\(\int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx\) [337]

Optimal result

Integrand size = 22, antiderivative size = 347 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx=-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{15 e}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}+\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{15 e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {16 a \left (a^2-b^2-c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{15 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \] Output:

-16/15*(a*c*cos(e*x+d)-a*b*sin(e*x+d))*(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2) 
/e-2/5*(c*cos(e*x+d)-b*sin(e*x+d))*(a+b*cos(e*x+d)+c*sin(e*x+d))^(3/2)/e+2 
/15*(23*a^2+9*b^2+9*c^2)*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(c,b)),2^(1 
/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*(a+b*cos(e*x+d)+c*sin(e*x 
+d))^(1/2)/e/((a+b*cos(e*x+d)+c*sin(e*x+d))/(a+(b^2+c^2)^(1/2)))^(1/2)-16/ 
15*a*(a^2-b^2-c^2)*InverseJacobiAM(1/2*d+1/2*e*x-1/2*arctan(c,b),2^(1/2)*( 
(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*((a+b*cos(e*x+d)+c*sin(e*x+d)) 
/(a+(b^2+c^2)^(1/2)))^(1/2)/e/(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.53 (sec) , antiderivative size = 3767, normalized size of antiderivative = 10.86 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]
 

Output:

(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((2*b*(23*a^2 + 9*b^2 + 9*c^2)) 
/(15*c) - (22*a*c*Cos[d + e*x])/15 - (2*b*c*Cos[2*(d + e*x)])/5 + (22*a*b* 
Sin[d + e*x])/15 + ((b^2 - c^2)*Sin[2*(d + e*x)])/5))/e + (2*a^3*AppellF1[ 
1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]]) 
/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2 
/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b 
^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] 
- c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c 
^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sq 
rt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan 
[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*e) + (34*a*b 
^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + A 
rcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + 
Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a 
/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 
+ c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*S 
qrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcT 
an[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + 
e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(15*Sqrt[1 + b^2/c^2] 
*c*e) + (34*a*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c...
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {3042, 3599, 27, 3042, 3625, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}

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.

\(\displaystyle \int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}dx\)

\(\Big \downarrow \) 3599

\(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \left (5 a^2+8 b \cos (d+e x) a+8 c \sin (d+e x) a+3 \left (b^2+c^2\right )\right )dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \left (5 a^2+8 b \cos (d+e x) a+8 c \sin (d+e x) a+3 \left (b^2+c^2\right )\right )dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \left (5 a^2+8 b \cos (d+e x) a+8 c \sin (d+e x) a+3 \left (b^2+c^2\right )\right )dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3625

\(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {\left (15 a^2+17 \left (b^2+c^2\right )\right ) a^2+b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x) a+c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x) a}{2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (\frac {\int \frac {\left (15 a^2+17 \left (b^2+c^2\right )\right ) a^2+b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x) a+c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x) a}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {\int \frac {\left (15 a^2+17 \left (b^2+c^2\right )\right ) a^2+b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x) a+c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x) a}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {1}{5} \left (\frac {a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}dx-8 a^2 \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}dx-8 a^2 \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {1}{5} \left (\frac {\frac {a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-8 a^2 \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {\frac {a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-8 a^2 \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{5} \left (\frac {\frac {2 a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-8 a^2 \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {1}{5} \left (\frac {\frac {2 a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {8 a^2 \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (\frac {\frac {2 a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {8 a^2 \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{5} \left (\frac {\frac {2 a \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {16 a^2 \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 a}-\frac {16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}\)

Input:

Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]
 

Output:

(-2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x] 
)^(3/2))/(5*e) + ((-16*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x])*Sqrt[a + b*Co 
s[d + e*x] + c*Sin[d + e*x]])/(3*e) + ((2*a*(23*a^2 + 9*(b^2 + c^2))*Ellip 
ticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2]) 
]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*Sqrt[(a + b*Cos[d + e*x] + 
 c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (16*a^2*(a^2 - b^2 - c^2)*Ellip 
ticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2]) 
]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(e*Sq 
rt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]))/(3*a))/5
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3598

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

rule 3599

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[1/n   Int[Simp[n*a^2 + ( 
n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x 
], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, 
 c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
 

rule 3606

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

rule 3625

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

rule 3628

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) 
/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] 
, x_Symbol] :> Simp[B/b   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] 
, x] + Simp[(A*b - a*B)/b   Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] 
, x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ 
A*b - a*B, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2259\) vs. \(2(325)=650\).

