Optimal. Leaf size=275 \[ \frac {3}{8} a \left (a^2+63 b^2\right ) x+\frac {3 b \left (3 a^2+5 b^2\right ) \log (\cosh (c+d x))}{d}-\frac {18 a b^2 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+10 b^2\right ) \tanh ^2(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^3(c+d x)}{d}-\frac {3 b^3 \tanh ^4(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^6(c+d x)}{2 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \tanh (c+d x)\right )}{4 d}-\frac {\cosh (c+d x) \sinh (c+d x) \left (a \left (5 a^2+51 b^2\right )+2 b \left (15 a^2+11 b^2\right ) \tanh (c+d x)\right )}{8 d} \]
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Rubi [A]
time = 0.34, antiderivative size = 306, normalized size of antiderivative = 1.11, number of steps
used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3744, 1818,
1816, 647, 31} \begin {gather*} -\frac {3 b \left (3 a^2+5 b^2\right ) \tanh ^2(c+d x)}{2 d}-\frac {3 a \left (a^2+63 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 (a+b) \left (a^2+23 a b+40 b^2\right ) \log (1-\tanh (c+d x))}{16 d}+\frac {3 (a-b) \left (a^2-23 a b+40 b^2\right ) \log (\tanh (c+d x)+1)}{16 d}+\frac {\sinh ^4(c+d x) \left (a \left (a^2+3 b^2\right ) \tanh (c+d x)+b \left (3 a^2+b^2\right )\right )}{4 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (4 b \left (6 a^2+5 b^2\right ) \tanh (c+d x)+a \left (a^2+39 b^2\right )\right )}{8 d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}-\frac {3 a b^2 \tanh ^3(c+d x)}{d}-\frac {b^3 \tanh ^8(c+d x)}{8 d}-\frac {b^3 \tanh ^6(c+d x)}{2 d}-\frac {3 b^3 \tanh ^4(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 647
Rule 1816
Rule 1818
Rule 3744
Rubi steps
\begin {align*} \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {x^4 \left (a+b x^3\right )^3}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\sinh ^4(c+d x) \left (b \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right ) \tanh (c+d x)\right )}{4 d}+\frac {\text {Subst}\left (\int \frac {x^3 \left (-4 b \left (3 a^2+b^2\right )-a \left (a^2+15 b^2\right ) x-4 b \left (3 a^2+b^2\right ) x^2-12 a b^2 x^3-4 b^3 x^4-12 a b^2 x^5-4 b^3 x^6-4 b^3 x^8\right )}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac {\sinh ^4(c+d x) \left (b \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right ) \tanh (c+d x)\right )}{4 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a \left (a^2+39 b^2\right )+4 b \left (6 a^2+5 b^2\right ) \tanh (c+d x)\right )}{8 d}+\frac {\text {Subst}\left (\int \frac {x^2 \left (3 a \left (a^2+39 b^2\right )+72 b \left (a^2+b^2\right ) x+48 a b^2 x^2+24 b^3 x^3+24 a b^2 x^4+16 b^3 x^5+8 b^3 x^7\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {\sinh ^4(c+d x) \left (b \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right ) \tanh (c+d x)\right )}{4 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a \left (a^2+39 b^2\right )+4 b \left (6 a^2+5 b^2\right ) \tanh (c+d x)\right )}{8 d}+\frac {\text {Subst}\left (\int \left (-3 a \left (a^2+63 b^2\right )-24 b \left (3 a^2+5 b^2\right ) x-72 a b^2 x^2-48 b^3 x^3-24 a b^2 x^4-24 b^3 x^5-8 b^3 x^7+\frac {3 \left (a^3+63 a b^2+8 b \left (3 a^2+5 b^2\right ) x\right )}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {3 a \left (a^2+63 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 b \left (3 a^2+5 b^2\right ) \tanh ^2(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^3(c+d x)}{d}-\frac {3 b^3 \tanh ^4(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^6(c+d x)}{2 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d}+\frac {\sinh ^4(c+d x) \left (b \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right ) \tanh (c+d x)\right )}{4 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a \left (a^2+39 b^2\right )+4 b \left (6 a^2+5 b^2\right ) \tanh (c+d x)\right )}{8 d}+\frac {3 \text {Subst}\left (\int \frac {a^3+63 a b^2+8 b \left (3 a^2+5 b^2\right ) x}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {3 a \left (a^2+63 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 b \left (3 a^2+5 b^2\right ) \tanh ^2(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^3(c+d x)}{d}-\frac {3 b^3 \tanh ^4(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^6(c+d x)}{2 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d}+\frac {\sinh ^4(c+d x) \left (b \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right ) \tanh (c+d x)\right )}{4 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a \left (a^2+39 b^2\right )+4 b \left (6 a^2+5 b^2\right ) \tanh (c+d x)\right )}{8 d}-\frac {\left (3 (a-b) \left (a^2-23 a b+40 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\tanh (c+d x)\right )}{16 d}+\frac {\left (3 (a+b) \left (a^2+23 a b+40 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=-\frac {3 (a+b) \left (a^2+23 a b+40 b^2\right ) \log (1-\tanh (c+d x))}{16 d}+\frac {3 (a-b) \left (a^2-23 a b+40 b^2\right ) \log (1+\tanh (c+d x))}{16 d}-\frac {3 a \left (a^2+63 b^2\right ) \tanh (c+d x)}{8 d}-\frac {3 b \left (3 a^2+5 b^2\right ) \tanh ^2(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^3(c+d x)}{d}-\frac {3 b^3 \tanh ^4(c+d x)}{2 d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^6(c+d x)}{2 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d}+\frac {\sinh ^4(c+d x) \left (b \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right ) \tanh (c+d x)\right )}{4 d}-\frac {\sinh ^2(c+d x) \tanh (c+d x) \left (a \left (a^2+39 b^2\right )+4 b \left (6 a^2+5 b^2\right ) \tanh (c+d x)\right )}{8 d}\\ \end {align*}
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Mathematica [A]
time = 6.20, size = 294, normalized size = 1.07 \begin {gather*} \frac {3 a \left (a^2+63 b^2\right ) (c+d x)}{8 d}-\frac {b \left (15 a^2+11 b^2\right ) \cosh (2 (c+d x))}{8 d}+\frac {b \left (3 a^2+b^2\right ) \cosh (4 (c+d x))}{32 d}+\frac {3 \left (3 a^2 b+5 b^3\right ) \log (\cosh (c+d x))}{d}+\frac {b \left (3 a^2+20 b^2\right ) \text {sech}^2(c+d x)}{2 d}-\frac {15 b^3 \text {sech}^4(c+d x)}{4 d}+\frac {b^3 \text {sech}^6(c+d x)}{d}-\frac {b^3 \text {sech}^8(c+d x)}{8 d}-\frac {a \left (a^2+12 b^2\right ) \sinh (2 (c+d x))}{4 d}+\frac {a \left (a^2+3 b^2\right ) \sinh (4 (c+d x))}{32 d}-\frac {108 a b^2 \tanh (c+d x)}{5 d}+\frac {21 a b^2 \text {sech}^2(c+d x) \tanh (c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^4(c+d x) \tanh (c+d x)}{5 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. \(678\) vs.
\(2(259)=518\).
time = 3.14, size = 679, normalized size = 2.47
method | result | size |
risch | \(-9 a^{2} b x -\frac {15 b \,{\mathrm e}^{2 d x +2 c} a^{2}}{16 d}-\frac {11 b^{3} {\mathrm e}^{-2 d x -2 c}}{16 d}+\frac {b^{3} {\mathrm e}^{-4 d x -4 c}}{64 d}-15 b^{3} x -\frac {{\mathrm e}^{-4 d x -4 c} a^{3}}{64 d}-\frac {30 b^{3} c}{d}+\frac {15 b^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} a^{2} b}{64 d}+\frac {9 b \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a^{2}}{d}+\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{2 d}-\frac {3 b^{2} {\mathrm e}^{-4 d x -4 c} a}{64 d}+\frac {3 a^{3} x}{8}+\frac {189 a \,b^{2} x}{8}-\frac {15 b \,{\mathrm e}^{-2 d x -2 c} a^{2}}{16 d}-\frac {18 a^{2} b c}{d}+\frac {b^{3} {\mathrm e}^{4 d x +4 c}}{64 d}-\frac {11 b^{3} {\mathrm e}^{2 d x +2 c}}{16 d}+\frac {{\mathrm e}^{4 d x +4 c} a^{3}}{64 d}-\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}+\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}+\frac {3 b^{2} {\mathrm e}^{4 d x +4 c} a}{64 d}-\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{2 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a^{2} b}{64 d}+\frac {2 b \left (15 a^{2} {\mathrm e}^{14 d x +14 c}+150 a b \,{\mathrm e}^{14 d x +14 c}+100 b^{2} {\mathrm e}^{14 d x +14 c}+90 a^{2} {\mathrm e}^{12 d x +12 c}+900 a b \,{\mathrm e}^{12 d x +12 c}+450 b^{2} {\mathrm e}^{12 d x +12 c}+225 a^{2} {\mathrm e}^{10 d x +10 c}+2430 a b \,{\mathrm e}^{10 d x +10 c}+1060 b^{2} {\mathrm e}^{10 d x +10 c}+300 a^{2} {\mathrm e}^{8 d x +8 c}+3780 a b \,{\mathrm e}^{8 d x +8 c}+1340 b^{2} {\mathrm e}^{8 d x +8 c}+225 a^{2} {\mathrm e}^{6 d x +6 c}+3618 a b \,{\mathrm e}^{6 d x +6 c}+1060 b^{2} {\mathrm e}^{6 d x +6 c}+90 a^{2} {\mathrm e}^{4 d x +4 c}+2124 a b \,{\mathrm e}^{4 d x +4 c}+450 b^{2} {\mathrm e}^{4 d x +4 c}+15 a^{2} {\mathrm e}^{2 d x +2 c}+714 a b \,{\mathrm e}^{2 d x +2 c}+100 b^{2} {\mathrm e}^{2 d x +2 c}+108 a b \right )}{5 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{8}}\) | \(679\) |
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. 647 vs.
