Optimal. Leaf size=94 \[ \frac{b^2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{b^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{b \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.118033, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3361, 3297, 3303, 3299, 3302} \[ \frac{b^2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{b^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{b \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3361
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \sin \left (a+\frac{b}{\sqrt{c+d x}}\right ) \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}-\frac{b \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=\frac{b \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=\frac{b \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{\left (b^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}+\frac{\left (b^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=\frac{b \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{b^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d}+\frac{(c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d}+\frac{b^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0762403, size = 99, normalized size = 1.05 \[ \frac{b^2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )+b^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )+c \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )+d x \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )+b \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 84, normalized size = 0.9 \begin{align*} -2\,{\frac{{b}^{2}}{d} \left ( -1/2\,{\frac{dx+c}{{b}^{2}}\sin \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }-1/2\,{\frac{\sqrt{dx+c}}{b}\cos \left ( a+{\frac{b}{\sqrt{dx+c}}} \right ) }-1/2\,{\it Si} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \cos \left ( a \right ) -1/2\,{\it Ci} \left ({\frac{b}{\sqrt{dx+c}}} \right ) \sin \left ( a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.20623, size = 167, normalized size = 1.78 \begin{align*} \frac{{\left ({\left (-i \,{\rm Ei}\left (\frac{i \, b}{\sqrt{d x + c}}\right ) + i \,{\rm Ei}\left (-\frac{i \, b}{\sqrt{d x + c}}\right )\right )} \cos \left (a\right ) +{\left ({\rm Ei}\left (\frac{i \, b}{\sqrt{d x + c}}\right ) +{\rm Ei}\left (-\frac{i \, b}{\sqrt{d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt{d x + c} b \cos \left (\frac{\sqrt{d x + c} a + b}{\sqrt{d x + c}}\right ) + 2 \,{\left (d x + c\right )} \sin \left (\frac{\sqrt{d x + c} a + b}{\sqrt{d x + c}}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.16723, size = 360, normalized size = 3.83 \begin{align*} \frac{b^{2} \operatorname{Ci}\left (\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) + b^{2} \operatorname{Ci}\left (-\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) + 2 \, b^{2} \cos \left (a\right ) \operatorname{Si}\left (\frac{b}{\sqrt{d x + c}}\right ) + 2 \, \sqrt{d x + c} b \cos \left (\frac{a d x + a c + \sqrt{d x + c} b}{d x + c}\right ) + 2 \,{\left (d x + c\right )} \sin \left (\frac{a d x + a c + \sqrt{d x + c} b}{d x + c}\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 \sin{\left (a + \frac{b}{\sqrt{c + d x}} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (a + \frac{b}{\sqrt{d x + c}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]