\(\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx\) [350]

Optimal result

Integrand size = 22, antiderivative size = 185 \[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx=\frac {6400 (3 \cos (d+e x)-4 \sin (d+e x))}{7 e \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac {320 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}}{7 e}+\frac {24 (3 \cos (d+e x)-4 \sin (d+e x)) (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{7 e}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}}{7 e} \] Output:

6400/7*(3*cos(e*x+d)-4*sin(e*x+d))/e/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2)- 
320/7*(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 
4/7*(3*cos(e*x+d)-4*sin(e*x+d))*(-5+4*cos(e*x+d)+3*sin(e*x+d))^(3/2)/e-2/7 
*(3*cos(e*x+d)-4*sin(e*x+d))*(-5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2)/e
 

Mathematica [A] (verified)

Time = 7.34 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.82 \[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx=\frac {(-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \left (91875 \cos \left (\frac {1}{2} (d+e x)\right )-11025 \cos \left (\frac {3}{2} (d+e x)\right )-147 \cos \left (\frac {5}{2} (d+e x)\right )+249 \cos \left (\frac {7}{2} (d+e x)\right )+30625 \sin \left (\frac {1}{2} (d+e x)\right )-15925 \sin \left (\frac {3}{2} (d+e x)\right )+3871 \sin \left (\frac {5}{2} (d+e x)\right )-307 \sin \left (\frac {7}{2} (d+e x)\right )\right )}{28 e \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )^7} \] Input:

Integrate[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(7/2),x]
 

Output:

((-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(7/2)*(91875*Cos[(d + e*x)/2] - 11 
025*Cos[(3*(d + e*x))/2] - 147*Cos[(5*(d + e*x))/2] + 249*Cos[(7*(d + e*x) 
)/2] + 30625*Sin[(d + e*x)/2] - 15925*Sin[(3*(d + e*x))/2] + 3871*Sin[(5*( 
d + e*x))/2] - 307*Sin[(7*(d + e*x))/2]))/(28*e*(Cos[(d + e*x)/2] - 3*Sin[ 
(d + e*x)/2])^7)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 3592, 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.

\(\displaystyle \int (3 \sin (d+e x)+4 \cos (d+e x)-5)^{7/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (3 \sin (d+e x)+4 \cos (d+e x)-5)^{7/2}dx\)

\(\Big \downarrow \) 3592

\(\displaystyle -\frac {60}{7} \int (4 \cos (d+e x)+3 \sin (d+e x)-5)^{5/2}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {60}{7} \int (4 \cos (d+e x)+3 \sin (d+e x)-5)^{5/2}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}\)

\(\Big \downarrow \) 3592

\(\displaystyle -\frac {60}{7} \left (-8 \int (4 \cos (d+e x)+3 \sin (d+e x)-5)^{3/2}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}{5 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {60}{7} \left (-8 \int (4 \cos (d+e x)+3 \sin (d+e x)-5)^{3/2}dx-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}{5 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}\)

\(\Big \downarrow \) 3592

\(\displaystyle -\frac {60}{7} \left (-8 \left (-\frac {20}{3} \int \sqrt {4 \cos (d+e x)+3 \sin (d+e x)-5}dx-\frac {2 \sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5} (3 \cos (d+e x)-4 \sin (d+e x))}{3 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}{5 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {60}{7} \left (-8 \left (-\frac {20}{3} \int \sqrt {4 \cos (d+e x)+3 \sin (d+e x)-5}dx-\frac {2 \sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5} (3 \cos (d+e x)-4 \sin (d+e x))}{3 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}{5 e}\right )-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}\)

\(\Big \downarrow \) 3591

\(\displaystyle -\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}-\frac {60}{7} \left (-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}{5 e}-8 \left (\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5}}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5}}{3 e}\right )\right )\)

Input:

Int[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(7/2),x]
 

Output:

(-2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x 
])^(5/2))/(7*e) - (60*((-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))) 
)/7
 

