Optimal. Leaf size=396 \[ -\frac {9 d^3 e^{-2 i e-2 i f x}}{64 a^3 f^4}-\frac {9 d^3 e^{-4 i e-4 i f x}}{1024 a^3 f^4}-\frac {d^3 e^{-6 i e-6 i f x}}{1728 a^3 f^4}-\frac {9 i d^2 e^{-2 i e-2 i f x} (c+d x)}{32 a^3 f^3}-\frac {9 i d^2 e^{-4 i e-4 i f x} (c+d x)}{256 a^3 f^3}-\frac {i d^2 e^{-6 i e-6 i f x} (c+d x)}{288 a^3 f^3}+\frac {9 d e^{-2 i e-2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac {9 d e^{-4 i e-4 i f x} (c+d x)^2}{128 a^3 f^2}+\frac {d e^{-6 i e-6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac {3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac {(c+d x)^4}{32 a^3 d} \]
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Rubi [A]
time = 0.27, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3810, 2207,
2225} \begin {gather*} -\frac {9 i d^2 (c+d x) e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac {9 i d^2 (c+d x) e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac {i d^2 (c+d x) e^{-6 i e-6 i f x}}{288 a^3 f^3}+\frac {9 d (c+d x)^2 e^{-2 i e-2 i f x}}{32 a^3 f^2}+\frac {9 d (c+d x)^2 e^{-4 i e-4 i f x}}{128 a^3 f^2}+\frac {d (c+d x)^2 e^{-6 i e-6 i f x}}{96 a^3 f^2}+\frac {3 i (c+d x)^3 e^{-2 i e-2 i f x}}{16 a^3 f}+\frac {3 i (c+d x)^3 e^{-4 i e-4 i f x}}{32 a^3 f}+\frac {i (c+d x)^3 e^{-6 i e-6 i f x}}{48 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}-\frac {9 d^3 e^{-2 i e-2 i f x}}{64 a^3 f^4}-\frac {9 d^3 e^{-4 i e-4 i f x}}{1024 a^3 f^4}-\frac {d^3 e^{-6 i e-6 i f x}}{1728 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+i a \tan (e+f x))^3} \, dx &=\int \left (\frac {(c+d x)^3}{8 a^3}+\frac {3 e^{-2 i e-2 i f x} (c+d x)^3}{8 a^3}+\frac {3 e^{-4 i e-4 i f x} (c+d x)^3}{8 a^3}+\frac {e^{-6 i e-6 i f x} (c+d x)^3}{8 a^3}\right ) \, dx\\ &=\frac {(c+d x)^4}{32 a^3 d}+\frac {\int e^{-6 i e-6 i f x} (c+d x)^3 \, dx}{8 a^3}+\frac {3 \int e^{-2 i e-2 i f x} (c+d x)^3 \, dx}{8 a^3}+\frac {3 \int e^{-4 i e-4 i f x} (c+d x)^3 \, dx}{8 a^3}\\ &=\frac {3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}-\frac {(i d) \int e^{-6 i e-6 i f x} (c+d x)^2 \, dx}{16 a^3 f}-\frac {(9 i d) \int e^{-4 i e-4 i f x} (c+d x)^2 \, dx}{32 a^3 f}-\frac {(9 i d) \int e^{-2 i e-2 i f x} (c+d x)^2 \, dx}{16 a^3 f}\\ &=\frac {9 d e^{-2 i e-2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac {9 d e^{-4 i e-4 i f x} (c+d x)^2}{128 a^3 f^2}+\frac {d e^{-6 i e-6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac {3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}-\frac {d^2 \int e^{-6 i e-6 i f x} (c+d x) \, dx}{48 a^3 f^2}-\frac {\left (9 d^2\right ) \int e^{-4 i e-4 i f x} (c+d x) \, dx}{64 a^3 f^2}-\frac {\left (9 d^2\right ) \int e^{-2 i e-2 i f x} (c+d x) \, dx}{16 a^3 f^2}\\ &=-\frac {9 i d^2 e^{-2 i e-2 i f x} (c+d x)}{32 a^3 f^3}-\frac {9 i d^2 e^{-4 i e-4 i f x} (c+d x)}{256 a^3 f^3}-\frac {i d^2 e^{-6 i e-6 i f x} (c+d x)}{288 a^3 f^3}+\frac {9 d e^{-2 i e-2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac {9 d e^{-4 i e-4 i f x} (c+d x)^2}{128 a^3 f^2}+\frac {d e^{-6 i e-6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac {3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}+\frac {\left (i d^3\right ) \int e^{-6 i e-6 i f x} \, dx}{288 a^3 f^3}+\frac {\left (9 i d^3\right ) \int e^{-4 i e-4 i f x} \, dx}{256 a^3 f^3}+\frac {\left (9 i d^3\right ) \int e^{-2 i e-2 i f x} \, dx}{32 a^3 f^3}\\ &=-\frac {9 d^3 e^{-2 i e-2 i f x}}{64 a^3 f^4}-\frac {9 d^3 e^{-4 i e-4 i f x}}{1024 a^3 f^4}-\frac {d^3 e^{-6 i e-6 i f x}}{1728 a^3 f^4}-\frac {9 i d^2 e^{-2 i e-2 i f x} (c+d x)}{32 a^3 f^3}-\frac {9 i d^2 e^{-4 i e-4 i f x} (c+d x)}{256 a^3 f^3}-\frac {i d^2 e^{-6 i e-6 i f x} (c+d x)}{288 a^3 f^3}+\frac {9 d e^{-2 i e-2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac {9 d e^{-4 i e-4 i f x} (c+d x)^2}{128 a^3 f^2}+\frac {d e^{-6 i e-6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac {3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 2.43, size = 667, normalized size = 1.68 \begin {gather*} \frac {i \sec ^3(e+f x) \left (243 \left (32 i c^3 f^3+8 c^2 d f^2 (5+12 i f x)+4 c d^2 f \left (-9 i+20 f x+24 i f^2 x^2\right )+d^3 \left (-17-36 i f x+40 f^2 x^2+32 i f^3 x^3\right )\right ) \cos (e+f x)+16 \left (36 c^3 f^3 (i+6 f x)+18 c^2 d f^2 \left (1+6 i f x+18 f^2 x^2\right )+6 c d^2 f \left (-i+6 f x+18 i f^2 x^2+36 f^3 x^3\right )+d^3 \left (-1-6 i f x+18 f^2 x^2+36 i f^3 x^3+54 f^4 x^4\right )\right ) \cos (3 (e+f x))-3645 i d^3 \sin (e+f x)+6804 c d^2 f \sin (e+f x)+5832 i c^2 d f^2 \sin (e+f x)-2592 c^3 f^3 \sin (e+f x)+6804 d^3 f x \sin (e+f x)+11664 i c d^2 f^2 x \sin (e+f x)-7776 c^2 d f^3 x \sin (e+f x)+5832 i d^3 f^2 x^2 \sin (e+f x)-7776 c d^2 f^3 x^2 \sin (e+f x)-2592 d^3 f^3 x^3 \sin (e+f x)+16 i d^3 \sin (3 (e+f x))-96 c d^2 f \sin (3 (e+f x))-288 i c^2 d f^2 \sin (3 (e+f x))+576 c^3 f^3 \sin (3 (e+f x))-96 d^3 f x \sin (3 (e+f x))-576 i c d^2 f^2 x \sin (3 (e+f x))+1728 c^2 d f^3 x \sin (3 (e+f x))+3456 i c^3 f^4 x \sin (3 (e+f x))-288 i d^3 f^2 x^2 \sin (3 (e+f x))+1728 c d^2 f^3 x^2 \sin (3 (e+f x))+5184 i c^2 d f^4 x^2 \sin (3 (e+f x))+576 d^3 f^3 x^3 \sin (3 (e+f x))+3456 i c d^2 f^4 x^3 \sin (3 (e+f x))+864 i d^3 f^4 x^4 \sin (3 (e+f x))\right )}{27648 a^3 f^4 (-i+\tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 3395 vs. \(2 (328 ) = 656\).
