Optimal. Leaf size=219 \[ -\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {3 f^3 \sin (c+d x) \cos (c+d x)}{8 a d^4}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 f (e+f x)^2 \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {3 f^3 x}{8 a d^3}+\frac {(e+f x)^3}{4 a d} \]
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Rubi [A] time = 0.24, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4523, 3296, 2638, 4404, 3311, 32, 2635, 8} \[ \frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 f (e+f x)^2 \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {3 f^3 \sin (c+d x) \cos (c+d x)}{8 a d^4}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {3 f^3 x}{8 a d^3}+\frac {(e+f x)^3}{4 a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 4404
Rule 4523
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cos (c+d x) \, dx}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x) \, dx}{a}\\ &=\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}+\frac {(3 f) \int (e+f x)^2 \sin ^2(c+d x) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d}\\ &=\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}+\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {3 f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a d^2}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}+\frac {(3 f) \int (e+f x)^2 \, dx}{4 a d}-\frac {\left (6 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}-\frac {\left (3 f^3\right ) \int \sin ^2(c+d x) \, dx}{4 a d^3}\\ &=\frac {(e+f x)^3}{4 a d}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {(e+f x)^3 \sin (c+d x)}{a d}+\frac {3 f^3 \cos (c+d x) \sin (c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a d^2}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}-\frac {\left (3 f^3\right ) \int 1 \, dx}{8 a d^3}+\frac {\left (6 f^3\right ) \int \sin (c+d x) \, dx}{a d^3}\\ &=-\frac {3 f^3 x}{8 a d^3}+\frac {(e+f x)^3}{4 a d}-\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {(e+f x)^3 \sin (c+d x)}{a d}+\frac {3 f^3 \cos (c+d x) \sin (c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a d^2}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 1.36, size = 132, normalized size = 0.60 \[ \frac {96 f \cos (c+d x) \left (d^2 (e+f x)^2-2 f^2\right )+4 d (e+f x) \cos (2 (c+d x)) \left (2 d^2 (e+f x)^2-3 f^2\right )+4 \sin (c+d x) \left (8 d (e+f x) \left (d^2 (e+f x)^2-6 f^2\right )-3 f \cos (c+d x) \left (2 d^2 (e+f x)^2-f^2\right )\right )}{32 a d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 270, normalized size = 1.23 \[ -\frac {2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} - 2 \, {\left (2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} + 2 \, d^{3} e^{3} - 3 \, d e f^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x - 24 \, {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} e f^{2} x + d^{2} e^{2} f - 2 \, f^{3}\right )} \cos \left (d x + c\right ) - {\left (8 \, d^{3} f^{3} x^{3} + 24 \, d^{3} e f^{2} x^{2} + 8 \, d^{3} e^{3} - 48 \, d e f^{2} + 24 \, {\left (d^{3} e^{2} f - 2 \, d f^{3}\right )} x - 3 \, {\left (2 \, d^{2} f^{3} x^{2} + 4 \, d^{2} e f^{2} x + 2 \, d^{2} e^{2} f - f^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, a d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 737, normalized size = 3.37 \[ -\frac {f^{3} \left (-\frac {\left (d x +c \right )^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {3 \left (d x +c \right )^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {3 \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{4}-\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}-\frac {\left (d x +c \right )^{3}}{2}\right )-3 c \,f^{3} \left (-\frac {\left (d x +c \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}\right )+3 f^{2} e d \left (-\frac {\left (d x +c \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}\right )+3 c^{2} f^{3} \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-6 c d e \,f^{2} \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+3 d^{2} e^{2} f \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+\frac {c^{3} f^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}-\frac {3 c^{2} d e \,f^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {3 c \,d^{2} e^{2} f \left (\cos ^{2}\left (d x +c \right )\right )}{2}-\frac {d^{3} e^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}-f^{3} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )+3 c \,f^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \left (d x +c \right ) \cos \left (d x +c \right )\right )-3 f^{2} e d \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \left (d x +c \right ) \cos \left (d x +c \right )\right )-3 c^{2} f^{3} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+6 c d e \,f^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 d^{2} e^{2} f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+\sin \left (d x +c \right ) c^{3} f^{3}-3 \sin \left (d x +c \right ) c^{2} d e \,f^{2}+3 \sin \left (d x +c \right ) c \,d^{2} e^{2} f -\sin \left (d x +c \right ) d^{3} e^{3}}{d^{4} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.79, size = 572, normalized size = 2.61 \[ -\frac {\frac {8 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} e^{3}}{a} - \frac {24 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c e^{2} f}{a d} + \frac {24 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c^{2} e f^{2}}{a d^{2}} - \frac {8 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c^{3} f^{3}}{a d^{3}} - \frac {6 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} e^{2} f}{a d} + \frac {12 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} c e f^{2}}{a d^{2}} - \frac {6 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} c^{2} f^{3}}{a d^{3}} - \frac {6 \, {\left ({\left (2 \, {\left (d x + c\right )}^{2} - 1\right )} \cos \left (2 \, d x + 2 \, c\right ) + 16 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (2 \, d x + 2 \, c\right ) + 8 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} e f^{2}}{a d^{2}} + \frac {6 \, {\left ({\left (2 \, {\left (d x + c\right )}^{2} - 1\right )} \cos \left (2 \, d x + 2 \, c\right ) + 16 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (2 \, d x + 2 \, c\right ) + 8 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} c f^{3}}{a d^{3}} - \frac {{\left (2 \, {\left (2 \, {\left (d x + c\right )}^{3} - 3 \, d x - 3 \, c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 48 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, {\left (d x + c\right )}^{2} - 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + 16 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} f^{3}}{a d^{3}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 339, normalized size = 1.55 \[ \frac {\frac {3\,f^3\,\sin \left (2\,c+2\,d\,x\right )}{2}-48\,f^3\,\cos \left (c+d\,x\right )+8\,d^3\,e^3\,\sin \left (c+d\,x\right )+2\,d^3\,e^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,e^2\,f\,\sin \left (2\,c+2\,d\,x\right )+24\,d^2\,f^3\,x^2\,\cos \left (c+d\,x\right )+8\,d^3\,f^3\,x^3\,\sin \left (c+d\,x\right )-48\,d\,e\,f^2\,\sin \left (c+d\,x\right )-48\,d\,f^3\,x\,\sin \left (c+d\,x\right )+2\,d^3\,f^3\,x^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,f^3\,x^2\,\sin \left (2\,c+2\,d\,x\right )-3\,d\,e\,f^2\,\cos \left (2\,c+2\,d\,x\right )+24\,d^2\,e^2\,f\,\cos \left (c+d\,x\right )-3\,d\,f^3\,x\,\cos \left (2\,c+2\,d\,x\right )+48\,d^2\,e\,f^2\,x\,\cos \left (c+d\,x\right )+24\,d^3\,e^2\,f\,x\,\sin \left (c+d\,x\right )+6\,d^3\,e^2\,f\,x\,\cos \left (2\,c+2\,d\,x\right )-6\,d^2\,e\,f^2\,x\,\sin \left (2\,c+2\,d\,x\right )+24\,d^3\,e\,f^2\,x^2\,\sin \left (c+d\,x\right )+6\,d^3\,e\,f^2\,x^2\,\cos \left (2\,c+2\,d\,x\right )}{8\,a\,d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.78, size = 2725, normalized size = 12.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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