\(2 (259) = 518\).
time = 0.50, size = 647, normalized size = 2.35 \begin {gather*} \frac {1}{64} \, a^{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 )} + \frac {3}{320} \, a b^{2} {\left (\frac {2520 \, {\left (d x + c\right )}}{d} + \frac {5 \, {\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{d} - \frac {135 \, e^{\left (-2 \, d x - 2 \, c\right )} + 5358 \, e^{\left (-4 \, d x - 4 \, c\right )} + 18190 \, e^{\left (-6 \, d x - 6 \, c\right )} + 28455 \, e^{\left (-8 \, d x - 8 \, c\right )} + 19995 \, e^{\left (-10 \, d x - 10 \, c\right )} + 6560 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 5 \, e^{\left (-6 \, d x - 6 \, c\right )} + 10 \, e^{\left (-8 \, d x - 8 \, c\right )} + 10 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )}\right )}}\right )} + \frac {1}{64} \, b^{3} {\left (\frac {960 \, {\left (d x + c\right )}}{d} - \frac {44 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}}{d} + \frac {960 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {36 \, e^{\left (-2 \, d x - 2 \, c\right )} + 324 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1384 \, e^{\left (-6 \, d x - 6 \, c\right )} - 9126 \, e^{\left (-8 \, d x - 8 \, c\right )} - 24112 \, e^{\left (-10 \, d x - 10 \, c\right )} - 31868 \, e^{\left (-12 \, d x - 12 \, c\right )} - 25912 \, e^{\left (-14 \, d x - 14 \, c\right )} - 11169 \, e^{\left (-16 \, d x - 16 \, c\right )} - 2516 \, e^{\left (-18 \, d x - 18 \, c\right )} - 1}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 8 \, e^{\left (-6 \, d x - 6 \, c\right )} + 28 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 70 \, e^{\left (-12 \, d x - 12 \, c\right )} + 56 \, e^{\left (-14 \, d x - 14 \, c\right )} + 28 \, e^{\left (-16 \, d x - 16 \, c\right )} + 8 \, e^{\left (-18 \, d x - 18 \, c\right )} + e^{\left (-20 \, d x - 20 \, c\right )}\right )}}\right )} + \frac {3}{64} \, a^{2} b {\left (\frac {192 \, {\left (d x + c\right )}}{d} - \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}}{d} + \frac {192 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {18 \, e^{\left (-2 \, d x - 2 \, c\right )} + 39 \, e^{\left (-4 \, d x - 4 \, c\right )} - 108 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\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. 12323 vs.
\(2 (259) = 518\).
time = 0.48, size = 12323, normalized size = 44.81 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 694 vs.
\(2 (259) = 518\).
time = 0.93, size = 694, normalized size = 2.52 \begin {gather*} \frac {35 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 105 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 105 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 35 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 280 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 2100 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 3360 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 1540 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 840 \, {\left (a^{3} - 24 \, a^{2} b + 63 \, a b^{2} - 40 \, b^{3}\right )} {\left (d x + c\right )} - 35 \, {\left (18 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 432 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 1134 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 720 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 60 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 96 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 44 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 6720 \, {\left (3 \, a^{2} b + 5 \, b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - \frac {8 \, {\left (6849 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 11415 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 53112 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 16800 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 80120 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 181692 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 100800 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 269220 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 358344 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 272160 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 520520 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 445830 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 423360 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 648970 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 358344 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 405216 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 520520 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 181692 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 237888 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 269220 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 53112 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 79968 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 80120 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 6849 \, a^{2} b - 12096 \, a b^{2} + 11415 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{2240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.70, size = 682, normalized size = 2.48 \begin {gather*} x\,\left (\frac {3\,a^3}{8}-9\,a^2\,b+\frac {189\,a\,b^2}{8}-15\,b^3\right )-\frac {4\,\left (71\,b^3+12\,a\,b^2\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {256\,b^3}{d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,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\right )}+\frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )\,\left (9\,a^2\,b+15\,b^3\right )}{d}+\frac {2\,\left (3\,a^2\,b+30\,a\,b^2+20\,b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {32\,\left (50\,b^3+3\,a\,b^2\right )}{5\,d\,\left (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\right )}+\frac {128\,b^3}{d\,\left (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\right )}-\frac {2\,\left (3\,a^2\,b+30\,a\,b^2+50\,b^3\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {32\,b^3}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^3}{64\,d}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,{\left (a-b\right )}^3}{64\,d}+\frac {8\,\left (23\,b^3+9\,a\,b^2\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2\,\left (2\,a+11\,b\right )}{16\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a-b\right )}^2\,\left (2\,a-11\,b\right )}{16\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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