Optimal. Leaf size=110 \[ a^3 x-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4226, 1816,
212} \begin {gather*} -\frac {a^3 \tanh ^3(c+d x)}{3 d}-\frac {a^3 \tanh (c+d x)}{d}+a^3 x+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1816
Rule 4226
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^4(c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^4 \left (a+b \left (1-x^2\right )\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-a^3-a^3 x^2+b \left (3 a^2+3 a b+b^2\right ) x^4-b^2 (3 a+2 b) x^6+b^3 x^8+\frac {a^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^3 x-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(301\) vs. \(2(110)=220\).
time = 3.84, size = 301, normalized size = 2.74 \begin {gather*} \frac {8 \left (b+a \cosh ^2(c+d x)\right )^3 \text {sech}^9(c+d x) \left (315 a^3 d x \cosh ^9(c+d x)+35 b^3 \text {sech}(c) \sinh (d x)+5 (27 a-10 b) b^2 \cosh ^2(c+d x) \text {sech}(c) \sinh (d x)+3 b \left (63 a^2-72 a b+b^2\right ) \cosh ^4(c+d x) \text {sech}(c) \sinh (d x)+\left (105 a^3-378 a^2 b+27 a b^2+4 b^3\right ) \cosh ^6(c+d x) \text {sech}(c) \sinh (d x)-\left (420 a^3-189 a^2 b-54 a b^2-8 b^3\right ) \cosh ^8(c+d x) \text {sech}(c) \sinh (d x)+35 b^3 \cosh (c+d x) \tanh (c)+5 (27 a-10 b) b^2 \cosh ^3(c+d x) \tanh (c)+3 b \left (63 a^2-72 a b+b^2\right ) \cosh ^5(c+d x) \tanh (c)+\left (105 a^3-378 a^2 b+27 a b^2+4 b^3\right ) \cosh ^7(c+d x) \tanh (c)\right )}{315 d (a+2 b+a \cosh (2 (c+d x)))^3} \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. \(468\) vs.
\(2(102)=204\).
time = 2.72, size = 469, normalized size = 4.26
method | result | size |
risch | \(a^{3} x +\frac {-\frac {12 a \,b^{2}}{35}-12 a \,b^{2} {\mathrm e}^{14 d x +14 c}-48 a^{2} b \,{\mathrm e}^{12 d x +12 c}-12 a \,b^{2} {\mathrm e}^{12 d x +12 c}-72 a^{2} b \,{\mathrm e}^{10 d x +10 c}-12 a \,b^{2} {\mathrm e}^{10 d x +10 c}-6 a^{2} b \,{\mathrm e}^{16 d x +16 c}-24 a^{2} b \,{\mathrm e}^{14 d x +14 c}-\frac {96 a^{2} b \,{\mathrm e}^{4 d x +4 c}}{5}-\frac {24 a^{2} b \,{\mathrm e}^{2 d x +2 c}}{5}-\frac {6 a^{2} b}{5}+\frac {8 a^{3}}{3}-\frac {16 b^{3}}{315}-\frac {156 a \,b^{2} {\mathrm e}^{8 d x +8 c}}{5}-\frac {84 a \,b^{2} {\mathrm e}^{6 d x +6 c}}{5}-\frac {12 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{35}-\frac {396 a^{2} b \,{\mathrm e}^{8 d x +8 c}}{5}-\frac {108 a \,b^{2} {\mathrm e}^{2 d x +2 c}}{35}-\frac {264 a^{2} b \,{\mathrm e}^{6 d x +6 c}}{5}+28 a^{3} {\mathrm e}^{14 d x +14 c}+20 a^{3} {\mathrm e}^{2 d x +2 c}+\frac {260 a^{3} {\mathrm e}^{12 d x +12 c}}{3}-\frac {32 b^{3} {\mathrm e}^{12 d x +12 c}}{3}+156 a^{3} {\mathrm e}^{10 d x +10 c}-\frac {16 b^{3} {\mathrm e}^{2 d x +2 c}}{35}+4 a^{3} {\mathrm e}^{16 d x +16 c}-\frac {112 b^{3} {\mathrm e}^{8 d x +8 c}}{5}+68 a^{3} {\mathrm e}^{4 d x +4 c}-\frac {64 b^{3} {\mathrm e}^{4 d x +4 c}}{35}+16 b^{3} {\mathrm e}^{10 d x +10 c}+180 a^{3} {\mathrm e}^{8 d x +8 c}+\frac {412 a^{3} {\mathrm e}^{6 d x +6 c}}{3}+\frac {32 b^{3} {\mathrm e}^{6 d x +6 c}}{5}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{9}}\) | \(469\) |
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. 1453 vs.
