Optimal. Leaf size=100 \[ \frac{\cos \left (\frac{b}{d}\right ) (b c-a d) \text{CosIntegral}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}+\frac{\sin \left (\frac{b}{d}\right ) (b c-a d) \text{Si}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \sin \left (\frac{a+b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.163526, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {4563, 3297, 3303, 3299, 3302} \[ \frac{\cos \left (\frac{b}{d}\right ) (b c-a d) \text{CosIntegral}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}+\frac{\sin \left (\frac{b}{d}\right ) (b c-a d) \text{Si}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \sin \left (\frac{a+b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4563
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \sin \left (\frac{a+b x}{c+d x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sin \left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \sin \left (\frac{a+b x}{c+d x}\right )}{d}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sin \left (\frac{a+b x}{c+d x}\right )}{d}+\frac{\left ((b c-a d) \cos \left (\frac{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}+\frac{\left ((b c-a d) \sin \left (\frac{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(b c-a d) \cos \left (\frac{b}{d}\right ) \text{Ci}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \sin \left (\frac{a+b x}{c+d x}\right )}{d}+\frac{(b c-a d) \sin \left (\frac{b}{d}\right ) \text{Si}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}\\ \end{align*}
Mathematica [C] time = 5.51914, size = 272, normalized size = 2.72 \[ \frac{2 \cos \left (\frac{b}{d}\right ) (b c-a d) \text{CosIntegral}\left (\frac{a d-b c}{d (c+d x)}\right )+d \exp \left (-\frac{i (a d+2 b c+b d x)}{d (c+d x)}\right ) \left (i c \left (e^{\frac{2 i b c}{d (c+d x)}}-e^{2 i \left (\frac{a}{c+d x}+\frac{b}{d}\right )}\right )+d x \sin \left (\frac{b}{d}\right ) \left (e^{i \left (\frac{2 a}{c+d x}+\frac{b}{d}\right )}+e^{\frac{i b (3 c+d x)}{d (c+d x)}}\right )+2 d x \cos \left (\frac{b}{d}\right ) e^{\frac{i (a d+2 b c+b d x)}{d (c+d x)}} \sin \left (\frac{a d-b c}{d (c+d x)}\right )\right )-2 \sin \left (\frac{b}{d}\right ) (b c-a d) \text{Si}\left (\frac{a d-b c}{d (c+d x)}\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.013, size = 142, normalized size = 1.4 \begin{align*} - \left ( ad-cb \right ) \left ( -{\frac{1}{d}\sin \left ({\frac{b}{d}}+{\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) \left ( d \left ({\frac{b}{d}}+{\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) -b \right ) ^{-1}}+{\frac{1}{d} \left ( -{\frac{1}{d}{\it Si} \left ({\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) \sin \left ({\frac{b}{d}} \right ) }+{\frac{1}{d}{\it Ci} \left ({\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) \cos \left ({\frac{b}{d}} \right ) } \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (\frac{b x + a}{d x + c}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2517, size = 323, normalized size = 3.23 \begin{align*} -\frac{2 \,{\left (b c - a d\right )} \sin \left (\frac{b}{d}\right ) \operatorname{Si}\left (-\frac{b c - a d}{d^{2} x + c d}\right ) -{\left ({\left (b c - a d\right )} \operatorname{Ci}\left (\frac{b c - a d}{d^{2} x + c d}\right ) +{\left (b c - a d\right )} \operatorname{Ci}\left (-\frac{b c - a d}{d^{2} x + c d}\right )\right )} \cos \left (\frac{b}{d}\right ) - 2 \,{\left (d^{2} x + c d\right )} \sin \left (\frac{b x + a}{d x + c}\right )}{2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (\frac{b x + a}{d x + c}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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