Optimal. Leaf size=336 \[ \frac {d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac {9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}+\frac {9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4}+\frac {d^2 e^{-6 e-6 f x} (c+d x)}{288 a^3 f^3}-\frac {9 d^2 e^{-4 e-4 f x} (c+d x)}{256 a^3 f^3}+\frac {9 d^2 e^{-2 e-2 f x} (c+d x)}{32 a^3 f^3}+\frac {d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac {9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}+\frac {9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d} \]
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Rubi [A]
time = 0.26, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3810, 2207,
2225} \begin {gather*} \frac {d^2 (c+d x) e^{-6 e-6 f x}}{288 a^3 f^3}-\frac {9 d^2 (c+d x) e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {9 d^2 (c+d x) e^{-2 e-2 f x}}{32 a^3 f^3}+\frac {d (c+d x)^2 e^{-6 e-6 f x}}{96 a^3 f^2}-\frac {9 d (c+d x)^2 e^{-4 e-4 f x}}{128 a^3 f^2}+\frac {9 d (c+d x)^2 e^{-2 e-2 f x}}{32 a^3 f^2}+\frac {(c+d x)^3 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^3 e^{-4 e-4 f x}}{32 a^3 f}+\frac {3 (c+d x)^3 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}+\frac {d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac {9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}+\frac {9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 3810
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+a \coth (e+f x))^3} \, dx &=\int \left (\frac {(c+d x)^3}{8 a^3}-\frac {e^{-6 e-6 f x} (c+d x)^3}{8 a^3}+\frac {3 e^{-4 e-4 f x} (c+d x)^3}{8 a^3}-\frac {3 e^{-2 e-2 f x} (c+d x)^3}{8 a^3}\right ) \, dx\\ &=\frac {(c+d x)^4}{32 a^3 d}-\frac {\int e^{-6 e-6 f x} (c+d x)^3 \, dx}{8 a^3}+\frac {3 \int e^{-4 e-4 f x} (c+d x)^3 \, dx}{8 a^3}-\frac {3 \int e^{-2 e-2 f x} (c+d x)^3 \, dx}{8 a^3}\\ &=\frac {e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}-\frac {d \int e^{-6 e-6 f x} (c+d x)^2 \, dx}{16 a^3 f}+\frac {(9 d) \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{32 a^3 f}-\frac {(9 d) \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{16 a^3 f}\\ &=\frac {d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac {9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}+\frac {9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}-\frac {d^2 \int e^{-6 e-6 f x} (c+d x) \, dx}{48 a^3 f^2}+\frac {\left (9 d^2\right ) \int e^{-4 e-4 f x} (c+d x) \, dx}{64 a^3 f^2}-\frac {\left (9 d^2\right ) \int e^{-2 e-2 f x} (c+d x) \, dx}{16 a^3 f^2}\\ &=\frac {d^2 e^{-6 e-6 f x} (c+d x)}{288 a^3 f^3}-\frac {9 d^2 e^{-4 e-4 f x} (c+d x)}{256 a^3 f^3}+\frac {9 d^2 e^{-2 e-2 f x} (c+d x)}{32 a^3 f^3}+\frac {d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac {9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}+\frac {9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}-\frac {d^3 \int e^{-6 e-6 f x} \, dx}{288 a^3 f^3}+\frac {\left (9 d^3\right ) \int e^{-4 e-4 f x} \, dx}{256 a^3 f^3}-\frac {\left (9 d^3\right ) \int e^{-2 e-2 f x} \, dx}{32 a^3 f^3}\\ &=\frac {d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac {9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}+\frac {9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4}+\frac {d^2 e^{-6 e-6 f x} (c+d x)}{288 a^3 f^3}-\frac {9 d^2 e^{-4 e-4 f x} (c+d x)}{256 a^3 f^3}+\frac {9 d^2 e^{-2 e-2 f x} (c+d x)}{32 a^3 f^3}+\frac {d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac {9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}+\frac {9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 1.42, size = 615, normalized size = 1.83 \begin {gather*} \frac {\text {csch}^3(e+f x) \left (81 \left (32 c^3 f^3+24 c^2 d f^2 (3+4 f x)+12 c d^2 f \left (7+12 f x+8 f^2 x^2\right )+d^3 \left (45+84 f x+72 f^2 x^2+32 f^3 x^3\right )\right ) \cosh (e+f x)+16 \left (36 c^3 f^3 (1+6 f x)+18 c^2 d f^2 \left (1+6 f x+18 f^2 x^2\right )+6 c d^2 f \left (1+6 f x+18 f^2 x^2+36 f^3 x^3\right )+d^3 \left (1+6 f x+18 f^2 x^2+36 f^3 x^3+54 f^4 x^4\right )\right ) \cosh (3 (e+f x))+4131 d^3 \sinh (e+f x)+8748 c d^2 f \sinh (e+f x)+9720 c^2 d f^2 \sinh (e+f x)+7776 c^3 f^3 \sinh (e+f x)+8748 d^3 f x \sinh (e+f x)+19440 c d^2 f^2 x \sinh (e+f x)+23328 c^2 d f^3 x \sinh (e+f x)+9720 d^3 f^2 x^2 \sinh (e+f x)+23328 c d^2 f^3 x^2 \sinh (e+f x)+7776 d^3 f^3 x^3 \sinh (e+f x)-16 d^3 \sinh (3 (e+f x))-96 c d^2 f \sinh (3 (e+f x))-288 c^2 d f^2 \sinh (3 (e+f x))-576 c^3 f^3 \sinh (3 (e+f x))-96 d^3 f x \sinh (3 (e+f x))-576 c d^2 f^2 x \sinh (3 (e+f x))-1728 c^2 d f^3 x \sinh (3 (e+f x))+3456 c^3 f^4 x \sinh (3 (e+f x))-288 d^3 f^2 x^2 \sinh (3 (e+f x))-1728 c d^2 f^3 x^2 \sinh (3 (e+f x))+5184 c^2 d f^4 x^2 \sinh (3 (e+f x))-576 d^3 f^3 x^3 \sinh (3 (e+f x))+3456 c d^2 f^4 x^3 \sinh (3 (e+f x))+864 d^3 f^4 x^4 \sinh (3 (e+f x))\right )}{27648 a^3 f^4 (1+\coth (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.01, size = 379, normalized size = 1.13
