\(\int x \cos (a+b \sqrt {c+d x}) \, dx\) [91]

Optimal result

Integrand size = 16, antiderivative size = 167 \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=-\frac {12 \cos \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}-\frac {2 c \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {6 (c+d x) \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {12 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2} \] Output:

-12*cos(a+b*(d*x+c)^(1/2))/b^4/d^2-2*c*cos(a+b*(d*x+c)^(1/2))/b^2/d^2+6*(d 
*x+c)*cos(a+b*(d*x+c)^(1/2))/b^2/d^2-12*(d*x+c)^(1/2)*sin(a+b*(d*x+c)^(1/2 
))/b^3/d^2-2*c*(d*x+c)^(1/2)*sin(a+b*(d*x+c)^(1/2))/b/d^2+2*(d*x+c)^(3/2)* 
sin(a+b*(d*x+c)^(1/2))/b/d^2
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.43 \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \left (\left (-6+b^2 (2 c+3 d x)\right ) \cos \left (a+b \sqrt {c+d x}\right )+b \sqrt {c+d x} \left (-6+b^2 d x\right ) \sin \left (a+b \sqrt {c+d x}\right )\right )}{b^4 d^2} \] Input:

Integrate[x*Cos[a + b*Sqrt[c + d*x]],x]
 

Output:

(2*((-6 + b^2*(2*c + 3*d*x))*Cos[a + b*Sqrt[c + d*x]] + b*Sqrt[c + d*x]*(- 
6 + b^2*d*x)*Sin[a + b*Sqrt[c + d*x]]))/(b^4*d^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3913, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx\)

\(\Big \downarrow \) 3913

\(\displaystyle \frac {2 \int \left (\frac {(c+d x)^{3/2} \cos \left (a+b \sqrt {c+d x}\right )}{d}-\frac {c \sqrt {c+d x} \cos \left (a+b \sqrt {c+d x}\right )}{d}\right )d\sqrt {c+d x}}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {6 \cos \left (a+b \sqrt {c+d x}\right )}{b^4 d}-\frac {6 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b^3 d}+\frac {3 (c+d x) \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d}-\frac {c \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d}+\frac {(c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d}-\frac {c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d}\right )}{d}\)

Input:

Int[x*Cos[a + b*Sqrt[c + d*x]],x]
 

Output:

(2*((-6*Cos[a + b*Sqrt[c + d*x]])/(b^4*d) - (c*Cos[a + b*Sqrt[c + d*x]])/( 
b^2*d) + (3*(c + d*x)*Cos[a + b*Sqrt[c + d*x]])/(b^2*d) - (6*Sqrt[c + d*x] 
*Sin[a + b*Sqrt[c + d*x]])/(b^3*d) - (c*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d 
*x]])/(b*d) + ((c + d*x)^(3/2)*Sin[a + b*Sqrt[c + d*x]])/(b*d)))/d
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3913

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_ 
.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[1/(n*f)   Subst[Int[ExpandIntegra 
nd[(a + b*Cos[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m, x], 
 x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p 
, 0] && IntegerQ[1/n]
 

