\(\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx\) [351]

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 = 5.75 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\frac {(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \left (11250 \cos \left (\frac {1}{2} (d+e x)\right )-1125 \cos \left (\frac {3}{2} (d+e x)\right )-9 \cos \left (\frac {5}{2} (d+e x)\right )+3750 \sin \left (\frac {1}{2} (d+e x)\right )-1625 \sin \left (\frac {3}{2} (d+e x)\right )+237 \sin \left (\frac {5}{2} (d+e x)\right )\right )}{30 e \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )^5} \] Input:

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

Output:

((-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)*(11250*Cos[(d + e*x)/2] - 11 
25*Cos[(3*(d + e*x))/2] - 9*Cos[(5*(d + e*x))/2] + 3750*Sin[(d + e*x)/2] - 
 1625*Sin[(3*(d + e*x))/2] + 237*Sin[(5*(d + e*x))/2]))/(30*e*(Cos[(d + e* 
x)/2] - 3*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

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3591

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

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])* 
Sqrt[-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 = 0.41 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.53

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

Input:

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

Output:

50/3*(sin(e*x+d+arctan(4/3))-1)*(1+sin(e*x+d+arctan(4/3)))*(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*si 
n(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 \[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {2 \, {\left (9 \, \cos \left (e x + d\right )^{3} + 567 \, \cos \left (e x + d\right )^{2} - {\left (237 \, \cos \left (e x + d\right )^{2} - 694 \, \cos \left (e x + d\right ) + 472\right )} \sin \left (e x + d\right ) - 2538 \, \cos \left (e x + d\right ) - 3096\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{15 \, {\left (e \cos \left (e x + d\right ) - 3 \, e \sin \left (e x + d\right ) + e\right )}} \] Input:

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

Output:

-2/15*(9*cos(e*x + d)^3 + 567*cos(e*x + d)^2 - (237*cos(e*x + d)^2 - 694*c 
os(e*x + d) + 472)*sin(e*x + d) - 2538*cos(e*x + d) - 3096)*sqrt(4*cos(e*x 
 + d) + 3*sin(e*x + d) - 5)/(e*cos(e*x + d) - 3*e*sin(e*x + d) + 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]

\[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\frac {1432704 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \cos \left (e x +d \right )^{2}+673056 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \cos \left (e x +d \right ) \sin \left (e x +d \right )-366560 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \cos \left (e x +d \right )+2773896 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \sin \left (e x +d \right )^{2}-3559920 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \sin \left (e x +d \right )+3262600 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}-9455625 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}d x \right ) e +9609375 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \sin \left (e x +d \right )^{3}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}d x \right ) e -21796875 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \sin \left (e x +d \right )^{2}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}d x \right ) e [phone] \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}\, \sin \left (e x +d \right )}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}d x \right ) e}{75645 e} \] 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)