Optimal. Leaf size=161 \[ \frac {f^2 \sin ^2(c+d x)}{4 a d^3}-\frac {2 f^2 \sin (c+d x)}{a d^3}+\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {f (e+f x) \sin (c+d x) \cos (c+d x)}{2 a d^2}-\frac {(e+f x)^2 \sin ^2(c+d x)}{2 a d}+\frac {(e+f x)^2 \sin (c+d x)}{a d}+\frac {e f x}{2 a d}+\frac {f^2 x^2}{4 a d} \]
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Rubi [A] time = 0.17, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4523, 3296, 2637, 4404, 3310} \[ \frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {f (e+f x) \sin (c+d x) \cos (c+d x)}{2 a d^2}+\frac {f^2 \sin ^2(c+d x)}{4 a d^3}-\frac {2 f^2 \sin (c+d x)}{a d^3}-\frac {(e+f x)^2 \sin ^2(c+d x)}{2 a d}+\frac {(e+f x)^2 \sin (c+d x)}{a d}+\frac {e f x}{2 a d}+\frac {f^2 x^2}{4 a d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3310
Rule 4404
Rule 4523
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \cos (c+d x) \, dx}{a}-\frac {\int (e+f x)^2 \cos (c+d x) \sin (c+d x) \, dx}{a}\\ &=\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {(e+f x)^2 \sin ^2(c+d x)}{2 a d}+\frac {f \int (e+f x) \sin ^2(c+d x) \, dx}{a d}-\frac {(2 f) \int (e+f x) \sin (c+d x) \, dx}{a d}\\ &=\frac {2 f (e+f x) \cos (c+d x)}{a d^2}+\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a d^2}+\frac {f^2 \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^2 \sin ^2(c+d x)}{2 a d}+\frac {f \int (e+f x) \, dx}{2 a d}-\frac {\left (2 f^2\right ) \int \cos (c+d x) \, dx}{a d^2}\\ &=\frac {e f x}{2 a d}+\frac {f^2 x^2}{4 a d}+\frac {2 f (e+f x) \cos (c+d x)}{a d^2}-\frac {2 f^2 \sin (c+d x)}{a d^3}+\frac {(e+f x)^2 \sin (c+d x)}{a d}-\frac {f (e+f x) \cos (c+d x) \sin (c+d x)}{2 a d^2}+\frac {f^2 \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^2 \sin ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 95, normalized size = 0.59 \[ \frac {\cos (2 (c+d x)) \left (2 d^2 (e+f x)^2-f^2\right )-4 \sin (c+d x) \left (d f (e+f x) \cos (c+d x)-2 \left (d^2 (e+f x)^2-2 f^2\right )\right )+16 d f (e+f x) \cos (c+d x)}{8 a d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 149, normalized size = 0.93 \[ -\frac {d^{2} f^{2} x^{2} + 2 \, d^{2} e f x - {\left (2 \, d^{2} f^{2} x^{2} + 4 \, d^{2} e f x + 2 \, d^{2} e^{2} - f^{2}\right )} \cos \left (d x + c\right )^{2} - 8 \, {\left (d f^{2} x + d e f\right )} \cos \left (d x + c\right ) - 2 \, {\left (2 \, d^{2} f^{2} x^{2} + 4 \, d^{2} e f x + 2 \, d^{2} e^{2} - 4 \, f^{2} - {\left (d f^{2} x + d e f\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, a d^{3}} \]
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 = 0.11, size = 339, normalized size = 2.11 \[ -\frac {f^{2} \left (-\frac {\left (d x +c \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}\right )-2 c \,f^{2} \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+2 d e f \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-\frac {c^{2} f^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+c d e f \left (\cos ^{2}\left (d x +c \right )\right )-\frac {d^{2} e^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}-f^{2} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \left (d x +c \right ) \cos \left (d x +c \right )\right )+2 c \,f^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-2 d e f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-\sin \left (d x +c \right ) c^{2} f^{2}+2 \sin \left (d x +c \right ) c d e f -\sin \left (d x +c \right ) d^{2} e^{2}}{d^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 289, normalized size = 1.80 \[ -\frac {\frac {4 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} e^{2}}{a} - \frac {8 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c e f}{a d} + \frac {4 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c^{2} f^{2}}{a d^{2}} - \frac {2 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} e f}{a d} + \frac {2 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} c f^{2}}{a d^{2}} - \frac {{\left ({\left (2 \, {\left (d x + c\right )}^{2} - 1\right )} \cos \left (2 \, d x + 2 \, c\right ) + 16 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (2 \, d x + 2 \, c\right ) + 8 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} f^{2}}{a d^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.20, size = 187, normalized size = 1.16 \[ \frac {8\,d^2\,e^2\,\sin \left (c+d\,x\right )-f^2\,\cos \left (2\,c+2\,d\,x\right )-16\,f^2\,\sin \left (c+d\,x\right )+2\,d^2\,e^2\,\cos \left (2\,c+2\,d\,x\right )+8\,d^2\,f^2\,x^2\,\sin \left (c+d\,x\right )-2\,d\,e\,f\,\sin \left (2\,c+2\,d\,x\right )+16\,d\,f^2\,x\,\cos \left (c+d\,x\right )+2\,d^2\,f^2\,x^2\,\cos \left (2\,c+2\,d\,x\right )-2\,d\,f^2\,x\,\sin \left (2\,c+2\,d\,x\right )+16\,d\,e\,f\,\cos \left (c+d\,x\right )+4\,d^2\,e\,f\,x\,\cos \left (2\,c+2\,d\,x\right )+16\,d^2\,e\,f\,x\,\sin \left (c+d\,x\right )}{8\,a\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.35, size = 1528, normalized size = 9.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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