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

Optimal. Leaf size=74 \[ a x+\frac{\sqrt{\frac{\pi }{2}} b \sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )}{\sqrt{d}}+\frac{\sqrt{\frac{\pi }{2}} b \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.042957, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3353, 3352, 3351} \[ a x+\frac{\sqrt{\frac{\pi }{2}} b \sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )}{\sqrt{d}}+\frac{\sqrt{\frac{\pi }{2}} b \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sin[c + d*x^2],x]

[Out]

a*x + (b*Sqrt[Pi/2]*Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x])/Sqrt[d] + (b*Sqrt[Pi/2]*FresnelC[Sqrt[d]*Sqrt[2/Pi]
*x]*Sin[c])/Sqrt[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 \left (a+b \sin \left (c+d x^2\right )\right ) \, dx &=a x+b \int \sin \left (c+d x^2\right ) \, dx\\ &=a x+(b \cos (c)) \int \sin \left (d x^2\right ) \, dx+(b \sin (c)) \int \cos \left (d x^2\right ) \, dx\\ &=a x+\frac{b \sqrt{\frac{\pi }{2}} \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{d}}+\frac{b \sqrt{\frac{\pi }{2}} C\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right ) \sin (c)}{\sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.142439, size = 61, normalized size = 0.82 \[ a x+\frac{\sqrt{\frac{\pi }{2}} b \left (\sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )+\cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sin[c + d*x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 48, normalized size = 0.7 \begin{align*} ax+{\frac{b\sqrt{2}\sqrt{\pi }}{2} \left ( \cos \left ( c \right ){\it FresnelS} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{d}} \right ) +\sin \left ( c \right ){\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{d}} \right ) \right ){\frac{1}{\sqrt{d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.63441, size = 315, normalized size = 4.26 \begin{align*} a x - \frac{\sqrt{\pi }{\left ({\left ({\left (-i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \cos \left (c\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \sin \left (c\right )\right )} \operatorname{erf}\left (\sqrt{i \, d} x\right ) +{\left ({\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \cos \left (c\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \sin \left (c\right )\right )} \operatorname{erf}\left (\sqrt{-i \, d} x\right )\right )} b}{8 \, \sqrt{{\left | d \right |}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x - 1/8*sqrt(pi)*(((-I*cos(1/4*pi + 1/2*arctan2(0, d)) - I*cos(-1/4*pi + 1/2*arctan2(0, d)) - sin(1/4*pi + 1
/2*arctan2(0, d)) + sin(-1/4*pi + 1/2*arctan2(0, d)))*cos(c) - (cos(1/4*pi + 1/2*arctan2(0, d)) + cos(-1/4*pi
+ 1/2*arctan2(0, d)) - I*sin(1/4*pi + 1/2*arctan2(0, d)) + I*sin(-1/4*pi + 1/2*arctan2(0, d)))*sin(c))*erf(sqr
t(I*d)*x) + ((I*cos(1/4*pi + 1/2*arctan2(0, d)) + I*cos(-1/4*pi + 1/2*arctan2(0, d)) - sin(1/4*pi + 1/2*arctan
2(0, d)) + sin(-1/4*pi + 1/2*arctan2(0, d)))*cos(c) - (cos(1/4*pi + 1/2*arctan2(0, d)) + cos(-1/4*pi + 1/2*arc
tan2(0, d)) + I*sin(1/4*pi + 1/2*arctan2(0, d)) - I*sin(-1/4*pi + 1/2*arctan2(0, d)))*sin(c))*erf(sqrt(-I*d)*x
))*b/sqrt(abs(d))

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Fricas [A]  time = 1.98925, size = 204, normalized size = 2.76 \begin{align*} \frac{\sqrt{2} \pi b \sqrt{\frac{d}{\pi }} \cos \left (c\right ) \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{d}{\pi }}\right ) + \sqrt{2} \pi b \sqrt{\frac{d}{\pi }} \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{d}{\pi }}\right ) \sin \left (c\right ) + 2 \, a d x}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.526034, size = 66, normalized size = 0.89 \begin{align*} a x + \frac{\sqrt{2} \sqrt{\pi } b \left (\sin{\left (c \right )} C\left (\frac{\sqrt{2} \sqrt{d} x}{\sqrt{\pi }}\right ) + \cos{\left (c \right )} S\left (\frac{\sqrt{2} \sqrt{d} x}{\sqrt{\pi }}\right )\right ) \sqrt{\frac{1}{d}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.11975, size = 138, normalized size = 1.86 \begin{align*} -\frac{1}{4} \,{\left (-\frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (-\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}\right ) e^{\left (i \, c\right )}}{{\left (-\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}} + \frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}\right ) e^{\left (-i \, c\right )}}{{\left (\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}}\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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