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.78

method	result	size
parts	\(\frac {2 x \sqrt {d x +c}\, \sin \left (a +\sqrt {d x +c}\, b \right )}{d b}+\frac {2 x \cos \left (a +\sqrt {d x +c}\, b \right )}{d \,b^{2}}-\frac {2 \left (\frac {-2 \left (a +\sqrt {d x +c}\, b \right )^{2} \cos \left (a +\sqrt {d x +c}\, b \right )+4 \cos \left (a +\sqrt {d x +c}\, b \right )+4 \left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )-2 a \left (\sin \left (a +\sqrt {d x +c}\, b \right )-\left (a +\sqrt {d x +c}\, b \right ) \cos \left (a +\sqrt {d x +c}\, b \right )\right )}{b^{2} d}-\frac {2 a \left (\sin \left (a +\sqrt {d x +c}\, b \right )-\left (a +\sqrt {d x +c}\, b \right ) \cos \left (a +\sqrt {d x +c}\, b \right )+\cos \left (a +\sqrt {d x +c}\, b \right ) a \right )}{d \,b^{2}}+\frac {2 \cos \left (a +\sqrt {d x +c}\, b \right )+2 \left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )-2 a \sin \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d}\right )}{d \,b^{2}}\)	\(297\)
derivativedivides	\(\frac {2 a c \sin \left (a +\sqrt {d x +c}\, b \right )-2 c \left (\cos \left (a +\sqrt {d x +c}\, b \right )+\left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )\right )-\frac {2 a^{3} \sin \left (a +\sqrt {d x +c}\, b \right )}{b^{2}}+\frac {6 a^{2} \left (\cos \left (a +\sqrt {d x +c}\, b \right )+\left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )\right )}{b^{2}}-\frac {6 a \left (\left (a +\sqrt {d x +c}\, b \right )^{2} \sin \left (a +\sqrt {d x +c}\, b \right )-2 \sin \left (a +\sqrt {d x +c}\, b \right )+2 \left (a +\sqrt {d x +c}\, b \right ) \cos \left (a +\sqrt {d x +c}\, b \right )\right )}{b^{2}}+\frac {2 \left (\left (a +\sqrt {d x +c}\, b \right )^{3} \sin \left (a +\sqrt {d x +c}\, b \right )+3 \left (a +\sqrt {d x +c}\, b \right )^{2} \cos \left (a +\sqrt {d x +c}\, b \right )-6 \cos \left (a +\sqrt {d x +c}\, b \right )-6 \left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )\right )}{b^{2}}}{b^{2} d^{2}}\)	\(299\)
default	\(\frac {2 a c \sin \left (a +\sqrt {d x +c}\, b \right )-2 c \left (\cos \left (a +\sqrt {d x +c}\, b \right )+\left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )\right )-\frac {2 a^{3} \sin \left (a +\sqrt {d x +c}\, b \right )}{b^{2}}+\frac {6 a^{2} \left (\cos \left (a +\sqrt {d x +c}\, b \right )+\left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )\right )}{b^{2}}-\frac {6 a \left (\left (a +\sqrt {d x +c}\, b \right )^{2} \sin \left (a +\sqrt {d x +c}\, b \right )-2 \sin \left (a +\sqrt {d x +c}\, b \right )+2 \left (a +\sqrt {d x +c}\, b \right ) \cos \left (a +\sqrt {d x +c}\, b \right )\right )}{b^{2}}+\frac {2 \left (\left (a +\sqrt {d x +c}\, b \right )^{3} \sin \left (a +\sqrt {d x +c}\, b \right )+3 \left (a +\sqrt {d x +c}\, b \right )^{2} \cos \left (a +\sqrt {d x +c}\, b \right )-6 \cos \left (a +\sqrt {d x +c}\, b \right )-6 \left (a +\sqrt {d x +c}\, b \right ) \sin \left (a +\sqrt {d x +c}\, b \right )\right )}{b^{2}}}{b^{2} d^{2}}\)	\(299\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.40 \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left ({\left (b^{3} d x - 6 \, b\right )} \sqrt {d x + c} \sin \left (\sqrt {d x + c} b + a\right ) + {\left (3 \, b^{2} d x + 2 \, b^{2} c - 6\right )} \cos \left (\sqrt {d x + c} b + a\right )\right )}}{b^{4} d^{2}} \] Input:

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

Output:

2*((b^3*d*x - 6*b)*sqrt(d*x + c)*sin(sqrt(d*x + c)*b + a) + (3*b^2*d*x + 2 
*b^2*c - 6)*cos(sqrt(d*x + c)*b + a))/(b^4*d^2)
 