time = 0.62, size = 3396, normalized size = 8.58
method | result | size |
risch | \(\frac {d^{3} x^{4}}{32 a^{3}}+\frac {d^{2} c \,x^{3}}{8 a^{3}}+\frac {3 d \,c^{2} x^{2}}{16 a^{3}}+\frac {c^{3} x}{8 a^{3}}+\frac {c^{4}}{32 a^{3} d}+\frac {3 i \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}-6 i d^{3} f^{2} x^{2}+12 c^{2} d \,f^{3} x -12 i c \,d^{2} f^{2} x +4 c^{3} f^{3}-6 i c^{2} d \,f^{2}-6 d^{3} f x -6 c \,d^{2} f +3 i d^{3}\right ) {\mathrm e}^{-2 i \left (f x +e \right )}}{64 a^{3} f^{4}}+\frac {3 i \left (32 d^{3} x^{3} f^{3}+96 c \,d^{2} f^{3} x^{2}-24 i d^{3} f^{2} x^{2}+96 c^{2} d \,f^{3} x -48 i c \,d^{2} f^{2} x +32 c^{3} f^{3}-24 i c^{2} d \,f^{2}-12 d^{3} f x -12 c \,d^{2} f +3 i d^{3}\right ) {\mathrm e}^{-4 i \left (f x +e \right )}}{1024 a^{3} f^{4}}+\frac {i \left (36 d^{3} x^{3} f^{3}+108 c \,d^{2} f^{3} x^{2}-18 i d^{3} f^{2} x^{2}+108 c^{2} d \,f^{3} x -36 i c \,d^{2} f^{2} x +36 c^{3} f^{3}-18 i c^{2} d \,f^{2}-6 d^{3} f x -6 c \,d^{2} f +i d^{3}\right ) {\mathrm e}^{-6 i \left (f x +e \right )}}{1728 a^{3} f^{4}}\) | \(396\) |
default | \(\text {Expression too large to display}\) | \(3396\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 374, normalized size = 0.94 \begin {gather*} \frac {{\left (576 i \, d^{3} f^{3} x^{3} + 576 i \, c^{3} f^{3} + 288 \, c^{2} d f^{2} - 96 i \, c d^{2} f - 16 \, d^{3} - 288 \, {\left (-6 i \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} - 96 \, {\left (-18 i \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} + i \, d^{3} f\right )} x + 864 \, {\left (d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} + 6 \, c^{2} d f^{4} x^{2} + 4 \, c^{3} f^{4} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 1296 \, {\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, c^{3} f^{3} - 6 \, c^{2} d f^{2} + 6 i \, c d^{2} f + 3 \, d^{3} + 6 \, {\left (-2 i \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 6 \, {\left (-2 i \, c^{2} d f^{3} - 2 \, c d^{2} f^{2} + i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 81 \, {\left (-32 i \, d^{3} f^{3} x^{3} - 32 i \, c^{3} f^{3} - 24 \, c^{2} d f^{2} + 12 i \, c d^{2} f + 3 \, d^{3} + 24 \, {\left (-4 i \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 12 \, {\left (-8 i \, c^{2} d f^{3} - 4 \, c d^{2} f^{2} + i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{27648 \, a^{3} f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 945, normalized size = 2.39 \begin {gather*} \begin {cases} \frac {\left (\left (2359296 i a^{6} c^{3} f^{11} e^{6 i e} + 7077888 i a^{6} c^{2} d f^{11} x e^{6 i e} + 1179648 a^{6} c^{2} d f^{10} e^{6 i e} + 7077888 i a^{6} c d^{2} f^{11} x^{2} e^{6 i e} + 2359296 a^{6} c d^{2} f^{10} x e^{6 i e} - 393216 i a^{6} c d^{2} f^{9} e^{6 i e} + 2359296 i a^{6} d^{3} f^{11} x^{3} e^{6 i e} + 1179648 a^{6} d^{3} f^{10} x^{2} e^{6 i e} - 393216 i a^{6} d^{3} f^{9} x e^{6 i e} - 65536 a^{6} d^{3} f^{8} e^{6 i e}\right ) e^{- 6 i f x} + \left (10616832 i a^{6} c^{3} f^{11} e^{8 i e} + 31850496 i a^{6} c^{2} d f^{11} x e^{8 i e} + 7962624 a^{6} c^{2} d f^{10} e^{8 i e} + 31850496 i a^{6} c d^{2} f^{11} x^{2} e^{8 i e} + 15925248 a^{6} c d^{2} f^{10} x e^{8 i e} - 3981312 i a^{6} c d^{2} f^{9} e^{8 i e} + 10616832 i a^{6} d^{3} f^{11} x^{3} e^{8 i e} + 7962624 a^{6} d^{3} f^{10} x^{2} e^{8 i e} - 3981312 i a^{6} d^{3} f^{9} x e^{8 i e} - 995328 a^{6} d^{3} f^{8} e^{8 i e}\right ) e^{- 4 i f x} + \left (21233664 i a^{6} c^{3} f^{11} e^{10 i e} + 63700992 i a^{6} c^{2} d f^{11} x e^{10 i e} + 31850496 a^{6} c^{2} d f^{10} e^{10 i e} + 63700992 i a^{6} c d^{2} f^{11} x^{2} e^{10 i e} + 63700992 a^{6} c d^{2} f^{10} x e^{10 i e} - 31850496 i a^{6} c d^{2} f^{9} e^{10 i e} + 21233664 i a^{6} d^{3} f^{11} x^{3} e^{10 i e} + 31850496 a^{6} d^{3} f^{10} x^{2} e^{10 i e} - 31850496 i a^{6} d^{3} f^{9} x e^{10 