3.1.61 \(\int \sqrt {d x} \sin (f x) \, dx\) [61]

Optimal. Leaf size=65 \[ -\frac {\sqrt {d x} \cos (f x)}{f}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{f^{3/2}} \]

[Out]

1/2*FresnelC(f^(1/2)*2^(1/2)/Pi^(1/2)*(d*x)^(1/2)/d^(1/2))*d^(1/2)*2^(1/2)*Pi^(1/2)/f^(3/2)-cos(f*x)*(d*x)^(1/
2)/f

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Rubi [A]
time = 0.04, antiderivative size = 65, 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 = {3377, 3385, 3433} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{f^{3/2}}-\frac {\sqrt {d x} \cos (f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*Sin[f*x],x]

[Out]

-((Sqrt[d*x]*Cos[f*x])/f) + (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/f^(3/2)

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 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 \sqrt {d x} \sin (f x) \, dx &=-\frac {\sqrt {d x} \cos (f x)}{f}+\frac {d \int \frac {\cos (f x)}{\sqrt {d x}} \, dx}{2 f}\\ &=-\frac {\sqrt {d x} \cos (f x)}{f}+\frac {\text {Subst}\left (\int \cos \left (\frac {f x^2}{d}\right ) \, dx,x,\sqrt {d x}\right )}{f}\\ &=-\frac {\sqrt {d x} \cos (f x)}{f}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{f^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.01, size = 69, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {d x} \Gamma \left (\frac {3}{2},-i f x\right )}{2 f \sqrt {-i f x}}-\frac {\sqrt {d x} \Gamma \left (\frac {3}{2},i f x\right )}{2 f \sqrt {i f x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*x]*Sin[f*x],x]

[Out]

-1/2*(Sqrt[d*x]*Gamma[3/2, (-I)*f*x])/(f*Sqrt[(-I)*f*x]) - (Sqrt[d*x]*Gamma[3/2, I*f*x])/(2*f*Sqrt[I*f*x])

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Maple [A]
time = 0.02, size = 65, normalized size = 1.00

method	result	size
meijerg	\(\frac {\sqrt {d x}\, \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {\sqrt {x}\, \sqrt {2}\, \sqrt {f}\, \cos \left (f x \right )}{2 \sqrt {\pi }}+\frac {\FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}\, \sqrt {f}}{\sqrt {\pi }}\right )}{2}\right )}{\sqrt {x}\, f^{\frac {3}{2}}}\)	\(54\)
derivativedivides	\(\frac {-\frac {d \sqrt {d x}\, \cos \left (f x \right )}{f}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{2 f \sqrt {\frac {f}{d}}}}{d}\)	\(65\)
default	\(\frac {-\frac {d \sqrt {d x}\, \cos \left (f x \right )}{f}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{2 f \sqrt {\frac {f}{d}}}}{d}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(1/2)*sin(f*x),x,method=_RETURNVERBOSE)

[Out]

2/d*(-1/2/f*d*(d*x)^(1/2)*cos(f*x)+1/4/f*d*2^(1/2)*Pi^(1/2)/(f/d)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)/(f/d)^(1/2)*
f*(d*x)^(1/2)/d))

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Maxima [C] Result contains complex when optimal does not.
time = 0.33, size = 84, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {d x} f \cos \left (f x\right ) + \left (i - 1\right ) \, \sqrt {\pi } d \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {i \, f}{d}}\right ) - \left (i + 1\right ) \, \sqrt {\pi } d \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {i \, f}{d}}\right )\right )}}{8 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*sin(f*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 54, normalized size = 0.83 \begin {gather*} \frac {\sqrt {2} \pi d \sqrt {\frac {f}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right ) - 2 \, \sqrt {d x} f \cos \left (f x\right )}{2 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*sin(f*x),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*pi*d*sqrt(f/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d))) - 2*sqrt(d*x)*f*cos(f*x))/f^2

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Sympy [A]
time = 1.08, size = 85, normalized size = 1.31 \begin {gather*} - \frac {5 \sqrt {d} \sqrt {x} \cos {\left (f x \right )} \Gamma \left (\frac {5}{4}\right )}{4 f \Gamma \left (\frac {9}{4}\right )} + \frac {5 \sqrt {2} \sqrt {\pi } \sqrt {d} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{8 f^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(1/2)*sin(f*x),x)

[Out]

-5*sqrt(d)*sqrt(x)*cos(f*x)*gamma(5/4)/(4*f*gamma(9/4)) + 5*sqrt(2)*sqrt(pi)*sqrt(d)*fresnelc(sqrt(2)*sqrt(f)*
sqrt(x)/sqrt(pi))*gamma(5/4)/(8*f**(3/2)*gamma(9/4))

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Giac [C] Result contains complex when optimal does not.
time = 3.55, size = 176, normalized size = 2.71 \begin {gather*} -\frac {\frac {\sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d f} \sqrt {d x} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )} f} + \frac {\sqrt {2} \sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d f} \sqrt {d x} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )} f} + \frac {2 \, \sqrt {d x} d e^{\left (i \, f x\right )}}{f} + \frac {2 \, \sqrt {d x} d e^{\left (-i \, f x\right )}}{f}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*sin(f*x),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x)*(d*x)^(1/2),x)

[Out]

int(sin(f*x)*(d*x)^(1/2), x)

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