Optimal. Leaf size=64 \[ \frac{a \log (c+d x)}{d}+\frac{b \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.123518, 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} \[ \frac{a \log (c+d x)}{d}+\frac{b \text{CosIntegral}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (x f+\frac{c f}{d}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3317
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{a+b \sin (e+f x)}{c+d x} \, dx &=\int \left (\frac{a}{c+d x}+\frac{b \sin (e+f x)}{c+d x}\right ) \, dx\\ &=\frac{a \log (c+d x)}{d}+b \int \frac{\sin (e+f x)}{c+d x} \, dx\\ &=\frac{a \log (c+d x)}{d}+\left (b \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 (b \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{b \text{Ci}\left (\frac{c f}{d}+f x\right ) \sin \left (e-\frac{c f}{d}\right )}{d}+\frac{b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (\frac{c f}{d}+f x\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.150375, size = 57, normalized size = 0.89 \[ \frac{a \log (c+d x)+b \text{CosIntegral}\left (f \left (\frac{c}{d}+x\right )\right ) \sin \left (e-\frac{c f}{d}\right )+b \cos \left (e-\frac{c f}{d}\right ) \text{Si}\left (f \left (\frac{c}{d}+x\right )\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 96, normalized size = 1.5 \begin{align*}{\frac{a\ln \left ( \left ( fx+e \right ) d+cf-de \right ) }{d}}+{\frac{b}{d}{\it Si} \left ( fx+e+{\frac{cf-de}{d}} \right ) \cos \left ({\frac{cf-de}{d}} \right ) }-{\frac{b}{d}{\it Ci} \left ( fx+e+{\frac{cf-de}{d}} \right ) \sin \left ({\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.23759, 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 )} b}{d}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.76993, size = 234, normalized size = 3.66 \begin{align*} \frac{2 \, b \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 (b \operatorname{Ci}\left (\frac{d f x + c f}{d}\right ) + b \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \sin{\left (e + f x \right )}}{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.92385, 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.
[In]
[Out]