3.1.9 \(\int (a+b \sin (c+d x^2)) \, dx\) [9]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3434, 3433, 3432} \begin {gather*} 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}} \end {gather*}

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 3432

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

Rule 3433

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

Rule 3434

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 \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*}

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Mathematica [A]
time = 0.10, size = 61, normalized size = 0.82 \begin {gather*} a x+\frac {b \sqrt {\frac {\pi }{2}} \left (\cos (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)\right )}{\sqrt {d}} \end {gather*}

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 = 0.03, size = 48, normalized size = 0.65

method	result	size
default	\(a x +\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (c \right ) \mathrm {S}\left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (c \right ) \FresnelC \left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{2 \sqrt {d}}\)	\(48\)
risch	\(a x +\frac {i b \,{\mathrm e}^{-i c} \sqrt {\pi }\, \erf \left (\sqrt {i d}\, x \right )}{4 \sqrt {i d}}-\frac {i b \,{\mathrm e}^{i c} \sqrt {\pi }\, \erf \left (\sqrt {-i d}\, x \right )}{4 \sqrt {-i d}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Result contains complex when optimal does not.
time = 0.30, size = 53, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (c\right ) + \left (i - 1\right ) \, \sin \left (c\right )\right )} \operatorname {erf}\left (\sqrt {i \, d} x\right ) + {\left (\left (i - 1\right ) \, \cos \left (c\right ) - \left (i + 1\right ) \, \sin \left (c\right )\right )} \operatorname {erf}\left (\sqrt {-i \, d} x\right )\right )} b}{8 \, \sqrt {d}} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*sqrt(pi)*((-(I + 1)*cos(c) + (I - 1)*sin(c))*erf(sqrt(I*d)*x) + ((I - 1)*cos(c) - (I + 1)*sin(c))
*erf(sqrt(-I*d)*x))*b/sqrt(d) + a*x

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Fricas [A]
time = 0.37, size = 67, normalized size = 0.91 \begin {gather*} \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 {gather*}

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 = 0.25, size = 66, normalized size = 0.89 \begin {gather*} 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 {gather*}

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] Result contains complex when optimal does not.
time = 5.29, size = 102, normalized size = 1.38 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 4.75, size = 56, normalized size = 0.76 \begin {gather*} a\,x+\frac {\sqrt {2}\,b\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {d}\,x}{\sqrt {\pi }}\right )\,\cos \left (c\right )}{2\,\sqrt {d}}+\frac {\sqrt {2}\,b\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {d}\,x}{\sqrt {\pi }}\right )\,\sin \left (c\right )}{2\,\sqrt {d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*sin(c + d*x^2),x)

[Out]

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

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