Optimal. Leaf size=140 \[ -\frac {b^3 x}{2}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3296, 1273,
1816, 213} \begin {gather*} -\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}+\frac {b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {b^3 x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 1273
Rule 1816
Rule 3296
Rubi steps
\begin {align*} \int \text {csch}^{10}(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^{10} \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {\text {Subst}\left (\int \frac {-2 a^3+10 a^3 x^2-2 a^2 (10 a+3 b) x^4+2 a^2 (10 a+9 b) x^6-2 a \left (5 a^2+9 a b+3 b^2\right ) x^8+\left (2 a^3+6 a^2 b+6 a b^2+b^3\right ) x^{10}}{x^{10} \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {\text {Subst}\left (\int \left (-\frac {2 a^3}{x^{10}}+\frac {8 a^3}{x^8}-\frac {6 a^2 (2 a+b)}{x^6}+\frac {4 a^2 (2 a+3 b)}{x^4}-\frac {2 a \left (a^2+3 a b+3 b^2\right )}{x^2}-\frac {b^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac {b^3 x}{2}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 115, normalized size = 0.82 \begin {gather*} \frac {-4 a \coth (c+d x) \left (128 a^2+504 a b+945 b^2-4 a (16 a+63 b) \text {csch}^2(c+d x)+3 a (16 a+63 b) \text {csch}^4(c+d x)-40 a^2 \text {csch}^6(c+d x)+35 a^2 \text {csch}^8(c+d x)\right )+315 b^3 (-2 (c+d x)+\sinh (2 (c+d x)))}{1260 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. \(321\) vs.
\(2(128)=256\).
time = 1.57, size = 322, normalized size = 2.30
method | result | size |
risch | \(-\frac {b^{3} x}{2}+\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{8 d}-\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {2 a \left (945 b^{2} {\mathrm e}^{16 d x +16 c}-7560 b^{2} {\mathrm e}^{14 d x +14 c}+5040 a b \,{\mathrm e}^{12 d x +12 c}+26460 b^{2} {\mathrm e}^{12 d x +12 c}-22680 a b \,{\mathrm e}^{10 d x +10 c}-52920 b^{2} {\mathrm e}^{10 d x +10 c}+16128 a^{2} {\mathrm e}^{8 d x +8 c}+40824 a b \,{\mathrm e}^{8 d x +8 c}+66150 b^{2} {\mathrm e}^{8 d x +8 c}-10752 a^{2} {\mathrm e}^{6 d x +6 c}-37296 a b \,{\mathrm e}^{6 d x +6 c}-52920 b^{2} {\mathrm e}^{6 d x +6 c}+4608 a^{2} {\mathrm e}^{4 d x +4 c}+18144 a b \,{\mathrm e}^{4 d x +4 c}+26460 b^{2} {\mathrm e}^{4 d x +4 c}-1152 a^{2} {\mathrm e}^{2 d x +2 c}-4536 a b \,{\mathrm e}^{2 d x +2 c}-7560 b^{2} {\mathrm e}^{2 d x +2 c}+128 a^{2}+504 a b +945 b^{2}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{9}}\) | \(322\) |
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. 842 vs.
\(2 (128) = 256\).
time = 0.27, size = 842, normalized size = 6.01 \begin {gather*} -\frac {1}{8} \, b^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {256}{315} \, a^{3} {\left (\frac {9 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} - \frac {36 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} + \frac {84 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} - \frac {126 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} - \frac {1}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}}\right )} - \frac {16}{5} \, a^{2} b {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + \frac {6 \, a b^{2}}{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. 1314 vs.
\(2 (128) = 256\).
