Optimal. Leaf size=133 \[ \frac {3}{8} b^2 (8 a+b) x+\frac {a^2 (a+3 b) \coth (c+d x)}{d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^3 \coth ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d} \]
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Rubi [A]
time = 0.20, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3296, 1273,
1819, 1816, 213} \begin {gather*} -\frac {a^3 \coth ^7(c+d x)}{7 d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {a^2 (a+3 b) \coth (c+d x)}{d}+\frac {3}{8} b^2 x (8 a+b)+\frac {b^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}-\frac {5 b^3 \sinh (c+d x) \cosh (c+d x)}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 1273
Rule 1816
Rule 1819
Rule 3296
Rubi steps
\begin {align*} \int \text {csch}^8(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^3}{x^8 \left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {\text {Subst}\left (\int \frac {4 a^3-20 a^3 x^2+4 a^2 (10 a+3 b) x^4-4 a^2 (10 a+9 b) x^6+\left (20 a^3+36 a^2 b+12 a b^2-b^3\right ) x^8-4 (a+b)^3 x^{10}}{x^8 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \frac {-8 a^3+32 a^3 x^2-24 a^2 (2 a+b) x^4+16 a^2 (2 a+3 b) x^6-\left (8 a^3+24 a^2 b+24 a b^2+3 b^3\right ) x^8}{x^8 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \left (-\frac {8 a^3}{x^8}+\frac {24 a^3}{x^6}-\frac {24 a^2 (a+b)}{x^4}+\frac {8 a^2 (a+3 b)}{x^2}+\frac {3 b^2 (8 a+b)}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {a^2 (a+3 b) \coth (c+d x)}{d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^3 \coth ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {\left (3 b^2 (8 a+b)\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {3}{8} b^2 (8 a+b) x+\frac {a^2 (a+3 b) \coth (c+d x)}{d}-\frac {a^2 (a+b) \coth ^3(c+d x)}{d}+\frac {3 a^3 \coth ^5(c+d x)}{5 d}-\frac {a^3 \coth ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 106, normalized size = 0.80 \begin {gather*} \frac {-32 a^2 \coth (c+d x) \left (-2 (8 a+35 b)+(8 a+35 b) \text {csch}^2(c+d x)-6 a \text {csch}^4(c+d x)+5 a \text {csch}^6(c+d x)\right )+35 b^2 (12 (8 a+b) (c+d x)-8 b \sinh (2 (c+d x))+b \sinh (4 (c+d x)))}{1120 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.54, size = 207, normalized size = 1.56
method | result | size |
risch | \(3 a \,b^{2} x +\frac {3 b^{3} x}{8}+\frac {b^{3} {\mathrm e}^{4 d x +4 c}}{64 d}-\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{8 d}+\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {b^{3} {\mathrm e}^{-4 d x -4 c}}{64 d}-\frac {4 a^{2} \left (105 b \,{\mathrm e}^{10 d x +10 c}-455 b \,{\mathrm e}^{8 d x +8 c}+280 a \,{\mathrm e}^{6 d x +6 c}+770 b \,{\mathrm e}^{6 d x +6 c}-168 a \,{\mathrm e}^{4 d x +4 c}-630 b \,{\mathrm e}^{4 d x +4 c}+56 a \,{\mathrm e}^{2 d x +2 c}+245 b \,{\mathrm e}^{2 d x +2 c}-8 a -35 b \right )}{35 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{7}}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 537 vs.
\(2 (123) = 246\).
time = 0.28, size = 537, normalized size = 4.04 \begin {gather*} \frac {1}{64} \, b^{3} {\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 )} + 3 \, a b^{2} x + \frac {32}{35} \, a^{3} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} - \frac {21 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} + \frac {35 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}} - \frac {1}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}}\right )} + 4 \, a^{2} b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\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 )}} - \frac {1}{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 )} \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. 928 vs.
\(2 (123) = 246\).
time = 0.42, size = 928, normalized size = 6.98 \begin {gather*} \frac {35 \, b^{3} \cosh \left (d x + c\right )^{11} + 385 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 525 \, b^{3} \cosh \left (d x + c\right )^{9} + 525 \, {\left (11 \, b^{3} \cosh \left (d x + c\right )^{3} - 9 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 8 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (2310 \, b^{3} \cosh \left (d x + c\right )^{5} - 6300 \, b^{3} \cosh \left (d x + c\right )^{3} + {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 7 \, {\left (1024 \, a^{3} + 4480 \, a^{2} b + 975 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 56 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x - 3 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left (330 \, b^{3} \cosh \left (d x + c\right )^{7} - 1890 \, b^{3} \cosh \left (d x + c\right )^{5} + {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (1024 \, a^{3} + 4480 \, a^{2} b + 975 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 42 \, {\left (512 \, a^{3} + 1600 \, a^{2} b + 215 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 56 \, {\left (5 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{4} + 384 \, a^{3} + 1680 \, a^{2} b - 315 \, {\left (8 \, a b^{2} + b^{3}\right )} d x - 10 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (275 \, b^{3} \cosh \left (d x + c\right )^{9} - 2700 \, b^{3} \cosh \left (d x + c\right )^{7} + 3 \, {\left (1024 \, a^{3} + 4480 \, a^{2} b + 2695 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (1024 \, a^{3} + 4480 \, a^{2} b + 975 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 18 \, {\left (512 \, a^{3} + 1600 \, a^{2} b + 215 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 70 \, {\left (512 \, a^{3} + 576 \, a^{2} b + 63 \, b^{3}\right )} \cosh \left (d x + c\right ) - 56 \, {\left ({\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{6} - 5 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{4} - 640 \, a^{3} - 2800 \, a^{2} b + 525 \, {\left (8 \, a b^{2} + b^{3}\right )} d x + 9 \, {\left (128 \, a^{3} + 560 \, a^{2} b - 105 \, {\left (8 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{2240 \, {\left (d \sinh \left (d x + c\right )^{7} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{5} + 7 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} - 10 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (d \cosh \left (d x + c\right )^{6} - 5 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} - 5 \, d\right )} \sinh \left (d x + c\right )\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs.
\(2 (123) = 246\).
time = 0.58, size = 253, normalized size = 1.90 \begin {gather*} \frac {35 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 280 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 840 \, {\left (8 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} - 35 \, {\left (144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - \frac {256 \, {\left (105 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 455 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 280 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 770 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 168 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 245 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a^{3} - 35 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}}}{2240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 749, normalized size = 5.63 \begin {gather*} \frac {\frac {32\,a^2\,b}{35\,d}-\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{35\,d}+\frac {192\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{35\,d}-\frac {16\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{7\,d}+\frac {24\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}-\frac {96\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}-\frac {96\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}+\frac {24\,a^2\,b\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1}-\frac {\frac {4\,a^2\,b}{7\,d}+\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{7\,d}-\frac {32\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}-\frac {64\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}+\frac {20\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}}{15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}+\frac {\frac {32\,a^2\,b}{35\,d}-\frac {8\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,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}-\frac {\frac {4\,\left (8\,a^3+9\,b\,a^2\right )}{35\,d}-\frac {96\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{35\,d}+\frac {12\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {3\,b^2\,x\,\left (8\,a+b\right )}{8}+\frac {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d}+\frac {b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d}-\frac {4\,a^2\,b}{7\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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