\(\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx\) [345]

Optimal result

Integrand size = 22, antiderivative size = 93 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=-\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e} \] Output:

1/3*(-120*cos(e*x+d)+160*sin(e*x+d))/e/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2) 
-2/3*(3*cos(e*x+d)-4*sin(e*x+d))*(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2)/e
 

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.12 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\frac {(5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \left (-45 \cos \left (\frac {1}{2} (d+e x)\right )-13 \cos \left (\frac {3}{2} (d+e x)\right )+9 \left (15 \sin \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {3}{2} (d+e x)\right )\right )\right )}{3 e \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^3} \] Input:

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

Output:

((5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2)*(-45*Cos[(d + e*x)/2] - 13*Co 
s[(3*(d + e*x))/2] + 9*(15*Sin[(d + e*x)/2] + Sin[(3*(d + e*x))/2])))/(3*e 
*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^3)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3592

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3591

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

Input:

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

Output:

(-40*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 4*Cos[d + e*x] + 3*S 
in[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 = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {50 \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+5\right )}{3 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\)	\(60\)

Input:

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

Output:

50/3*(1+sin(e*x+d+arctan(4/3)))*(sin(e*x+d+arctan(4/3))-1)*(sin(e*x+d+arct 
an(4/3))+5)/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 = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=-\frac {2 \, {\left (13 \, \cos \left (e x + d\right )^{2} - 9 \, {\left (\cos \left (e x + d\right ) + 8\right )} \sin \left (e x + d\right ) + 29 \, \cos \left (e x + d\right ) + 16\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}{3 \, {\left (3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e\right )}} \] Input:

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

Output:

-2/3*(13*cos(e*x + d)^2 - 9*(cos(e*x + d) + 8)*sin(e*x + d) + 29*cos(e*x + 
 d) + 16)*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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75207 vs. \(2 (85) = 170\).

Time = 1.34 (sec) , antiderivative size = 75207, normalized size of antiderivative = 808.68 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*(54*tan(1/2*e*x + 1/2*d)^2*tan(1/4*e*x + 1/4*d)^2*tan(1/4*e*x + 1/12* 
d)^6*tan(-1/4*e*x + 1/4*d)^2*tan(1/2*d)^2*tan(1/6*d)^5 - 18*tan(1/2*e*x + 
1/2*d)^2*tan(1/4*e*x + 1/4*d)^2*tan(1/4*e*x + 1/12*d)^6*tan(-1/4*e*x + 1/4 
*d)^2*tan(1/2*d)*tan(1/6*d)^6 + 18*tan(1/2*e*x + 1/2*d)^2*tan(1/4*e*x + 1/ 
4*d)^2*tan(1/4*e*x + 1/12*d)^6*tan(-1/4*e*x + 1/4*d)*tan(1/2*d)^2*tan(1/6* 
d)^6 + 54*tan(1/2*e*x + 1/2*d)^2*tan(1/4*e*x + 1/4*d)^2*tan(1/4*e*x + 1/12 
*d)^5*tan(-1/4*e*x + 1/4*d)^2*tan(1/2*d)^2*tan(1/6*d)^6 + 90*tan(1/2*e*x + 
 1/2*d)^2*tan(1/4*e*x + 1/4*d)*tan(1/4*e*x + 1/12*d)^6*tan(-1/4*e*x + 1/4* 
d)^2*tan(1/2*d)^2*tan(1/6*d)^6 - 63*tan(1/2*e*x + 1/2*d)*tan(1/4*e*x + 1/4 
*d)^2*tan(1/4*e*x + 1/12*d)^6*tan(-1/4*e*x + 1/4*d)^2*tan(1/2*d)^2*tan(1/6 
*d)^6 - 9*tan(1/2*e*x + 1/2*d)^2*tan(1/4*e*x + 1/4*d)^2*tan(1/4*e*x + 1/12 
*d)^6*tan(-1/4*e*x + 1/4*d)^2*tan(1/2*d)^2*tan(1/6*d)^4 - 18*tan(1/2*e*x + 
 1/2*d)^2*tan(1/4*e*x + 1/4*d)^2*tan(1/4*e*x + 1/12*d)^5*tan(-1/4*e*x + 1/ 
4*d)^2*tan(1/2*d)^2*tan(1/6*d)^5 + 45*tan(1/2*e*x + 1/2*d)*tan(1/4*e*x + 1 
/4*d)^2*tan(1/4*e*x + 1/12*d)^6*tan(-1/4*e*x + 1/4*d)^2*tan(1/2*d)^2*tan(1 
/6*d)^5 + tan(1/2*e*x + 1/2*d)^2*tan(1/4*e*x + 1/4*d)^2*tan(1/4*e*x + 1/12 
*d)^6*tan(-1/4*e*x + 1/4*d)^2*tan(1/6*d)^6 - 2*tan(1/2*e*x + 1/2*d)^2*tan( 
1/4*e*x + 1/4*d)^2*tan(1/4*e*x + 1/12*d)^6*tan(-1/4*e*x + 1/4*d)*tan(1/2*d 
)*tan(1/6*d)^6 - 13*tan(1/2*e*x + 1/2*d)*tan(1/4*e*x + 1/4*d)^2*tan(1/4*e* 
x + 1/12*d)^6*tan(-1/4*e*x + 1/4*d)^2*tan(1/2*d)*tan(1/6*d)^6 + tan(1/2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\frac {-96 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \cos \left (e x +d \right )+528 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )+1520 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}+3075 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e +1875 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )^{2}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e +6250 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \sin \left (e x +d \right )}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e}{123 e} \] Input:

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

Output:

( - 96*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5)*cos(d + e*x) + 528*sqrt(4 
*cos(d + e*x) + 3*sin(d + e*x) + 5)*sin(d + e*x) + 1520*sqrt(4*cos(d + e*x 
) + 3*sin(d + e*x) + 5) + 3075*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 + 1875*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 + 6250*int((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)/(123*e)