\(\int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx\) [165]

Optimal result

Integrand size = 21, antiderivative size = 75 \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=\frac {2\ 13^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\arctan \left (\frac {3}{2}\right )\right ),2\right )}{3 d}-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt {2 \cos (c+d x)+3 \sin (c+d x)}}{3 d} \] Output:

2/3*13^(3/4)*InverseJacobiAM(1/2*c+1/2*d*x-1/2*arctan(3/2),2^(1/2))/d-2/3* 
(3*cos(d*x+c)-2*sin(d*x+c))*(2*cos(d*x+c)+3*sin(d*x+c))^(1/2)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.77 \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=\frac {2 (-3 \cos (c+d x)+2 \sin (c+d x)) \sqrt {2 \cos (c+d x)+3 \sin (c+d x)}+2\ 13^{3/4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (c+d x+\arctan \left (\frac {2}{3}\right )\right )\right ) \sec \left (c+d x+\arctan \left (\frac {2}{3}\right )\right ) \sqrt {-\left (\left (-1+\sin \left (c+d x+\arctan \left (\frac {2}{3}\right )\right )\right ) \sin \left (c+d x+\arctan \left (\frac {2}{3}\right )\right )\right )} \sqrt {1+\sin \left (c+d x+\arctan \left (\frac {2}{3}\right )\right )}}{3 d} \] Input:

Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2),x]
 

Output:

(2*(-3*Cos[c + d*x] + 2*Sin[c + d*x])*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x] 
] + 2*13^(3/4)*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[2 
/3]]^2]*Sec[c + d*x + ArcTan[2/3]]*Sqrt[-((-1 + Sin[c + d*x + ArcTan[2/3]] 
)*Sin[c + d*x + ArcTan[2/3]])]*Sqrt[1 + Sin[c + d*x + ArcTan[2/3]]])/(3*d)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3552, 3042, 3556, 3042, 3120}

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.

\(\displaystyle \int (3 \sin (c+d x)+2 \cos (c+d x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}dx\)

\(\Big \downarrow \) 3552

\(\displaystyle \frac {13}{3} \int \frac {1}{\sqrt {2 \cos (c+d x)+3 \sin (c+d x)}}dx-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {13}{3} \int \frac {1}{\sqrt {2 \cos (c+d x)+3 \sin (c+d x)}}dx-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3556

\(\displaystyle \frac {1}{3} 13^{3/4} \int \frac {1}{\sqrt {\cos \left (c+d x-\arctan \left (\frac {3}{2}\right )\right )}}dx-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} 13^{3/4} \int \frac {1}{\sqrt {\sin \left (c+d x-\arctan \left (\frac {3}{2}\right )+\frac {\pi }{2}\right )}}dx-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {2\ 13^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\arctan \left (\frac {3}{2}\right )\right ),2\right )}{3 d}-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}{3 d}\)

Input:

Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2),x]
 

Output:

(2*13^(3/4)*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(3*d) - (2*(3*Cos[c + 
 d*x] - 2*Sin[c + d*x])*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])/(3*d)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3552

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x 
_Symbol] :> Simp[(-(b*Cos[c + d*x] - a*Sin[c + d*x]))*((a*Cos[c + d*x] + b* 
Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[(n - 1)*((a^2 + b^2)/n)   Int[(a*Co 
s[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d}, x] && N 
eQ[a^2 + b^2, 0] &&  !IntegerQ[(n - 1)/2] && GtQ[n, 1]
 

rule 3556

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x 
_Symbol] :> Simp[(a^2 + b^2)^(n/2)   Int[Cos[c + d*x - ArcTan[a, b]]^n, x], 
 x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) && GtQ[a^2 
+ b^2, 0]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.44

method	result	size
default	\(\frac {\frac {13 \sqrt {\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+1}\, \sqrt {-2 \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+2}\, \sqrt {-\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {26 \cos \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )^{2} \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}{3}}{\cos \left (d x +c +\arctan \left (\frac {2}{3}\right )\right ) \sqrt {\sqrt {13}\, \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, d}\)	\(108\)

Input:

int((2*cos(d*x+c)+3*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

(13/3*(sin(d*x+c+arctan(2/3))+1)^(1/2)*(-2*sin(d*x+c+arctan(2/3))+2)^(1/2) 
*(-sin(d*x+c+arctan(2/3)))^(1/2)*EllipticF((sin(d*x+c+arctan(2/3))+1)^(1/2 
),1/2*2^(1/2))-26/3*cos(d*x+c+arctan(2/3))^2*sin(d*x+c+arctan(2/3)))/cos(d 
*x+c+arctan(2/3))/(13^(1/2)*sin(d*x+c+arctan(2/3)))^(1/2)/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.19 \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left ({\left (3 \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right )\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )} - \left (2 i + 3\right ) \, \sqrt {\frac {3}{2} i + 1} {\rm weierstrassPInverse}\left (\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + \left (2 i - 3\right ) \, \sqrt {-\frac {3}{2} i + 1} {\rm weierstrassPInverse}\left (-\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right )}}{3 \, d} \] Input:

integrate((2*cos(d*x+c)+3*sin(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

-2/3*((3*cos(d*x + c) - 2*sin(d*x + c))*sqrt(2*cos(d*x + c) + 3*sin(d*x + 
c)) - (2*I + 3)*sqrt(3/2*I + 1)*weierstrassPInverse(48/13*I + 20/13, 0, co 
s(d*x + c) - I*sin(d*x + c)) + (2*I - 3)*sqrt(-3/2*I + 1)*weierstrassPInve 
rse(-48/13*I + 20/13, 0, cos(d*x + c) + I*sin(d*x + c)))/d
 

Sympy [F(-1)]

Timed out. \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate((2*cos(d*x+c)+3*sin(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((2*cos(d*x+c)+3*sin(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate((2*cos(d*x + c) + 3*sin(d*x + c))^(3/2), x)
 

Giac [F]

\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((2*cos(d*x+c)+3*sin(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate((2*cos(d*x + c) + 3*sin(d*x + c))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=\int {\left (2\,\cos \left (c+d\,x\right )+3\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:

int((2*cos(c + d*x) + 3*sin(c + d*x))^(3/2),x)
 

Output:

int((2*cos(c + d*x) + 3*sin(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx=\frac {-88 \sqrt {2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )}\, \cos \left (d x +c \right )+102 \sqrt {2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )}\, \sin \left (d x +c \right )-507 \left (\int \frac {\sqrt {2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )}d x \right ) d}{18 d} \] Input:

int((2*cos(d*x+c)+3*sin(d*x+c))^(3/2),x)
                                                                                    
                                                                                    
 

Output:

( - 88*sqrt(2*cos(c + d*x) + 3*sin(c + d*x))*cos(c + d*x) + 102*sqrt(2*cos 
(c + d*x) + 3*sin(c + d*x))*sin(c + d*x) - 507*int((sqrt(2*cos(c + d*x) + 
3*sin(c + d*x))*cos(c + d*x)*sin(c + d*x))/(2*cos(c + d*x) + 3*sin(c + d*x 
)),x)*d)/(18*d)