3.168 \(\int \sin (a+b (c+d x)^2) \, dx\)

Optimal. Leaf size=83 \[ \frac{\sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right )}{\sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d} \]

[Out]

(Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d) + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/P
i]*(c + d*x)]*Sin[a])/(Sqrt[b]*d)

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Rubi [A]  time = 0.0424326, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3353, 3352, 3351} \[ \frac{\sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right )}{\sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^2],x]

[Out]

(Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d) + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/P
i]*(c + d*x)]*Sin[a])/(Sqrt[b]*d)

Rule 3353

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int \sin \left (a+b (c+d x)^2\right ) \, dx &=\cos (a) \int \sin \left (b (c+d x)^2\right ) \, dx+\sin (a) \int \cos \left (b (c+d x)^2\right ) \, dx\\ &=\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt{b} d}\\ \end{align*}

Mathematica [A]  time = 0.0708423, size = 67, normalized size = 0.81 \[ \frac{\sqrt{\frac{\pi }{2}} \left (\sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right )+\cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )\right )}{\sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^2],x]

[Out]

(Sqrt[Pi/2]*(Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)] + FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a]))/(
Sqrt[b]*d)

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Maple [B]  time = 0.009, size = 136, normalized size = 1.6 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{2} \left ( \cos \left ({\frac{{b}^{2}{c}^{2}{d}^{2}-{d}^{2}b \left ({c}^{2}b+a \right ) }{{d}^{2}b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2} \left ( b{d}^{2}x+bcd \right ) }{\sqrt{\pi }}{\frac{1}{\sqrt{{d}^{2}b}}}} \right ) -\sin \left ({\frac{{b}^{2}{c}^{2}{d}^{2}-{d}^{2}b \left ({c}^{2}b+a \right ) }{{d}^{2}b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2} \left ( b{d}^{2}x+bcd \right ) }{\sqrt{\pi }}{\frac{1}{\sqrt{{d}^{2}b}}}} \right ) \right ){\frac{1}{\sqrt{{d}^{2}b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+(d*x+c)^2*b),x)

[Out]

1/2*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*(cos((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/b)*FresnelS(2^(1/2)/Pi^(1/2)/(d^2*b)
^(1/2)*(b*d^2*x+b*c*d))-sin((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/b)*FresnelC(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^
2*x+b*c*d)))

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Maxima [C]  time = 1.79814, size = 335, normalized size = 4.04 \begin{align*} -\frac{\sqrt{\pi }{\left ({\left ({\left (-i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \operatorname{erf}\left (\frac{i \, b d x + i \, b c}{\sqrt{i \, b}}\right ) +{\left ({\left (-i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) +{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \operatorname{erf}\left (\frac{i \, b d x + i \, b c}{\sqrt{-i \, b}}\right )\right )}}{8 \, d \sqrt{{\left | b \right |}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(pi)*(((-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) - sin(1/4*pi + 1/2*ar
ctan2(0, b)) + sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (cos(1/4*pi + 1/2*arctan2(0, b)) + cos(-1/4*pi + 1/2
*arctan2(0, b)) - I*sin(1/4*pi + 1/2*arctan2(0, b)) + I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((I*b*d*x
+ I*b*c)/sqrt(I*b)) + ((-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi
+ 1/2*arctan2(0, b)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) + (cos(1/4*pi + 1/2*arctan2(0, b)) + cos(-1/4*
pi + 1/2*arctan2(0, b)) + I*sin(1/4*pi + 1/2*arctan2(0, b)) - I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf(
(I*b*d*x + I*b*c)/sqrt(-I*b)))/(d*sqrt(abs(b)))

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Fricas [A]  time = 1.64334, size = 242, normalized size = 2.92 \begin{align*} \frac{\sqrt{2} \pi \sqrt{\frac{b d^{2}}{\pi }} \cos \left (a\right ) \operatorname{S}\left (\frac{\sqrt{2} \sqrt{\frac{b d^{2}}{\pi }}{\left (d x + c\right )}}{d}\right ) + \sqrt{2} \pi \sqrt{\frac{b d^{2}}{\pi }} \operatorname{C}\left (\frac{\sqrt{2} \sqrt{\frac{b d^{2}}{\pi }}{\left (d x + c\right )}}{d}\right ) \sin \left (a\right )}{2 \, b d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*pi*sqrt(b*d^2/pi)*cos(a)*fresnel_sin(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d) + sqrt(2)*pi*sqrt(b*d^2/
pi)*fresnel_cos(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d)*sin(a))/(b*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (a + b \left (c + d x\right )^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*(d*x+c)**2),x)

[Out]

Integral(sin(a + b*(c + d*x)**2), x)

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Giac [C]  time = 1.13234, size = 204, normalized size = 2.46 \begin{align*} \frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{b d^{2}}{\left (-\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}{\left (x + \frac{c}{d}\right )}\right ) e^{\left (i \, a\right )}}{4 \, \sqrt{b d^{2}}{\left (-\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}} - \frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{b d^{2}}{\left (\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}{\left (x + \frac{c}{d}\right )}\right ) e^{\left (-i \, a\right )}}{4 \, \sqrt{b d^{2}}{\left (\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(I*a)/(sqrt(b*d^
2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)) - 1/4*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4)
+ 1)*(x + c/d))*e^(-I*a)/(sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4) + 1))