method | result | size |
risch | \(\frac {d^{3} x^{4}}{32 a^{3}}+\frac {d^{2} c \,x^{3}}{8 a^{3}}+\frac {3 d \,c^{2} x^{2}}{16 a^{3}}+\frac {c^{3} x}{8 a^{3}}+\frac {c^{4}}{32 a^{3} d}+\frac {3 \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{64 a^{3} f^{4}}-\frac {3 \left (32 d^{3} x^{3} f^{3}+96 c \,d^{2} f^{3} x^{2}+96 c^{2} d \,f^{3} x +24 d^{3} f^{2} x^{2}+32 c^{3} f^{3}+48 c \,d^{2} f^{2} x +24 c^{2} d \,f^{2}+12 d^{3} f x +12 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-4 f x -4 e}}{1024 a^{3} f^{4}}+\frac {\left (36 d^{3} x^{3} f^{3}+108 c \,d^{2} f^{3} x^{2}+108 c^{2} d \,f^{3} x +18 d^{3} f^{2} x^{2}+36 c^{3} f^{3}+36 c \,d^{2} f^{2} x +18 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +d^{3}\right ) {\mathrm e}^{-6 f x -6 e}}{1728 a^{3} f^{4}}\) | \(379\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.55, size = 434, normalized size = 1.29 \begin {gather*} \frac {1}{96} \, c^{3} {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} + \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} - 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} + 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c^{2} d e^{\left (-6 \, e\right )}}{384 \, a^{3} f^{2}} + \frac {{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} + 648 \, {\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 8 \, {\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d^{2} e^{\left (-6 \, e\right )}}{2304 \, a^{3} f^{3}} + \frac {{\left (864 \, f^{4} x^{4} e^{\left (6 \, e\right )} + 1296 \, {\left (4 \, f^{3} x^{3} e^{\left (4 \, e\right )} + 6 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 6 \, f x e^{\left (4 \, e\right )} + 3 \, e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (32 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 24 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 12 \, f x e^{\left (2 \, e\right )} + 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 16 \, {\left (36 \, f^{3} x^{3} + 18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{3} e^{\left (-6 \, e\right )}}{27648 \, a^{3} f^{4}} \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. 883 vs.
\(2 (310) = 620\).
time = 0.39, size = 883, normalized size = 2.63 \begin {gather*} \frac {16 \, {\left (54 \, d^{3} f^{4} x^{4} + 36 \, c^{3} f^{3} + 18 \, c^{2} d f^{2} + 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} + 6 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} + 18 \, c^{2} d f^{3} + 6 \, c d^{2} f^{2} + d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 48 \, {\left (54 \, d^{3} f^{4} x^{4} + 36 \, c^{3} f^{3} + 18 \, c^{2} d f^{2} + 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} + 6 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} + 18 \, c^{2} d f^{3} + 6 \, c d^{2} f^{2} + d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 16 \, {\left (54 \, d^{3} f^{4} x^{4} - 36 \, c^{3} f^{3} - 18 \, c^{2} d f^{2} - 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} - 6 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} - 18 \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} - d^{3} f\right )} x\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 81 \, {\left (32 \, d^{3} f^{3} x^{3} + 32 \, c^{3} f^{3} + 72 \, c^{2} d f^{2} + 84 \, c d^{2} f + 45 \, d^{3} + 24 \, {\left (4 \, c d^{2} f^{3} + 3 \, d^{3} f^{2}\right )} x^{2} + 12 \, {\left (8 \, c^{2} d f^{3} + 12 \, c d^{2} f^{2} + 7 \, d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 3 \, {\left (2592 \, d^{3} f^{3} x^{3} + 2592 \, c^{3} f^{3} + 3240 \, c^{2} d f^{2} + 2916 \, c d^{2} f + 1377 \, d^{3} + 648 \, {\left (12 \, c d^{2} f^{3} + 5 \, d^{3} f^{2}\right )} x^{2} + 16 \, {\left (54 \, d^{3} f^{4} x^{4} - 36 \, c^{3} f^{3} - 18 \, c^{2} d f^{2} - 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} - 6 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} - 18 \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} - d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 324 \, {\left (24 \, c^{2} d f^{3} + 20 \, c d^{2} f^{2} + 9 \, d^{3} f\right )} x\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )}{27648 \, {\left (a^{3} f^{4} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 3 \, a^{3} f^{4} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 3 \, a^{3} f^{4} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + a^{3} f^{4} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3918 vs.
