (2\,a+b\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \coth ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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