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90 \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\begin {cases} \frac {x^{2} \cos {\left (a \right )}}{2} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{2} \cos {\left (a + b \sqrt {c} \right )}}{2} & \text {for}\: d = 0 \\\frac {2 x \sqrt {c + d x} \sin {\left (a + b \sqrt {c + d x} \right )}}{b d} + \frac {4 c \cos {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} + \frac {6 x \cos {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} - \frac {12 \sqrt {c + d x} \sin {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {12 \cos {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((x**2*cos(a)/2, Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x**2*cos(a + 
 b*sqrt(c))/2, Eq(d, 0)), (2*x*sqrt(c + d*x)*sin(a + b*sqrt(c + d*x))/(b*d 
) + 4*c*cos(a + b*sqrt(c + d*x))/(b**2*d**2) + 6*x*cos(a + b*sqrt(c + d*x) 
)/(b**2*d) - 12*sqrt(c + d*x)*sin(a + b*sqrt(c + d*x))/(b**3*d**2) - 12*co 
s(a + b*sqrt(c + d*x))/(b**4*d**2), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.57 \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left (a c \sin \left (\sqrt {d x + c} b + a\right ) - {\left ({\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right ) + \cos \left (\sqrt {d x + c} b + a\right )\right )} c - \frac {a^{3} \sin \left (\sqrt {d x + c} b + a\right )}{b^{2}} + \frac {3 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right ) + \cos \left (\sqrt {d x + c} b + a\right )\right )} a^{2}}{b^{2}} - \frac {3 \, {\left (2 \, {\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) + {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} a}{b^{2}} + \frac {3 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \cos \left (\sqrt {d x + c} b + a\right ) + {\left ({\left (\sqrt {d x + c} b + a\right )}^{3} - 6 \, \sqrt {d x + c} b - 6 \, a\right )} \sin \left (\sqrt {d x + c} b + a\right )}{b^{2}}\right )}}{b^{2} d^{2}} \] Input:

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

Output:

2*(a*c*sin(sqrt(d*x + c)*b + a) - ((sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c) 
*b + a) + cos(sqrt(d*x + c)*b + a))*c - a^3*sin(sqrt(d*x + c)*b + a)/b^2 + 
 3*((sqrt(d*x + c)*b + a)*sin(sqrt(d*x + c)*b + a) + cos(sqrt(d*x + c)*b + 
 a))*a^2/b^2 - 3*(2*(sqrt(d*x + c)*b + a)*cos(sqrt(d*x + c)*b + a) + ((sqr 
t(d*x + c)*b + a)^2 - 2)*sin(sqrt(d*x + c)*b + a))*a/b^2 + (3*((sqrt(d*x + 
 c)*b + a)^2 - 2)*cos(sqrt(d*x + c)*b + a) + ((sqrt(d*x + c)*b + a)^3 - 6* 
sqrt(d*x + c)*b - 6*a)*sin(sqrt(d*x + c)*b + a))/b^2)/(b^2*d^2)
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99 \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=-\frac {2 \, {\left (\frac {{\left (b^{2} c - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 6 \, {\left (\sqrt {d x + c} b + a\right )} a - 3 \, a^{2} + 6\right )} \cos \left (\sqrt {d x + c} b + a\right )}{b^{2}} + \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} + 6 \, \sqrt {d x + c} b\right )} \sin \left (\sqrt {d x + c} b + a\right )}{b^{2}}\right )}}{b^{2} d^{2}} \] Input:

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

Output:

-2*((b^2*c - 3*(sqrt(d*x + c)*b + a)^2 + 6*(sqrt(d*x + c)*b + a)*a - 3*a^2 
 + 6)*cos(sqrt(d*x + c)*b + a)/b^2 + ((sqrt(d*x + c)*b + a)*b^2*c - a*b^2* 
c - (sqrt(d*x + c)*b + a)^3 + 3*(sqrt(d*x + c)*b + a)^2*a - 3*(sqrt(d*x + 
c)*b + a)*a^2 + a^3 + 6*sqrt(d*x + c)*b)*sin(sqrt(d*x + c)*b + a)/b^2)/(b^ 
2*d^2)
 

Mupad [F(-1)]

Timed out. \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\int x\,\cos \left (a+b\,\sqrt {c+d\,x}\right ) \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.60 \[ \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {4 \cos \left (\sqrt {d x +c}\, b +a \right ) b^{2} c +6 \cos \left (\sqrt {d x +c}\, b +a \right ) b^{2} d x -12 \cos \left (\sqrt {d x +c}\, b +a \right )+2 \sqrt {d x +c}\, \sin \left (\sqrt {d x +c}\, b +a \right ) b^{3} d x -12 \sqrt {d x +c}\, \sin \left (\sqrt {d x +c}\, b +a \right ) b}{b^{4} d^{2}} \] Input:

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

Output:

(2*(2*cos(sqrt(c + d*x)*b + a)*b**2*c + 3*cos(sqrt(c + d*x)*b + a)*b**2*d* 
x - 6*cos(sqrt(c + d*x)*b + a) + sqrt(c + d*x)*sin(sqrt(c + d*x)*b + a)*b* 
*3*d*x - 6*sqrt(c + d*x)*sin(sqrt(c + d*x)*b + a)*b))/(b**4*d**2)