3.1.4 \(\int \frac {\sin (x)}{\sqrt {1+x}} \, dx\) [4]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3387, 3386, 3432, 3385, 3433} \begin {gather*} \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )-\sqrt {2 \pi } \sin (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right ) \end {gather*}

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 3385

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 3386

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 3387

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 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]

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)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1+x}\right )-(2 \sin (1)) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.01, size = 68, normalized size = 1.17 \begin {gather*} -\frac {e^{-i} \left (\sqrt {-i (1+x)} \Gamma \left (\frac {1}{2},-i (1+x)\right )+e^{2 i} \sqrt {i (1+x)} \Gamma \left (\frac {1}{2},i (1+x)\right )\right )}{2 \sqrt {1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/Sqrt[1 + x],x]

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sin[x]/Sqrt[1 + x],x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.06, size = 42, normalized size = 0.72

method	result	size
derivativedivides	\(\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (1\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {1+x}}{\sqrt {\pi }}\right )-\sin \left (1\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {1+x}}{\sqrt {\pi }}\right )\right )\)	\(42\)
default	\(\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (1\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {1+x}}{\sqrt {\pi }}\right )-\sin \left (1\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {1+x}}{\sqrt {\pi }}\right )\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[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] Result contains complex when optimal does not.
time = 0.37, size = 112, normalized size = 1.93 \begin {gather*} \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 {gather*}

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 = 0.45, size = 46, normalized size = 0.79 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (x \right )}}{\sqrt {x + 1}}\, dx \end {gather*}

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] Result contains complex when optimal does not.
time = 0.00, size = 80, normalized size = 1.38 \begin {gather*} 2 \left (\frac {2 \sqrt {\pi } \mathrm {e}^{-\mathrm {i}} \mathrm {erf}\left (\frac {\mathrm {i} \sqrt {2} \sqrt {x+1}}{1-\mathrm {i}}\right )}{\left (-1+\mathrm {i}\right ) \sqrt {2}\cdot 2 2 \mathrm {i}}+\frac {2 \mathrm {i} \sqrt {\pi } \mathrm {e}^{\mathrm {i}} \mathrm {erf}\left (-\frac {\sqrt {2} \sqrt {x+1}}{1-\mathrm {i}}\right )}{\left (-1-\mathrm {i}\right ) \sqrt {2}\cdot 2\cdot 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sin \left (x\right )}{\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(x + 1)^(1/2),x)

[Out]

int(sin(x)/(x + 1)^(1/2), x)

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