\(\int \cos (a+b \sqrt {c+d x}) \, dx\) [92]

Optimal result

Integrand size = 14, antiderivative size = 54 \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d}+\frac {2 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3443, 3377, 2718} \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d}+\frac {2 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d} \]

[In]

Int[Cos[a + b*Sqrt[c + d*x]],x]

[Out]

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

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 3443

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Cos[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && In
tegerQ[1/n]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = \frac {2 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d}-\frac {2 \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d} \\ & = \frac {2 \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d}+\frac {2 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \left (\cos \left (a+b \sqrt {c+d x}\right )+b \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )\right )}{b^2 d} \]

[In]

Integrate[Cos[a + b*Sqrt[c + d*x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(\frac {2 \cos \left (a +b \sqrt {d x +c}\right )+2 \left (a +b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {d x +c}\right )-2 a \sin \left (a +b \sqrt {d x +c}\right )}{b^{2} d}\)	\(61\)
default	\(\frac {2 \cos \left (a +b \sqrt {d x +c}\right )+2 \left (a +b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {d x +c}\right )-2 a \sin \left (a +b \sqrt {d x +c}\right )}{b^{2} d}\)	\(61\)

[In]

int(cos(a+b*(d*x+c)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

2/d/b^2*(cos(a+b*(d*x+c)^(1/2))+(a+b*(d*x+c)^(1/2))*sin(a+b*(d*x+c)^(1/2))-a*sin(a+b*(d*x+c)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left (\sqrt {d x + c} b \sin \left (\sqrt {d x + c} b + a\right ) + \cos \left (\sqrt {d x + c} b + a\right )\right )}}{b^{2} d} \]

[In]

integrate(cos(a+b*(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

2*(sqrt(d*x + c)*b*sin(sqrt(d*x + c)*b + a) + cos(sqrt(d*x + c)*b + a))/(b^2*d)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.20 \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\begin {cases} x \cos {\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \cos {\left (a + b \sqrt {c} \right )} & \text {for}\: d = 0 \\\frac {2 \sqrt {c + d x} \sin {\left (a + b \sqrt {c + d x} \right )}}{b d} + \frac {2 \cos {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(a+b*(d*x+c)**(1/2)),x)

[Out]

Piecewise((x*cos(a), Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x*cos(a + b*sqrt(c)), Eq(d, 0)), (2*sqrt(c + d*x)*sin
(a + b*sqrt(c + d*x))/(b*d) + 2*cos(a + b*sqrt(c + d*x))/(b**2*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right ) - a \sin \left (\sqrt {d x + c} b + a\right ) + \cos \left (\sqrt {d x + c} b + a\right )\right )}}{b^{2} d} \]

[In]

integrate(cos(a+b*(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

2*((sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c)*b + a) - a*sin(sqrt(d*x + c)*b + a) + cos(sqrt(d*x + c)*b + a))/(b^
2*d)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left (\sqrt {d x + c} b \sin \left (\sqrt {d x + c} b + a\right ) + \cos \left (\sqrt {d x + c} b + a\right )\right )}}{b^{2} d} \]

[In]

integrate(cos(a+b*(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

2*(sqrt(d*x + c)*b*sin(sqrt(d*x + c)*b + a) + cos(sqrt(d*x + c)*b + a))/(b^2*d)

Mupad [B] (verification not implemented)

Time = 13.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2\,\left (\cos \left (a+b\,\sqrt {c+d\,x}\right )+b\,\sin \left (a+b\,\sqrt {c+d\,x}\right )\,\sqrt {c+d\,x}\right )}{b^2\,d} \]

[In]

int(cos(a + b*(c + d*x)^(1/2)),x)

[Out]

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