i e} - 15925248 a^{6} d^{3} f^{8} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{113246208 a^{9} f^{12}} & \text {for}\: a^{9} f^{12} e^{12 i e} \neq 0 \\\frac {x^{4} \cdot \left (3 d^{3} e^{4 i e} + 3 d^{3} e^{2 i e} + d^{3}\right ) e^{- 6 i e}}{32 a^{3}} + \frac {x^{3} \cdot \left (3 c d^{2} e^{4 i e} + 3 c d^{2} e^{2 i e} + c d^{2}\right ) e^{- 6 i e}}{8 a^{3}} + \frac {x^{2} \cdot \left (9 c^{2} d e^{4 i e} + 9 c^{2} d e^{2 i e} + 3 c^{2} d\right ) e^{- 6 i e}}{16 a^{3}} + \frac {x \left (3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3}\right ) e^{- 6 i e}}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c^{3} x}{8 a^{3}} + \frac {3 c^{2} d x^{2}}{16 a^{3}} + \frac {c d^{2} x^{3}}{8 a^{3}} + \frac {d^{3} x^{4}}{32 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.75, size = 573, normalized size = 1.45 \begin {gather*} \frac {{\left (864 \, d^{3} f^{4} x^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 3456 \, c d^{2} f^{4} x^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 5184 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 576 i \, d^{3} f^{3} x^{3} + 3456 \, c^{3} f^{4} x e^{\left (6 i \, f x + 6 i \, e\right )} + 15552 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 1728 i \, c d^{2} f^{3} x^{2} + 15552 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} + 1944 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 1728 i \, c^{2} d f^{3} x + 288 \, d^{3} f^{2} x^{2} + 5184 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 15552 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3888 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 576 i \, c^{3} f^{3} + 576 \, c d^{2} f^{2} x + 7776 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 7776 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 1944 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 972 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 288 \, c^{2} d f^{2} - 96 i \, d^{3} f x - 7776 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 972 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 96 i \, c d^{2} f - 3888 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 243 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 16 \, d^{3}\right )} e^{\left (-6 i \, f x - 6 i \, 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 = 3.85, size = 411, normalized size = 1.04 \begin {gather*} {\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (12\,c^3\,f^3-c^2\,d\,f^2\,18{}\mathrm {i}-18\,c\,d^2\,f+d^3\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^3\,f^4}+\frac {d^3\,x^3\,3{}\mathrm {i}}{16\,a^3\,f}-\frac {d\,x\,\left (-2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}+d^2\right )\,9{}\mathrm {i}}{32\,a^3\,f^3}-\frac {d^2\,x^2\,\left (-2\,c\,f+d\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{32\,a^3\,f^2}\right )+{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (96\,c^3\,f^3-c^2\,d\,f^2\,72{}\mathrm {i}-36\,c\,d^2\,f+d^3\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{1024\,a^3\,f^4}+\frac {d^3\,x^3\,3{}\mathrm {i}}{32\,a^3\,f}-\frac {d\,x\,\left (-8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}+d^2\right )\,9{}\mathrm {i}}{256\,a^3\,f^3}-\frac {d^2\,x^2\,\left (-4\,c\,f+d\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{128\,a^3\,f^2}\right )+{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (36\,c^3\,f^3-c^2\,d\,f^2\,18{}\mathrm {i}-6\,c\,d^2\,f+d^3\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{1728\,a^3\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{48\,a^3\,f}-\frac {d\,x\,\left (-18\,c^2\,f^2+c\,d\,f\,6{}\mathrm {i}+d^2\right )\,1{}\mathrm {i}}{288\,a^3\,f^3}-\frac {d^2\,x^2\,\left (-6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{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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