\(2 (347) = 694\).
time = 1.55, size = 3918, normalized size = 11.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 548, normalized size = 1.63 \begin {gather*} \frac {{\left (864 \, d^{3} f^{4} x^{4} e^{\left (6 \, f x + 6 \, e\right )} + 3456 \, c d^{2} f^{4} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 5184 \, d^{3} f^{3} x^{3} e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, d^{3} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 576 \, d^{3} f^{3} x^{3} + 3456 \, c^{3} f^{4} x e^{\left (6 \, f x + 6 \, e\right )} + 15552 \, c d^{2} f^{3} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c d^{2} f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 1728 \, c d^{2} f^{3} x^{2} + 15552 \, c^{2} d f^{3} x e^{\left (4 \, f x + 4 \, e\right )} + 7776 \, d^{3} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c^{2} d f^{3} x e^{\left (2 \, f x + 2 \, e\right )} - 1944 \, d^{3} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 1728 \, c^{2} d f^{3} x + 288 \, d^{3} f^{2} x^{2} + 5184 \, c^{3} f^{3} e^{\left (4 \, f x + 4 \, e\right )} + 15552 \, c d^{2} f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, c^{3} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3888 \, c d^{2} f^{2} x e^{\left (2 \, f x + 2 \, e\right )} + 576 \, c^{3} f^{3} + 576 \, c d^{2} f^{2} x + 7776 \, c^{2} d f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 7776 \, d^{3} f x e^{\left (4 \, f x + 4 \, e\right )} - 1944 \, c^{2} d f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 972 \, d^{3} f x e^{\left (2 \, f x + 2 \, e\right )} + 288 \, c^{2} d f^{2} + 96 \, d^{3} f x + 7776 \, c d^{2} f e^{\left (4 \, f x + 4 \, e\right )} - 972 \, c d^{2} f e^{\left (2 \, f x + 2 \, e\right )} + 96 \, c d^{2} f + 3888 \, d^{3} e^{\left (4 \, f x + 4 \, e\right )} - 243 \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} + 16 \, d^{3}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{27648 \, a^{3} f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.44, size = 374, normalized size = 1.11 \begin {gather*} {\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {12\,c^3\,f^3+18\,c^2\,d\,f^2+18\,c\,d^2\,f+9\,d^3}{64\,a^3\,f^4}+\frac {3\,d^3\,x^3}{16\,a^3\,f}+\frac {9\,d\,x\,\left (2\,c^2\,f^2+2\,c\,d\,f+d^2\right )}{32\,a^3\,f^3}+\frac {9\,d^2\,x^2\,\left (d+2\,c\,f\right )}{32\,a^3\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {96\,c^3\,f^3+72\,c^2\,d\,f^2+36\,c\,d^2\,f+9\,d^3}{1024\,a^3\,f^4}+\frac {3\,d^3\,x^3}{32\,a^3\,f}+\frac {9\,d\,x\,\left (8\,c^2\,f^2+4\,c\,d\,f+d^2\right )}{256\,a^3\,f^3}+\frac {9\,d^2\,x^2\,\left (d+4\,c\,f\right )}{128\,a^3\,f^2}\right )+{\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {36\,c^3\,f^3+18\,c^2\,d\,f^2+6\,c\,d^2\,f+d^3}{1728\,a^3\,f^4}+\frac {d^3\,x^3}{48\,a^3\,f}+\frac {d\,x\,\left (18\,c^2\,f^2+6\,c\,d\,f+d^2\right )}{288\,a^3\,f^3}+\frac {d^2\,x^2\,\left (d+6\,c\,f\right )}{96\,a^3\,f^2}\right )+\frac {c^3\,x}{8\,a^3}+\frac {d^3\,x^4}{32\,a^3}+\frac {3\,c^2\,d\,x^2}{16\,a^3}+\frac {c\,d^2\,x^3}{8\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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