3.200 \(\int \frac{\sin (a+\frac{b}{\sqrt{c+d x}})}{e+f x} \, dx\)

Optimal. Leaf size=276 \[ \frac{\sin \left (a-\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}+\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\sin \left (a+\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}-\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{\cos \left (a+\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{Si}\left (\frac{b \sqrt{f}}{\sqrt{c f-d e}}-\frac{b}{\sqrt{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt{f}}{\sqrt{c f-d e}}\right ) \text{Si}\left (\frac{\sqrt{f} b}{\sqrt{c f-d e}}+\frac{b}{\sqrt{c+d x}}\right )}{f}-\frac{2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{f} \]

[Out]

(-2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/f + (CosIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] + b/Sqrt[c + d*x]]*Si
n[a - (b*Sqrt[f])/Sqrt[-(d*e) + c*f]])/f + (CosIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + d*x]]*Sin[
a + (b*Sqrt[f])/Sqrt[-(d*e) + c*f]])/f - (2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/f - (Cos[a + (b*Sqrt[f])/Sqrt
[-(d*e) + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + d*x]])/f + (Cos[a - (b*Sqrt[f])/Sqrt[-
(d*e) + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] + b/Sqrt[c + d*x]])/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.

[In]

Int[Sin[a + b/Sqrt[c + d*x]]/(e + f*x),x]

[Out]

(-2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/f + (CosIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] + b/Sqrt[c + d*x]]*Si
n[a - (b*Sqrt[f])/Sqrt[-(d*e) + c*f]])/f + (CosIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + d*x]]*Sin[
a + (b*Sqrt[f])/Sqrt[-(d*e) + c*f]])/f - (2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/f - (Cos[a + (b*Sqrt[f])/Sqrt
[-(d*e) + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] - b/Sqrt[c + d*x]])/f + (Cos[a - (b*Sqrt[f])/Sqrt[-
(d*e) + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[-(d*e) + c*f] + b/Sqrt[c + d*x]])/f

Rule 3431

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

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.

[In]

Integrate[Sin[a + b/Sqrt[c + d*x]]/(e + f*x),x]

[Out]

Integrate[Sin[a + b/Sqrt[c + d*x]]/(e + f*x), x]

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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.

[In]

int(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x)

[Out]

-2*b^2*(-1/2/b^2/f*(Si(b/(d*x+c)^(1/2)+a-(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*cos((a*c*f-a*d*e
+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))+Ci(b/(d*x+c)^(1/2)+a-(a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d
*e))*sin((a*c*f-a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e)))-1/2/b^2/f*(Si(b/(d*x+c)^(1/2)+a+(-a*c*f+a*d*e+(
b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*cos((-a*c*f+a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))-Ci(b/(d*x+c)^
(1/2)+a+(-a*c*f+a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(c*f-d*e))*sin((-a*c*f+a*d*e+(b^2*c*f^2-b^2*d*e*f)^(1/2))/(
c*f-d*e)))+1/b^2/f*(Si(b/(d*x+c)^(1/2))*cos(a)+Ci(b/(d*x+c)^(1/2))*sin(a)))

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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.

[In]

integrate(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x, algorithm="maxima")

[Out]

integrate(sin(a + b/sqrt(d*x + c))/(f*x + e), x)

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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.

[In]

integrate(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x, algorithm="fricas")

[Out]

1/2*(2*I*Ei(I*b/sqrt(d*x + c))*e^(I*a) - 2*I*Ei(-I*b/sqrt(d*x + c))*e^(-I*a) - I*Ei(-1/2*(2*sqrt(b^2*f/(d*e -
c*f))*(d*x + c) - 2*I*sqrt(d*x + c)*b)/(d*x + c))*e^(I*a + sqrt(b^2*f/(d*e - c*f))) - I*Ei(1/2*(2*sqrt(b^2*f/(
d*e - c*f))*(d*x + c) + 2*I*sqrt(d*x + c)*b)/(d*x + c))*e^(I*a - sqrt(b^2*f/(d*e - c*f))) + I*Ei(-1/2*(2*sqrt(
b^2*f/(d*e - c*f))*(d*x + c) + 2*I*sqrt(d*x + c)*b)/(d*x + c))*e^(-I*a + sqrt(b^2*f/(d*e - c*f))) + I*Ei(1/2*(
2*sqrt(b^2*f/(d*e - c*f))*(d*x + c) - 2*I*sqrt(d*x + c)*b)/(d*x + c))*e^(-I*a - sqrt(b^2*f/(d*e - c*f))))/f

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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.

[In]

integrate(sin(a+b/(d*x+c)**(1/2))/(f*x+e),x)

[Out]

Timed out

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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.

[In]

integrate(sin(a+b/(d*x+c)^(1/2))/(f*x+e),x, algorithm="giac")

[Out]

integrate(sin(a + b/sqrt(d*x + c))/(f*x + e), x)