Optimal. Leaf size=276 \[ \frac{\sin \left (a-\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}+\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\sin \left (a+\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}-\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{\cos \left (a+\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{Si}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}-\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{Si}\left (\frac{\sqrt{f} b}{\sqrt{c f-d e}}+\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{f} \]
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Rubi [A] time = 1.20285, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {3431, 3303, 3299, 3302, 3345} \[ \frac{\sin \left (a-\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}+\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\sin \left (a+\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}-\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{\cos \left (a+\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{Si}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}-\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{Si}\left (\frac{\sqrt{f} b}{\sqrt{c f-d e}}+\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3431
Rule 3303
Rule 3299
Rule 3302
Rule 3345
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{e+f x} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \left (\frac{d \sin (a+b x)}{f x}+\frac{d (-d e+c f) x \sin (a+b x)}{f \left (f+(d e-c f) x^2\right )}\right ) \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}+\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int \frac{x \sin (a+b x)}{f+(d e-c f) x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}\\ &=\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d e+c f} \sin (a+b x)}{2 (d e-c f) \left (\sqrt{f}-\sqrt{-d e+c f} x\right )}+\frac{\sqrt{-d e+c f} \sin (a+b x)}{2 (d e-c f) \left (\sqrt{f}+\sqrt{-d e+c f} x\right )}\right ) \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}-\frac{(2 \cos (a)) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}-\frac{(2 \sin (a)) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}\\ &=-\frac{2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{f}-\frac{2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{\sqrt{-d e+c f} \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\sqrt{f}-\sqrt{-d e+c f} x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}+\frac{\sqrt{-d e+c f} \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\sqrt{f}+\sqrt{-d e+c f} x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}\\ &=-\frac{2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{f}-\frac{2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\left (\sqrt{-d e+c f} \cos \left (a-\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}+b x\right )}{\sqrt{f}+\sqrt{-d e+c f} x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}+\frac{\left (\sqrt{-d e+c f} \cos \left (a+\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}-b x\right )}{\sqrt{f}-\sqrt{-d e+c f} x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}+\frac{\left (\sqrt{-d e+c f} \sin \left (a-\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}+b x\right )}{\sqrt{f}+\sqrt{-d e+c f} x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}-\frac{\left (\sqrt{-d e+c f} \sin \left (a+\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}-b x\right )}{\sqrt{f}-\sqrt{-d e+c f} x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{f}\\ &=-\frac{2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{f}+\frac{\text{Ci}\left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}+\frac{b}{\sqrt{c+d x}}\right ) \sin \left (a-\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right )}{f}+\frac{\text{Ci}\left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}-\frac{b}{\sqrt{c+d x}}\right ) \sin \left (a+\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right )}{f}-\frac{2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{\cos \left (a+\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right ) \text{Si}\left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}-\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt{f}}{\sqrt{-d e+c f}}\right ) \text{Si}\left (\frac{b \sqrt{f}}{\sqrt{-d e+c f}}+\frac{b}{\sqrt{c+d x}}\right )}{f}\\ \end{align*}
Mathematica [F] time = 15.3157, size = 0, normalized size = 0. \[ \int \frac{\sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{e+f x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.029, size = 438, normalized size = 1.6 \begin{align*} -2\,{b}^{2} \left ( -1/2\,{\frac{1}{{b}^{2}f} \left ({\it Si} \left ({\frac{b}{\sqrt{dx+c}}}+a-{\frac{acf-ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) \cos \left ({\frac{acf-ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) +{\it Ci} \left ({\frac{b}{\sqrt{dx+c}}}+a-{\frac{acf-ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) \sin \left ({\frac{acf-ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) \right ) }-1/2\,{\frac{1}{{b}^{2}f} \left ({\it Si} \left ({\frac{b}{\sqrt{dx+c}}}+a+{\frac{-acf+ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) \cos \left ({\frac{-acf+ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) -{\it Ci} \left ({\frac{b}{\sqrt{dx+c}}}+a+{\frac{-acf+ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) \sin \left ({\frac{-acf+ade+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{cf-de}} \right ) \right ) }+{\frac{1}{{b}^{2}f} \left ({\it Si} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \cos \left ( a \right ) +{\it Ci} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \sin \left ( a \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 \frac{\sin \left (a + \frac{b}{\sqrt{d x + c}}\right )}{f x + e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.35873, size = 741, normalized size = 2.68 \begin{align*} \frac{2 i \,{\rm Ei}\left (\frac{i \, b}{\sqrt{d x + c}}\right ) e^{\left (i \, a\right )} - 2 i \,{\rm Ei}\left (-\frac{i \, b}{\sqrt{d x + c}}\right ) e^{\left (-i \, a\right )} - i \,{\rm Ei}\left (-\frac{2 \, \sqrt{\frac{b^{2} f}{d e - c f}}{\left (d x + c\right )} - 2 i \, \sqrt{d x + c} b}{2 \,{\left (d x + c\right )}}\right ) e^{\left (i \, a + \sqrt{\frac{b^{2} f}{d e - c f}}\right )} - i \,{\rm Ei}\left (\frac{2 \, \sqrt{\frac{b^{2} f}{d e - c f}}{\left (d x + c\right )} + 2 i \, \sqrt{d x + c} b}{2 \,{\left (d x + c\right )}}\right ) e^{\left (i \, a - \sqrt{\frac{b^{2} f}{d e - c f}}\right )} + i \,{\rm Ei}\left (-\frac{2 \, \sqrt{\frac{b^{2} f}{d e - c f}}{\left (d x + c\right )} + 2 i \, \sqrt{d x + c} b}{2 \,{\left (d x + c\right )}}\right ) e^{\left (-i \, a + \sqrt{\frac{b^{2} f}{d e - c f}}\right )} + i \,{\rm Ei}\left (\frac{2 \, \sqrt{\frac{b^{2} f}{d e - c f}}{\left (d x + c\right )} - 2 i \, \sqrt{d x + c} b}{2 \,{\left (d x + c\right )}}\right ) e^{\left (-i \, a - \sqrt{\frac{b^{2} f}{d e - c f}}\right )}}{2 \, f} \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 \frac{\sin \left (a + \frac{b}{\sqrt{d x + c}}\right )}{f x + e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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