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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.07, size = 67, normalized size = 0.81 \[ \frac {\sqrt {\frac {\pi }{2}} \left (\sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+\cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )\right )}{\sqrt {b} d} \]

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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fricas [A]  time = 0.76, size = 89, normalized size = 1.07 \[ \frac {\sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} \cos \relax (a) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + \sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) \sin \relax (a)}{2 \, b d^{2}} \]

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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giac [C]  time = 0.74, size = 151, normalized size = 1.82 \[ \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (i \, a\right )}}{4 \, \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} - \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (-i \, a\right )}}{4 \, \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} \]

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))

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maple [B]  time = 0.03, size = 136, normalized size = 1.64 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \left (b \,d^{2} x +b c d \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{2 \sqrt {d^{2} b}} \]

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 = 0.51, size = 69, normalized size = 0.83 \[ -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \relax (a) + \left (i - 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {i \, b}}\right ) + {\left (-\left (i - 1\right ) \, \cos \relax (a) + \left (i + 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {-i \, b}}\right )\right )}}{8 \, \sqrt {b} d} \]

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(2)*sqrt(pi)*((-(I + 1)*cos(a) + (I - 1)*sin(a))*erf((I*b*d*x + I*b*c)/sqrt(I*b)) + (-(I - 1)*cos(a)
+ (I + 1)*sin(a))*erf((I*b*d*x + I*b*c)/sqrt(-I*b)))/(sqrt(b)*d)

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mupad [B]  time = 0.05, size = 95, normalized size = 1.14 \[ \frac {\sqrt {2}\,\sqrt {\pi }\,\cos \relax (a)\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{b\,d^2}}\,\left (b\,x\,d^2+b\,c\,d\right )}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{b\,d^2}}}{2}+\frac {\sqrt {2}\,\sqrt {\pi }\,\sin \relax (a)\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{b\,d^2}}\,\left (b\,x\,d^2+b\,c\,d\right )}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{b\,d^2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*(c + d*x)^2),x)

[Out]

(2^(1/2)*pi^(1/2)*cos(a)*fresnels((2^(1/2)*(1/(b*d^2))^(1/2)*(b*c*d + b*d^2*x))/pi^(1/2))*(1/(b*d^2))^(1/2))/2
+ (2^(1/2)*pi^(1/2)*sin(a)*fresnelc((2^(1/2)*(1/(b*d^2))^(1/2)*(b*c*d + b*d^2*x))/pi^(1/2))*(1/(b*d^2))^(1/2)
)/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\left (a + b \left (c + d x\right )^{2} \right )}\, dx \]

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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