Time = 1.08 (sec) , antiderivative size = 2260, normalized size of antiderivative = 6.51

method	result	size
default	\(\text {Expression too large to display}\)	\(2260\)

Input:

int((a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

(-(-b^2*sin(e*x+d-arctan(-b,c))-c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2)^(1 
/2))*cos(e*x+d-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)*(2*a^3*(1/(b^2+c^2)^ 
(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2) 
))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)) 
)^(1/2)*(-(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c))-1)/(a+(b^2+c^2)^(1/2))) 
^(1/2)/(-(-b^2*sin(e*x+d-arctan(-b,c))-c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+ 
c^2)^(1/2))*cos(e*x+d-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)*EllipticF(((( 
b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a- 
(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+(b^2+c^2)^(3/2)*(-2/5/(b^2+c 
^2)^(1/2)*sin(e*x+d-arctan(-b,c))*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c) 
)+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)+8/15/(b^2+c^2)*a*(((b^2+c^2)^(1/2)*s 
in(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)+4/15/(b^2+c^2)^ 
(1/2)*a*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+ 
a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2) 
/(-a+(b^2+c^2)^(1/2)))^(1/2)*(-(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c))-1) 
/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*c 
os(e*x+d-arctan(-b,c))^2)^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(e*x+d-arct 
an(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2 
)^(1/2)))^(1/2))+2*(3/5+8/15/(b^2+c^2)*a^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2 
+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 1589, normalized size of antiderivative = 4.58 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),x, algorithm="fricas")
 

Output:

-2/45*((I*a^3*b - 33*I*a*b^3 - 33*I*a*b*c^2 - 33*a*c^3 + (a^3 - 33*a*b^2)* 
c)*sqrt(1/2*b + 1/2*I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^ 
2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^ 
4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a 
*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 
+ 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(e 
*x + d) - 3*(I*b^2 + I*c^2)*sin(e*x + d))/(b^2 + c^2)) + (-I*a^3*b + 33*I* 
a*b^3 + 33*I*a*b*c^2 - 33*a*c^3 + (a^3 - 33*a*b^2)*c)*sqrt(1/2*b - 1/2*I*c 
)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c 
^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 
9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b 
- 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 
 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(-I*b^2 - I 
*c^2)*sin(e*x + d))/(b^2 + c^2)) + 3*(23*I*a^2*b^2 + 9*I*b^4 + 9*I*c^4 + I 
*(23*a^2 + 18*b^2)*c^2)*sqrt(1/2*b + 1/2*I*c)*weierstrassZeta(4/3*(4*a^2*b 
^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 
 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 
 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 
9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), weierstrassPInverse(4/3*( 
4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^...
 

Sympy [F(-1)]

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

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))**(5/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),x, algorithm="maxima")
 

Output:

integrate((b*cos(e*x + d) + c*sin(e*x + d) + a)^(5/2), x)
 

Giac [F]

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

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),x, algorithm="giac")
 

Output:

integrate((b*cos(e*x + d) + c*sin(e*x + d) + a)^(5/2), x)
 

Mupad [F(-1)]

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

int((a + b*cos(d + e*x) + c*sin(d + e*x))^(5/2),x)
 

Output:

int((a + b*cos(d + e*x) + c*sin(d + e*x))^(5/2), x)
 

Reduce [F]

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

int((a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),x)
 

Output:

int((a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),x)