Optimal. Leaf size=64 \[ \frac{a \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{a \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d}+\frac{a \log (c+d x)}{d} \]
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Rubi [A] time = 0.150089, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3317, 3303, 3299, 3302} \[ \frac{a \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{a \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d}+\frac{a \log (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{a+a \sin (e+f x)}{c+d x} \, dx &=\int \left (\frac{a}{c+d x}+\frac{a \sin (e+f x)}{c+d x}\right ) \, dx\\ &=\frac{a \log (c+d x)}{d}+a \int \frac{\sin (e+f x)}{c+d x} \, dx\\ &=\frac{a \log (c+d x)}{d}+\left (a \cos \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sin \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx+\left (a \sin \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cos \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac{a \log (c+d x)}{d}+\frac{a \text{Ci}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{a \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (\frac{c f}{d}+f x\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.294371, size = 54, normalized size = 0.84 \[ \frac{a \left (\text{CosIntegral}\left (f \left (\frac{c}{d}+x\right )\right ) \sin \left (e-\frac{c f}{d}\right )+\cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (f \left (\frac{c}{d}+x\right )\right )+\log (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 96, normalized size = 1.5 \begin{align*}{\frac{a}{d}{\it Si} \left ( fx+e+{\frac{cf-de}{d}} \right ) \cos \left ({\frac{cf-de}{d}} \right ) }-{\frac{a}{d}{\it Ci} \left ( fx+e+{\frac{cf-de}{d}} \right ) \sin \left ({\frac{cf-de}{d}} \right ) }+{\frac{a\ln \left ( \left ( fx+e \right ) d+cf-de \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.20674, size = 231, normalized size = 3.61 \begin{align*} \frac{\frac{2 \, a f \log \left (c + \frac{{\left (f x + e\right )} d}{f} - \frac{d e}{f}\right )}{d} + \frac{{\left (f{\left (-i \, E_{1}\left (\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{1}\left (-\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac{d e - c f}{d}\right ) + f{\left (E_{1}\left (\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{1}\left (-\frac{i \,{\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac{d e - c f}{d}\right )\right )} a}{d}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66127, size = 234, normalized size = 3.66 \begin{align*} \frac{2 \, a \cos \left (-\frac{d e - c f}{d}\right ) \operatorname{Si}\left (\frac{d f x + c f}{d}\right ) + 2 \, a \log \left (d x + c\right ) -{\left (a \operatorname{Ci}\left (\frac{d f x + c f}{d}\right ) + a \operatorname{Ci}\left (-\frac{d f x + c f}{d}\right )\right )} \sin \left (-\frac{d e - c f}{d}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{\sin{\left (e + f x \right )}}{c + d x}\, dx + \int \frac{1}{c + d x}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.25651, size = 961, normalized size = 15.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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