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

Optimal. Leaf size=238 \[ \frac{\sin \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}+b \sqrt{c+d x}\right )}{f}+\frac{\sin \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{f}-\frac{\cos \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{\sqrt{c f-d e} b}{\sqrt{f}}+\sqrt{c+d x} b\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.748664, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3431, 3303, 3299, 3302} \[ \frac{\sin \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}+b \sqrt{c+d x}\right )}{f}+\frac{\sin \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{f}-\frac{\cos \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{c f-d e}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right ) \text{Si}\left (\frac{\sqrt{c f-d e} b}{\sqrt{f}}+\sqrt{c+d x} b\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sin \left (a+b \sqrt{c+d x}\right )}{e+f x} \, dx &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{d \sin (a+b x)}{2 \sqrt{f} \left (\sqrt{-d e+c f}-\sqrt{f} x\right )}+\frac{d \sin (a+b x)}{2 \sqrt{f} \left (\sqrt{-d e+c f}+\sqrt{f} x\right )}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\sqrt{-d e+c f}-\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{\sqrt{f}}+\frac{\operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\sqrt{-d e+c f}+\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{\sqrt{f}}\\ &=\frac{\cos \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b x\right )}{\sqrt{-d e+c f}+\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{\sqrt{f}}+\frac{\cos \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b x\right )}{\sqrt{-d e+c f}-\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{\sqrt{f}}+\frac{\sin \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b x\right )}{\sqrt{-d e+c f}+\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{\sqrt{f}}-\frac{\sin \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b x\right )}{\sqrt{-d e+c f}-\sqrt{f} x} \, dx,x,\sqrt{c+d x}\right )}{\sqrt{f}}\\ &=\frac{\text{Ci}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b \sqrt{c+d x}\right ) \sin \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right )}{f}+\frac{\text{Ci}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b \sqrt{c+d x}\right ) \sin \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right )}{f}-\frac{\cos \left (a+\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}-b \sqrt{c+d x}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt{-d e+c f}}{\sqrt{f}}\right ) \text{Si}\left (\frac{b \sqrt{-d e+c f}}{\sqrt{f}}+b \sqrt{c+d x}\right )}{f}\\ \end{align*}

Mathematica [C]  time = 1.46982, size = 238, normalized size = 1. \[ \frac{i e^{-i \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right )} \left (-e^{2 i \left (a+\frac{b \sqrt{c f-d e}}{\sqrt{f}}\right )} \text{Ei}\left (i b \left (\sqrt{c+d x}-\frac{\sqrt{c f-d e}}{\sqrt{f}}\right )\right )-e^{2 i a} \text{Ei}\left (i b \left (\frac{\sqrt{c f-d e}}{\sqrt{f}}+\sqrt{c+d x}\right )\right )+\text{Ei}\left (-i b \left (\sqrt{c+d x}-\frac{\sqrt{c f-d e}}{\sqrt{f}}\right )\right )+e^{\frac{2 i b \sqrt{c f-d e}}{\sqrt{f}}} \text{Ei}\left (-i b \left (\frac{\sqrt{c f-d e}}{\sqrt{f}}+\sqrt{c+d x}\right )\right )\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/2)*(ExpIntegralEi[(-I)*b*(-(Sqrt[-(d*e) + c*f]/Sqrt[f]) + Sqrt[c + d*x])] - E^((2*I)*(a + (b*Sqrt[-(d*e) +
c*f])/Sqrt[f]))*ExpIntegralEi[I*b*(-(Sqrt[-(d*e) + c*f]/Sqrt[f]) + Sqrt[c + d*x])] + E^(((2*I)*b*Sqrt[-(d*e)
+ c*f])/Sqrt[f])*ExpIntegralEi[(-I)*b*(Sqrt[-(d*e) + c*f]/Sqrt[f] + Sqrt[c + d*x])] - E^((2*I)*a)*ExpIntegralE
i[I*b*(Sqrt[-(d*e) + c*f]/Sqrt[f] + Sqrt[c + d*x])]))/(E^(I*(a + (b*Sqrt[-(d*e) + c*f])/Sqrt[f]))*f)