time = 0.40, size = 1314, normalized size = 9.39 \begin {gather*} \frac {315 \, b^{3} \cosh \left (d x + c\right )^{11} + 3465 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} - 4 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \sinh \left (d x + c\right )^{9} + 9 \, {\left (5775 \, b^{3} \cosh \left (d x + c\right )^{3} - {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 9 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 36 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2} - 4 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{7} + 21 \, {\left (6930 \, b^{3} \cosh \left (d x + c\right )^{5} - 4 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 9 \, {\left (4096 \, a^{3} + 16128 \, a^{2} b + 16800 \, a b^{2} + 2625 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 36 \, {\left (1260 \, b^{3} d x + 14 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} - 1024 \, a^{3} - 4032 \, a^{2} b - 7560 \, a b^{2} - 21 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 9 \, {\left (11550 \, b^{3} \cosh \left (d x + c\right )^{7} - 14 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 35 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 5 \, {\left (4096 \, a^{3} + 16128 \, a^{2} b + 16800 \, a b^{2} + 2625 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 42 \, {\left (2048 \, a^{3} + 6144 \, a^{2} b + 5040 \, a b^{2} + 675 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (28 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{6} - 8820 \, b^{3} d x - 105 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 7168 \, a^{3} + 28224 \, a^{2} b + 52920 \, a b^{2} + 120 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 9 \, {\left (1925 \, b^{3} \cosh \left (d x + c\right )^{9} - 4 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 21 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (4096 \, a^{3} + 16128 \, a^{2} b + 16800 \, a b^{2} + 2625 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 14 \, {\left (2048 \, a^{3} + 6144 \, a^{2} b + 5040 \, a b^{2} + 675 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 126 \, {\left (1024 \, a^{3} + 1152 \, a^{2} b + 840 \, a b^{2} + 105 \, b^{3}\right )} \cosh \left (d x + c\right ) - 36 \, {\left ({\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{8} - 7 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{6} + 4410 \, b^{3} d x + 20 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} - 3584 \, a^{3} - 14112 \, a^{2} b - 26460 \, a b^{2} - 28 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{2520 \, {\left (d \sinh \left (d x + c\right )^{9} + 9 \, {\left (4 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{7} + 9 \, {\left (14 \, d \cosh \left (d x + c\right )^{4} - 21 \, d \cosh \left (d x + c\right )^{2} + 4 \, d\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (28 \, d \cosh \left (d x + c\right )^{6} - 105 \, d \cosh \left (d x + c\right )^{4} + 120 \, d \cosh \left (d x + c\right )^{2} - 28 \, d\right )} \sinh \left (d x + c\right )^{3} + 9 \, {\left (d \cosh \left (d x + c\right )^{8} - 7 \, d \cosh \left (d x + c\right )^{6} + 20 \, d \cosh \left (d x + c\right )^{4} - 28 \, d \cosh \left (d x + c\right )^{2} + 14 \, 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. 360 vs.
\(2 (128) = 256\).
time = 0.59, size = 360, normalized size = 2.57 \begin {gather*} -\frac {1260 \, {\left (d x + c\right )} b^{3} - 315 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, {\left (2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + \frac {16 \, {\left (945 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} - 7560 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 26460 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 22680 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 52920 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 16128 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40824 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 66150 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 10752 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 37296 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 52920 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 4608 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18144 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 26460 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 1152 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 4536 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 7560 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 128 \, a^{3} + 504 \, a^{2} b + 945 \, a b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{9}}}{2520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 1500, normalized size = 10.71 \begin {gather*} \frac {\frac {2\,a\,b^2}{3\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{d}+\frac {10\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{3\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{d}-\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{9\,d}+\frac {14\,a\,b^2\,{\mathrm {e}}^{12\,c+12\,d\,x}}{3\,d}-\frac {2\,a\,b^2\,{\mathrm {e}}^{14\,c+14\,d\,x}}{3\,d}}{28\,{\mathrm {e}}^{4\,c+4\,d\,x}-8\,{\mathrm {e}}^{2\,c+2\,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 (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{315\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{7\,d}-\frac {8\,a\,b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}}{3\,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 {2\,\left (4\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {4\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{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}-\frac {\frac {2\,a\,b^2}{3\,d}+\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{3\,d}-\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{3\,d}-\frac {16\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{9\,d}-\frac {16\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}-\frac {16\,a\,b^2\,{\mathrm {e}}^{14\,c+14\,d\,x}}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{16\,c+16\,d\,x}}{3\,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 (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}+\frac {20\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {20\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{63\,d}+\frac {10\,a\,b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}}{3\,d}-\frac {2\,a\,b^2\,{\mathrm {e}}^{10\,c+10\,d\,x}}{3\,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 {2\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{7\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{d}-\frac {2\,a\,b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}}{3\,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 {b^3\,x}{2}-\frac {\frac {2\,\left (4\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{7\,d}-\frac {40\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}+\frac {10\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{7\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{21\,d}-\frac {4\,a\,b^2\,{\mathrm {e}}^{10\,c+10\,d\,x}}{d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{12\,c+12\,d\,x}}{3\,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 {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {4\,a\,b^2}{3\,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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