Optimal. Leaf size=91 \[ \frac {f \cos (c+d x)}{a d^2}-\frac {f \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}+\frac {(e+f x) \sin (c+d x)}{a d}+\frac {f x}{4 a d} \]
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Rubi [A] time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4523, 3296, 2638, 4404, 2635, 8} \[ \frac {f \cos (c+d x)}{a d^2}-\frac {f \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}+\frac {(e+f x) \sin (c+d x)}{a d}+\frac {f x}{4 a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 3296
Rule 4404
Rule 4523
Rubi steps
\begin {align*} \int \frac {(e+f x) \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x) \cos (c+d x) \, dx}{a}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x) \, dx}{a}\\ &=\frac {(e+f x) \sin (c+d x)}{a d}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}+\frac {f \int \sin ^2(c+d x) \, dx}{2 a d}-\frac {f \int \sin (c+d x) \, dx}{a d}\\ &=\frac {f \cos (c+d x)}{a d^2}+\frac {(e+f x) \sin (c+d x)}{a d}-\frac {f \cos (c+d x) \sin (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}+\frac {f \int 1 \, dx}{4 a d}\\ &=\frac {f x}{4 a d}+\frac {f \cos (c+d x)}{a d^2}+\frac {(e+f x) \sin (c+d x)}{a d}-\frac {f \cos (c+d x) \sin (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 52, normalized size = 0.57 \[ \frac {d (e+f x) (4 \sin (c+d x)+\cos (2 (c+d x)))-f (\sin (c+d x)-4) \cos (c+d x)}{4 a d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 67, normalized size = 0.74 \[ -\frac {d f x - 2 \, {\left (d f x + d e\right )} \cos \left (d x + c\right )^{2} - 4 \, f \cos \left (d x + c\right ) - {\left (4 \, d f x + 4 \, d e - f \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, a d^{2}} \]
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 [A] time = 0.11, size = 114, normalized size = 1.25 \[ -\frac {f \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+\frac {c f \left (\cos ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\cos ^{2}\left (d x +c \right )\right ) d e}{2}-f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+\sin \left (d x +c \right ) c f -\sin \left (d x +c \right ) d e}{d^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 114, normalized size = 1.25 \[ -\frac {\frac {4 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} e}{a} - \frac {4 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c f}{a d} - \frac {{\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} f}{a d}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.04, size = 84, normalized size = 0.92 \[ -\frac {\frac {f\,\sin \left (2\,c+2\,d\,x\right )}{2}+8\,f\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,d\,e\,\sin \left (c+d\,x\right )+2\,d\,e\,{\sin \left (c+d\,x\right )}^2-4\,d\,f\,x\,\sin \left (c+d\,x\right )+d\,f\,x\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )}{4\,a\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.99, size = 724, normalized size = 7.96 \[ \begin {cases} \frac {8 d e \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} - \frac {8 d e \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 d e \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {d f x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 d f x \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} - \frac {6 d f x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 d f x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {d f x}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {2 f \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 f \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} - \frac {2 f \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 f}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} & \text {for}\: d \neq 0 \\\frac {\left (e x + \frac {f x^{2}}{2}\right ) \cos ^{3}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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