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Maple [B]  time = 0.021, size = 785, normalized size = 3.3 \begin{align*} 2\,{\frac{1}{{b}^{2}} \left ( 1/2\,{\frac{{b}^{2} \left ( af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def} \right ) }{{f}^{2}} \left ({\it Si} \left ( b\sqrt{dx+c}+a-{\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \cos \left ({\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) +{\it Ci} \left ( b\sqrt{dx+c}+a-{\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \sin \left ({\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \right ) \left ({\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}}-a \right ) ^{-1}}-1/2\,{\frac{{b}^{2} \left ( -af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def} \right ) }{{f}^{2}} \left ({\it Si} \left ( b\sqrt{dx+c}+a+{\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \cos \left ({\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) -{\it Ci} \left ( b\sqrt{dx+c}+a+{\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \sin \left ({\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \right ) \left ( -{\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}}-a \right ) ^{-1}}-a{b}^{2} \left ( 1/2\,{\frac{1}{f} \left ({\it Si} \left ( b\sqrt{dx+c}+a-{\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \cos \left ({\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) +{\it Ci} \left ( b\sqrt{dx+c}+a-{\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \sin \left ({\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \right ) \left ({\frac{af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}}-a \right ) ^{-1}}+1/2\,{\frac{1}{f} \left ({\it Si} \left ( b\sqrt{dx+c}+a+{\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \cos \left ({\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) -{\it Ci} \left ( b\sqrt{dx+c}+a+{\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \sin \left ({\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}} \right ) \right ) \left ( -{\frac{-af+\sqrt{{b}^{2}c{f}^{2}-{b}^{2}def}}{f}}-a \right ) ^{-1}} \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*(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f^2/((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(Si(b*(d*x+c)^(1/
2)+a-(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+c)^(1/2)+a-(a*f+(
b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))-1/2*b^2*(-a*f+(b^2*c*f^2-b^2*d*e*f)^(
1/2))/f^2/(-(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)*(Si(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f
)*cos((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)-Ci(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((-a
*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))-a*b^2*(1/2/((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f-a)/f*(Si(b*(d*x+c)^(1/2)+a
-(a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)+Ci(b*(d*x+c)^(1/2)+a-(a*f+(b^2*
c*f^2-b^2*d*e*f)^(1/2))/f)*sin((a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))+1/2/(-(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/
f-a)/f*(Si(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*cos((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)-C
i(b*(d*x+c)^(1/2)+a+(-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f)*sin((-a*f+(b^2*c*f^2-b^2*d*e*f)^(1/2))/f))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (\sqrt{d x + c} b + a\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(sqrt(d*x + c)*b + a)/(f*x + e), x)

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Fricas [C]  time = 1.83603, size = 524, normalized size = 2.2 \begin{align*} \frac{-i \,{\rm Ei}\left (i \, \sqrt{d x + c} b - \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right )} - i \,{\rm Ei}\left (i \, \sqrt{d x + c} b + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (i \, a - \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right )} + i \,{\rm Ei}\left (-i \, \sqrt{d x + c} b - \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right )} + i \,{\rm Ei}\left (-i \, \sqrt{d x + c} b + \sqrt{\frac{b^{2} d e - b^{2} c f}{f}}\right ) e^{\left (-i \, a - \sqrt{\frac{b^{2} d e - b^{2} c f}{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*(-I*Ei(I*sqrt(d*x + c)*b - sqrt((b^2*d*e - b^2*c*f)/f))*e^(I*a + sqrt((b^2*d*e - b^2*c*f)/f)) - I*Ei(I*sqr
t(d*x + c)*b + sqrt((b^2*d*e - b^2*c*f)/f))*e^(I*a - sqrt((b^2*d*e - b^2*c*f)/f)) + I*Ei(-I*sqrt(d*x + c)*b -
sqrt((b^2*d*e - b^2*c*f)/f))*e^(-I*a + sqrt((b^2*d*e - b^2*c*f)/f)) + I*Ei(-I*sqrt(d*x + c)*b + sqrt((b^2*d*e
- b^2*c*f)/f))*e^(-I*a - sqrt((b^2*d*e - b^2*c*f)/f)))/f

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b \sqrt{c + d x} \right )}}{e + f x}\, dx \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]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (\sqrt{d x + c} b + a\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(sqrt(d*x + c)*b + a)/(f*x + e), x)