Optimal. Leaf size=152 \[ \frac {9}{8} a b^2 x+\frac {3 a^2 b \cosh (c+d x)}{d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}-\frac {a^3 \coth (c+d x)}{d}-\frac {9 a b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3299, 3852, 8,
2718, 2715, 2713} \begin {gather*} -\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \cosh (c+d x)}{d}+\frac {3 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {9 a b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {9}{8} a b^2 x+\frac {b^3 \cosh ^7(c+d x)}{7 d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^3(c+d x)}{d}-\frac {b^3 \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 3299
Rule 3852
Rubi steps
\begin {align*} \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=-\int \left (-a^3 \text {csch}^2(c+d x)-3 a^2 b \sinh (c+d x)-3 a b^2 \sinh ^4(c+d x)-b^3 \sinh ^7(c+d x)\right ) \, dx\\ &=a^3 \int \text {csch}^2(c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh (c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^4(c+d x) \, dx+b^3 \int \sinh ^7(c+d x) \, dx\\ &=\frac {3 a^2 b \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {1}{4} \left (9 a b^2\right ) \int \sinh ^2(c+d x) \, dx-\frac {\left (i a^3\right ) \text {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}-\frac {b^3 \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {3 a^2 b \cosh (c+d x)}{d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}-\frac {a^3 \coth (c+d x)}{d}-\frac {9 a b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac {1}{8} \left (9 a b^2\right ) \int 1 \, dx\\ &=\frac {9}{8} a b^2 x+\frac {3 a^2 b \cosh (c+d x)}{d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}-\frac {a^3 \coth (c+d x)}{d}-\frac {9 a b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 140, normalized size = 0.92 \begin {gather*} \frac {2520 a b^2 c+2520 a b^2 d x+35 b \left (192 a^2-35 b^2\right ) \cosh (c+d x)+245 b^3 \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))-1120 a^3 \coth \left (\frac {1}{2} (c+d x)\right )-1680 a b^2 \sinh (2 (c+d x))+210 a b^2 \sinh (4 (c+d x))-1120 a^3 \tanh \left (\frac {1}{2} (c+d x)\right )}{2240 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.90, size = 268, normalized size = 1.76
method | result | size |
risch | \(\frac {9 a \,b^{2} x}{8}+\frac {b^{3} {\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 b^{3} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {3 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{64 d}+\frac {7 b^{3} {\mathrm e}^{3 d x +3 c}}{128 d}-\frac {3 a \,{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}+\frac {3 b \,{\mathrm e}^{d x +c} a^{2}}{2 d}-\frac {35 b^{3} {\mathrm e}^{d x +c}}{128 d}+\frac {3 b \,{\mathrm e}^{-d x -c} a^{2}}{2 d}-\frac {35 b^{3} {\mathrm e}^{-d x -c}}{128 d}+\frac {3 a \,{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}+\frac {7 b^{3} {\mathrm e}^{-3 d x -3 c}}{128 d}-\frac {3 a \,b^{2} {\mathrm e}^{-4 d x -4 c}}{64 d}-\frac {7 b^{3} {\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b^{3} {\mathrm e}^{-7 d x -7 c}}{896 d}-\frac {2 a^{3}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(268\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 220, normalized size = 1.45 \begin {gather*} \frac {3}{64} \, a b^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{4480} \, b^{3} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, 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. 302 vs.
\(2 (142) = 284\).
time = 0.49, size = 302, normalized size = 1.99 \begin {gather*} \frac {20 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 105 \, a b^{2} \cosh \left (d x + c\right )^{5} + 525 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 945 \, a b^{2} \cosh \left (d x + c\right )^{3} + 2 \, {\left (70 \, b^{3} \cosh \left (d x + c\right )^{3} - 81 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 4 \, {\left (35 \, b^{3} \cosh \left (d x + c\right )^{5} - 135 \, b^{3} \cosh \left (d x + c\right )^{3} + 147 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 105 \, {\left (10 \, a b^{2} \cosh \left (d x + c\right )^{3} - 27 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 280 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cosh \left (d x + c\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{7} - 81 \, b^{3} \cosh \left (d x + c\right )^{5} + 294 \, b^{3} \cosh \left (d x + c\right )^{3} + 1260 \, a b^{2} d x + 1120 \, a^{3} + 105 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d \sinh \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 276, normalized size = 1.82 \begin {gather*} \frac {5040 \, {\left (d x + c\right )} a b^{2} + 5 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 49 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 210 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 245 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 1680 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6720 \, a^{2} b e^{\left (d x + c\right )} - 1225 \, b^{3} e^{\left (d x + c\right )} - \frac {{\left (1890 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 294 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 210 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 54 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} - 35 \, {\left (192 \, a^{2} b - 35 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, {\left (16 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )} + 210 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )} {\left (e^{\left (d x + c\right )} - 1\right )}}}{4480 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 252, normalized size = 1.66 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b-35\,b^3\right )}{128\,d}-\frac {2\,a^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {7\,b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{128\,d}+\frac {7\,b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{128\,d}-\frac {7\,b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{640\,d}-\frac {7\,b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{640\,d}+\frac {b^3\,{\mathrm {e}}^{-7\,c-7\,d\,x}}{896\,d}+\frac {b^3\,{\mathrm {e}}^{7\,c+7\,d\,x}}{896\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (192\,a^2\,b-35\,b^3\right )}{128\,d}+\frac {9\,a\,b^2\,x}{8}+\frac {3\,a\,b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {3\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {3\,a\,b^2\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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