\(\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx\) [344]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac {16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e} \] Output:

1/3*(-960*cos(e*x+d)+1280*sin(e*x+d))/e/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2 
)-16/3*(3*cos(e*x+d)-4*sin(e*x+d))*(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2)/e-2 
/5*(3*cos(e*x+d)-4*sin(e*x+d))*(5+4*cos(e*x+d)+3*sin(e*x+d))^(3/2)/e
 

Mathematica [A] (verified)

Time = 6.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {(5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \left (3750 \cos \left (\frac {1}{2} (d+e x)\right )+1625 \cos \left (\frac {3}{2} (d+e x)\right )+3 \left (79 \cos \left (\frac {5}{2} (d+e x)\right )-3750 \sin \left (\frac {1}{2} (d+e x)\right )-375 \sin \left (\frac {3}{2} (d+e x)\right )+3 \sin \left (\frac {5}{2} (d+e x)\right )\right )\right )}{30 e \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^5} \] Input:

Integrate[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]
 

Output:

-1/30*((5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)*(3750*Cos[(d + e*x)/2] 
+ 1625*Cos[(3*(d + e*x))/2] + 3*(79*Cos[(5*(d + e*x))/2] - 3750*Sin[(d + e 
*x)/2] - 375*Sin[(3*(d + e*x))/2] + 3*Sin[(5*(d + e*x))/2])))/(e*(3*Cos[(d 
 + e*x)/2] + Sin[(d + e*x)/2])^5)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3042, 3592, 3042, 3592, 3042, 3591}

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.

\(\displaystyle \int (3 \sin (d+e x)+4 \cos (d+e x)+5)^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (3 \sin (d+e x)+4 \cos (d+e x)+5)^{5/2}dx\)

\(\Big \downarrow \) 3592

\(\displaystyle 8 \int (4 \cos (d+e x)+3 \sin (d+e x)+5)^{3/2}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle 8 \int (4 \cos (d+e x)+3 \sin (d+e x)+5)^{3/2}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}\)

\(\Big \downarrow \) 3592

\(\displaystyle 8 \left (\frac {20}{3} \int \sqrt {4 \cos (d+e x)+3 \sin (d+e x)+5}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}}{3 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle 8 \left (\frac {20}{3} \int \sqrt {4 \cos (d+e x)+3 \sin (d+e x)+5}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}}{3 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}\)

\(\Big \downarrow \) 3591

\(\displaystyle 8 \left (-\frac {2 \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5} (3 \cos (d+e x)-4 \sin (d+e x))}{3 e}-\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}\)

Input:

Int[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]
 

Output:

(-2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x] 
)^(3/2))/(5*e) + 8*((-40*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 
4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*S 
qrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e))
 

Defintions of rubi rules used

rule 3042

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

rule 3591

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Simp[-2*((c*Cos[d + e*x] - b*Sin[d + e*x])/(e*Sqrt[a + b* 
Cos[d + e*x] + c*Sin[d + e*x]])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^ 
2 - b^2 - c^2, 0]
 

rule 3592

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_), x_Symbol] :> Simp[(-(c*Cos[d + e*x] - b*Sin[d + e*x]))*((a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Simp[a*((2*n - 1)/n)   Int[(a 
 + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[a^2 - b^2 - c^2, 0] && GtQ[n, 0]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {50 \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (3 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )^{2}+14 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+43\right )}{3 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\)	\(74\)

Input:

int((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

50/3*(1+sin(e*x+d+arctan(4/3)))*(sin(e*x+d+arctan(4/3))-1)*(3*sin(e*x+d+ar 
ctan(4/3))^2+14*sin(e*x+d+arctan(4/3))+43)/cos(e*x+d+arctan(4/3))/(5+5*sin 
(e*x+d+arctan(4/3)))^(1/2)/e
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {2 \, {\left (237 \, \cos \left (e x + d\right )^{3} + 931 \, \cos \left (e x + d\right )^{2} + 9 \, {\left (\cos \left (e x + d\right )^{2} - 62 \, \cos \left (e x + d\right ) - 344\right )} \sin \left (e x + d\right ) + 1166 \, \cos \left (e x + d\right ) + 472\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}{15 \, {\left (3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e\right )}} \] Input:

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x, algorithm="fricas")
 

Output:

-2/15*(237*cos(e*x + d)^3 + 931*cos(e*x + d)^2 + 9*(cos(e*x + d)^2 - 62*co 
s(e*x + d) - 344)*sin(e*x + d) + 1166*cos(e*x + d) + 472)*sqrt(4*cos(e*x + 
 d) + 3*sin(e*x + d) + 5)/(3*e*cos(e*x + d) + e*sin(e*x + d) + 3*e)
 

Sympy [F(-1)]

Timed out. \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x, algorithm="maxima")
 

Output:

integrate((4*cos(e*x + d) + 3*sin(e*x + d) + 5)^(5/2), x)
 

Giac [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x, algorithm="giac")
 

Output:

integrate((4*cos(e*x + d) + 3*sin(e*x + d) + 5)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\int {\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )+5\right )}^{5/2} \,d x \] Input:

int((4*cos(d + e*x) + 3*sin(d + e*x) + 5)^(5/2),x)
 

Output:

int((4*cos(d + e*x) + 3*sin(d + e*x) + 5)^(5/2), x)
 

Reduce [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\frac {1432704 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \cos \left (e x +d \right )^{2}+673056 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \cos \left (e x +d \right ) \sin \left (e x +d \right )+366560 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \cos \left (e x +d \right )+2773896 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )^{2}+3559920 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )+3262600 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}+9455625 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e +9609375 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )^{3}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e [phone] \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )^{2}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e [phone] \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e}{75645 e} \] Input:

int((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x)
 

Output:

(1432704*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5)*cos(d + e*x)**2 + 67305 
6*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5)*cos(d + e*x)*sin(d + e*x) + 36 
6560*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5)*cos(d + e*x) + 2773896*sqrt 
(4*cos(d + e*x) + 3*sin(d + e*x) + 5)*sin(d + e*x)**2 + 3559920*sqrt(4*cos 
(d + e*x) + 3*sin(d + e*x) + 5)*sin(d + e*x) + 3262600*sqrt(4*cos(d + e*x) 
 + 3*sin(d + e*x) + 5) + 9455625*int(sqrt(4*cos(d + e*x) + 3*sin(d + e*x) 
+ 5)/(4*cos(d + e*x) + 3*sin(d + e*x) + 5),x)*e + 9609375*int((sqrt(4*cos( 
d + e*x) + 3*sin(d + e*x) + 5)*sin(d + e*x)**3)/(4*cos(d + e*x) + 3*sin(d 
+ e*x) + 5),x)*e + 21796875*int((sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5) 
*sin(d + e*x)**2)/(4*cos(d + e*x) + 3*sin(d + e*x) + 5),x)*e + 25378125*in 
t((sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5)*sin(d + e*x))/(4*cos(d + e*x) 
 + 3*sin(d + e*x) + 5),x)*e)/(75645*e)