Optimal. Leaf size=51 \[ -\frac {f \sin (c+d x)}{a d^2}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {e x}{a}+\frac {f x^2}{2 a} \]
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Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {4523, 3296, 2637} \[ -\frac {f \sin (c+d x)}{a d^2}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {e x}{a}+\frac {f x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 4523
Rubi steps
\begin {align*} \int \frac {(e+f x) \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x) \, dx}{a}-\frac {\int (e+f x) \sin (c+d x) \, dx}{a}\\ &=\frac {e x}{a}+\frac {f x^2}{2 a}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {f \int \cos (c+d x) \, dx}{a d}\\ &=\frac {e x}{a}+\frac {f x^2}{2 a}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {f \sin (c+d x)}{a d^2}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 53, normalized size = 1.04 \[ -\frac {(c+d x) (c f-2 d e-d f x)-2 d (e+f x) \cos (c+d x)+2 f \sin (c+d x)}{2 a d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 49, normalized size = 0.96 \[ \frac {d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, {\left (d f x + d e\right )} \cos \left (d x + c\right ) - 2 \, f \sin \left (d x + c\right )}{2 \, a d^{2}} \]
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 = 0.12, size = 78, normalized size = 1.53 \[ -\frac {f \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )+c f \cos \left (d x +c \right )-d e \cos \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}+c f \left (d x +c \right )-d e \left (d x +c \right )}{d^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 151, normalized size = 2.96 \[ -\frac {4 \, c f {\left (\frac {1}{a d + \frac {a d \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a d}\right )} - 4 \, e {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}\right )} - \frac {{\left ({\left (d x + c\right )}^{2} + 2 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right )\right )} f}{a d}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.94, size = 53, normalized size = 1.04 \[ \frac {\frac {f\,x^2}{2}+e\,x}{a}-\frac {f\,\sin \left (c+d\,x\right )-d\,\left (e\,\cos \left (c+d\,x\right )+f\,x\,\cos \left (c+d\,x\right )\right )}{a\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.16, size = 326, normalized size = 6.39 \[ \begin {cases} \frac {2 d^{2} e x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 d^{2} e x}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {d^{2} f x^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {d^{2} f x^{2}}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {4 d e}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {2 d f x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 d f x}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {4 f \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} & \text {for}\: d \neq 0 \\\frac {\left (e x + \frac {f x^{2}}{2}\right ) \cos ^{2}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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