\(2 (102) = 204\).
time = 0.28, size = 1453, normalized size = 13.21 \begin {gather*} \text {Too large to display} \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. 1323 vs.
\(2 (102) = 204\).
time = 0.38, size = 1323, normalized size = 12.03 \begin {gather*} \frac {{\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} + 9 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{9} + 9 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 9 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3} + 4 \, {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{7} + 21 \, {\left (4 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 36 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 9 \, {\left (14 \, {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 700 \, a^{3} + 84 \, a^{2} b + 204 \, a b^{2} - 32 \, b^{3} + 21 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 9 \, {\left (14 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 35 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 20 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 84 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (28 \, {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 105 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 2660 \, a^{3} - 252 \, a^{2} b - 252 \, a b^{2} + 896 \, b^{3} + 120 \, {\left (175 \, a^{3} + 21 \, a^{2} b + 51 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 9 \, {\left (4 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 21 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 40 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 126 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right ) - 9 \, {\left ({\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{8} + 7 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 20 \, {\left (175 \, a^{3} + 21 \, a^{2} b + 51 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 420 \, a^{3} - 126 \, a^{2} b - 336 \, a b^{2} - 672 \, b^{3} + 28 \, {\left (95 \, a^{3} - 9 \, a^{2} b - 9 \, a b^{2} + 32 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{315 \, {\left (d \cosh \left (d x + c\right )^{9} + 9 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} + 9 \, d \cosh \left (d x + c\right )^{7} + 21 \, {\left (4 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 36 \, d \cosh \left (d x + c\right )^{5} + 9 \, {\left (14 \, d \cosh \left (d x + c\right )^{5} + 35 \, d \cosh \left (d x + c\right )^{3} + 20 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} + 9 \, {\left (4 \, d \cosh \left (d x + c\right )^{7} + 21 \, d \cosh \left (d x + c\right )^{5} + 40 \, d \cosh \left (d x + c\right )^{3} + 28 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 126 \, d \cosh \left (d x + c\right )\right )}} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \tanh ^{4}{\left (c + d x \right )}\, dx \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. 475 vs.
\(2 (102) = 204\).
time = 0.45, size = 475, normalized size = 4.32 \begin {gather*} \frac {315 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (630 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} - 945 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 4410 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} - 3780 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 1890 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 13650 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 7560 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 1890 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 1680 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 24570 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 11340 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 1890 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 2520 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 28350 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 12474 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 4914 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 3528 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 21630 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 8316 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 2646 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1008 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 10710 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 3024 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 54 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 288 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3150 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 756 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 486 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 72 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}}}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.62, size = 1834, normalized size = 16.67 \begin {gather*} \frac {\frac {13\,a^3+3\,a\,b^2+16\,b^3}{63\,d}+\frac {10\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{63\,d}+\frac {20\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{63\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{21\,d}-\frac {5\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{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}-\frac {\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}-\frac {2\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{63\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,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 {13\,a^3+3\,a\,b^2+16\,b^3}{63\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{21\,d}-\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b^2-a^3\right )}{d}-\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{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}-\frac {\frac {a\,b^2-a^3}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+a^3\,x+\frac {\frac {2\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{63\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{21\,d}+\frac {20\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{63\,d}+\frac {10\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{21\,d}-\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{7\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a\,b^2-a^3\right )}{d}-\frac {2\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,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 {a\,b^2-a^3}{3\,d}-\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{3\,d}-\frac {5\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{9\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{9\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{3\,d}+\frac {7\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{14\,c+14\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{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}-\frac {\frac {2\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{9\,d}+\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{14\,c+14\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{16\,c+16\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{9\,{\mathrm {e}}^{2\,c+2\,d\,x}+36\,{\mathrm {e}}^{4\,c+4\,d\,x}+84\,{\mathrm {e}}^{6\,c+6\,d\,x}+126\,{\mathrm {e}}^{8\,c+8\,d\,x}+126\,{\mathrm {e}}^{10\,c+10\,d\,x}+84\,{\mathrm {e}}^{12\,c+12\,d\,x}+36\,{\mathrm {e}}^{14\,c+14\,d\,x}+9\,{\mathrm {e}}^{16\,c+16\,d\,x}+{\mathrm {e}}^{18\,c+18\,d\,x}+1}-\frac {\frac {2\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{105\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{63\,d}-\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{21\,d}+\frac {4\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,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 {2\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d\,\left ({\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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