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 = 0.57 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.46

method	result	size
default	\(\frac {250 \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \left (5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )^{3}-27 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )^{2}+71 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-177\right )}{7 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\)	\(86\)

Input:

int((-5+4*cos(e*x+d)+3*sin(e*x+d))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

250/7*(sin(e*x+d+arctan(4/3))-1)*(1+sin(e*x+d+arctan(4/3)))*(5*sin(e*x+d+a 
rctan(4/3))^3-27*sin(e*x+d+arctan(4/3))^2+71*sin(e*x+d+arctan(4/3))-177)/c 
os(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 = 121, normalized size of antiderivative = 0.65 \[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx=-\frac {2 \, {\left (249 \, \cos \left (e x + d\right )^{4} + 51 \, \cos \left (e x + d\right )^{3} - 3042 \, \cos \left (e x + d\right )^{2} - {\left (307 \, \cos \left (e x + d\right )^{3} - 1782 \, \cos \left (e x + d\right )^{2} + 2860 \, \cos \left (e x + d\right ) - 1392\right )} \sin \left (e x + d\right ) + 10068 \, \cos \left (e x + d\right ) + 12912\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{7 \, {\left (e \cos \left (e x + d\right ) - 3 \, e \sin \left (e x + d\right ) + e\right )}} \] Input:

integrate((-5+4*cos(e*x+d)+3*sin(e*x+d))^(7/2),x, algorithm="fricas")
 

Output:

-2/7*(249*cos(e*x + d)^4 + 51*cos(e*x + d)^3 - 3042*cos(e*x + d)^2 - (307* 
cos(e*x + d)^3 - 1782*cos(e*x + d)^2 + 2860*cos(e*x + d) - 1392)*sin(e*x + 
 d) + 10068*cos(e*x + d) + 12912)*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. \[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx=\text {Timed out} \] Input:

integrate((-5+4*cos(e*x+d)+3*sin(e*x+d))**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {7}{2}} \,d x } \] Input:

integrate((-5+4*cos(e*x+d)+3*sin(e*x+d))^(7/2),x, algorithm="maxima")
 

Output:

integrate((4*cos(e*x + d) + 3*sin(e*x + d) - 5)^(7/2), x)
 

Giac [F]

\[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {7}{2}} \,d x } \] Input:

integrate((-5+4*cos(e*x+d)+3*sin(e*x+d))^(7/2),x, algorithm="giac")
 

Output:

integrate((4*cos(e*x + d) + 3*sin(e*x + d) - 5)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx=\int {\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5\right )}^{7/2} \,d x \] Input:

int((4*cos(d + e*x) + 3*sin(d + e*x) - 5)^(7/2),x)
 

Output:

int((4*cos(d + e*x) + 3*sin(d + e*x) - 5)^(7/2), x)
 

Reduce [F]

\[ \int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx =\text {Too large to display} \] Input:

int((-5+4*cos(e*x+d)+3*sin(e*x+d))^(7/2),x)
 

Output:

( - 224635392*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*cos(d + e*x)**3 + 
1092192768*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*cos(d + e*x)**2*sin(d 
 + e*x) - 3392454144*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*cos(d + e*x 
)**2 + 504461376*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*cos(d + e*x)*si 
n(d + e*x)**2 - 2402361216*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*cos(d 
 + e*x)*sin(d + e*x) + 1750890560*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5 
)*cos(d + e*x) + 1560344544*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*sin( 
d + e*x)**3 - 7507215456*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*sin(d + 
 e*x)**2 + 5790037920*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*sin(d + e* 
x) - 1698925600*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5) + 9843305625*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 + 5909765625*int((sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)*si 
n(d + e*x)**4)/(4*cos(d + e*x) + 3*sin(d + e*x) - 5),x)*e - 33248437500*in 
t((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 + 50106093750*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 - 35776237500*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)/(15749289*e)