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 = 0.0641802, antiderivative size = 51, normalized size of antiderivative = 1., 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 4523
Rule 3296
Rule 2637
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*}
Mathematica [A] time = 0.505234, 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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Maple [A] time = 0.058, size = 78, normalized size = 1.5 \begin{align*} -{\frac{1}{a{d}^{2}} \left ( f \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) +cf\cos \left ( dx+c \right ) -de\cos \left ( dx+c \right ) -{\frac{f \left ( dx+c \right ) ^{2}}{2}}+cf \left ( dx+c \right ) -de \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52251, size = 204, normalized size = 4. \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60205, size = 117, normalized size = 2.29 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.37991, size = 439, normalized size = 8.61 \begin{align*} \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{2 d e \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 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{2 f \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{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}} + \frac{2 f}{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}{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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