3.4 \(\int \frac{\sin (x)}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=58 \[ \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )-\sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right ) \]

[Out]

Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]] - Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1]

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Rubi [A]  time = 0.0815101, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3306, 3305, 3351, 3304, 3352} \[ \sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right )-\sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/Sqrt[1 + x],x]

[Out]

Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]] - Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1]

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

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 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

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]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{\sqrt{1+x}} \, dx &=\cos (1) \int \frac{\sin (1+x)}{\sqrt{1+x}} \, dx-\sin (1) \int \frac{\cos (1+x)}{\sqrt{1+x}} \, dx\\ &=(2 \cos (1)) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1+x}\right )-(2 \sin (1)) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1+x}\right )\\ &=\sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1+x}\right )-\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1+x}\right ) \sin (1)\\ \end{align*}

Mathematica [C]  time = 0.0161421, size = 68, normalized size = 1.17 \[ -\frac{e^{-i} \left (\sqrt{-i (x+1)} \text{Gamma}\left (\frac{1}{2},-i (x+1)\right )+e^{2 i} \sqrt{i (x+1)} \text{Gamma}\left (\frac{1}{2},i (x+1)\right )\right )}{2 \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/Sqrt[1 + x],x]

[Out]

-(Sqrt[(-I)*(1 + x)]*Gamma[1/2, (-I)*(1 + x)] + E^(2*I)*Sqrt[I*(1 + x)]*Gamma[1/2, I*(1 + x)])/(2*E^I*Sqrt[1 +
x])

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Maple [A]  time = 0.012, size = 42, normalized size = 0.7 \begin{align*} \sqrt{2}\sqrt{\pi } \left ( \cos \left ( 1 \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{1+x}} \right ) -\sin \left ( 1 \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{1+x}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(1+x)^(1/2),x)

[Out]

2^(1/2)*Pi^(1/2)*(cos(1)*FresnelS(2^(1/2)/Pi^(1/2)*(1+x)^(1/2))-sin(1)*FresnelC(2^(1/2)/Pi^(1/2)*(1+x)^(1/2)))

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Maxima [C]  time = 1.55022, size = 151, normalized size = 2.6 \begin{align*} \frac{1}{8} \, \sqrt{\pi }{\left ({\left (\left (i + 1\right ) \, \sqrt{2} \cos \left (1\right ) + \left (i - 1\right ) \, \sqrt{2} \sin \left (1\right )\right )} \operatorname{erf}\left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x + 1}\right ) +{\left (\left (i - 1\right ) \, \sqrt{2} \cos \left (1\right ) + \left (i + 1\right ) \, \sqrt{2} \sin \left (1\right )\right )} \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x + 1}\right ) +{\left (-\left (i - 1\right ) \, \sqrt{2} \cos \left (1\right ) - \left (i + 1\right ) \, \sqrt{2} \sin \left (1\right )\right )} \operatorname{erf}\left (\sqrt{-i} \sqrt{x + 1}\right ) +{\left (\left (i + 1\right ) \, \sqrt{2} \cos \left (1\right ) + \left (i - 1\right ) \, \sqrt{2} \sin \left (1\right )\right )} \operatorname{erf}\left (\left (-1\right )^{\frac{1}{4}} \sqrt{x + 1}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

1/8*sqrt(pi)*(((I + 1)*sqrt(2)*cos(1) + (I - 1)*sqrt(2)*sin(1))*erf((1/2*I + 1/2)*sqrt(2)*sqrt(x + 1)) + ((I -
1)*sqrt(2)*cos(1) + (I + 1)*sqrt(2)*sin(1))*erf((1/2*I - 1/2)*sqrt(2)*sqrt(x + 1)) + (-(I - 1)*sqrt(2)*cos(1)
- (I + 1)*sqrt(2)*sin(1))*erf(sqrt(-I)*sqrt(x + 1)) + ((I + 1)*sqrt(2)*cos(1) + (I - 1)*sqrt(2)*sin(1))*erf((
-1)^(1/4)*sqrt(x + 1)))

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Fricas [A]  time = 1.13812, size = 182, normalized size = 3.14 \begin{align*} \sqrt{2} \sqrt{\pi } \cos \left (1\right ) \operatorname{S}\left (\frac{\sqrt{2} \sqrt{x + 1}}{\sqrt{\pi }}\right ) - \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2} \sqrt{x + 1}}{\sqrt{\pi }}\right ) \sin \left (1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

sqrt(2)*sqrt(pi)*cos(1)*fresnel_sin(sqrt(2)*sqrt(x + 1)/sqrt(pi)) - sqrt(2)*sqrt(pi)*fresnel_cos(sqrt(2)*sqrt(
x + 1)/sqrt(pi))*sin(1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (x \right )}}{\sqrt{x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+x)**(1/2),x)

[Out]

Integral(sin(x)/sqrt(x + 1), x)

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Giac [C]  time = 1.09736, size = 58, normalized size = 1. \begin{align*} -\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x + 1}\right ) e^{i} + \left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x + 1}\right ) e^{\left (-i\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+x)^(1/2),x, algorithm="giac")

[Out]

-(1/4*I + 1/4)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(x + 1))*e^I + (1/4*I - 1/4)*sqrt(2)*sqrt(pi)*e
rf((1/2*I - 1/2)*sqrt(2)*sqrt(x + 1))*e^(-I)