3.5.15 \(\int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx\) [415]

3.5.15.1 Optimal result

Integrand size = 22, antiderivative size = 382 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {8 a E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{3 \left (a^2-b^2-c^2\right ) e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]

output

2/3*(c*cos(e*x+d)-b*sin(e*x+d))/(a^2-b^2-c^2)/e/(a+b*cos(e*x+d)+c*sin(e*x+ 
d))^(3/2)+8/3*(a*c*cos(e*x+d)-a*b*sin(e*x+d))/(a^2-b^2-c^2)^2/e/(a+b*cos(e 
*x+d)+c*sin(e*x+d))^(1/2)+8/3*a*(cos(1/2*d+1/2*e*x-1/2*arctan(b,c))^2)^(1/ 
2)/cos(1/2*d+1/2*e*x-1/2*arctan(b,c))*EllipticE(sin(1/2*d+1/2*e*x-1/2*arct 
an(b,c)),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)/(a^2-b^2-c^2)^2/e/((a+b*cos(e*x+d)+c*sin(e*x+d))/( 
a+(b^2+c^2)^(1/2)))^(1/2)-2/3*(cos(1/2*d+1/2*e*x-1/2*arctan(b,c))^2)^(1/2) 
/cos(1/2*d+1/2*e*x-1/2*arctan(b,c))*EllipticF(sin(1/2*d+1/2*e*x-1/2*arctan 
(b,c)),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)/(a^2-b^2-c^2)/e/(a+b*cos(e*x+d 
)+c*sin(e*x+d))^(1/2)
 

3.5.15.2 Mathematica [C] (warning: unable to verify)

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

Time = 6.69 (sec) , antiderivative size = 2408, normalized size of antiderivative = 6.30 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\text {Result too large to show} \]

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]]*((8*a*(b^2 + c^2))/(3*b*c*(a^2 
- b^2 - c^2)^2) + (2*(a*c + b^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(3*b*(-a 
^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(3*a^2*c + b 
^2*c + c^3 + 4*a*b^2*Sin[d + e*x] + 4*a*c^2*Sin[d + e*x]))/(3*b*(-a^2 + b^ 
2 + c^2)^2*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))))/e + (2*a^2*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]]]*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[1 + b^2/c^2]*c*(-a^2 + b^2 + 
c^2)^2*e) + (2*b^2*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]]*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[a + c*Sqrt[(b^2 + c^2)/c^2]*Si 
n[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 +...
 

3.5.15.3 Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {3042, 3608, 27, 3042, 3635, 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.

\(\displaystyle \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}dx\)

\(\Big \downarrow \) 3608

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3635

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3628

\(\displaystyle \frac {\frac {4 a \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}dx-\left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {4 a \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}dx-\left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {\frac {\frac {4 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {4 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3606

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {8 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{a^2-b^2-c^2}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}\)

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]))/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(3/2)) + ((8*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x] 
))/((a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + ((8*a 
*EllipticE[(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])]) - (2*(a^2 - b^2 - c^2)*Elli 
pticF[(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*S 
qrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]))/(a^2 - b^2 - c^2))/(3*(a^2 - b^ 
2 - c^2))
 

3.5.15.3.1 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 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 3608

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 + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 
1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c 
*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x 
] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && 
NeQ[n, -3/2]
 

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]
 

rule 3635

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

3.5.15.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3641\) vs. \(2(428)=856\).

Time = 9.28 (sec) , antiderivative size = 3642, normalized size of antiderivative = 9.53

method	result	size
default	\(\text {Expression too large to display}\)	\(3642\)

input

int(1/(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)*(b^2*sin(e*x+d-arct 
an(-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)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*(b^ 
4*sin(e*x+d-arctan(-b,c))^4+2*b^2*c^2*sin(e*x+d-arctan(-b,c))^4+c^4*sin(e* 
x+d-arctan(-b,c))^4-2*a^2*b^2*sin(e*x+d-arctan(-b,c))^2-2*a^2*c^2*sin(e*x+ 
d-arctan(-b,c))^2+a^4)/(b^4*sin(e*x+d-arctan(-b,c))^3+2*b^2*c^2*sin(e*x+d- 
arctan(-b,c))^3+c^4*sin(e*x+d-arctan(-b,c))^3+3*(b^2+c^2)^(1/2)*a*b^2*sin( 
e*x+d-arctan(-b,c))^2+3*(b^2+c^2)^(1/2)*a*c^2*sin(e*x+d-arctan(-b,c))^2+3* 
a^2*b^2*sin(e*x+d-arctan(-b,c))+3*a^2*c^2*sin(e*x+d-arctan(-b,c))+(b^2+c^2 
)^(1/2)*a^3)/(2*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arc 
tan(-b,c))^2)^(1/2)*sin(e*x+d-arctan(-b,c))*a*b^2+2*(((b^2+c^2)^(1/2)*sin( 
e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)*sin(e*x+d-arctan(- 
b,c))*a*c^2-(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(- 
b,c))+a)*(b^2+c^2))^(1/2)*sin(e*x+d-arctan(-b,c))^2*b^2-(cos(e*x+d-arctan( 
-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*sin( 
e*x+d-arctan(-b,c))^2*c^2-(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin( 
e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*a^2)*(-1/4/a/(a^2-b^2-c^2)*(b^2+c^ 
2)^(1/2)*(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c 
))+a)*(b^2+c^2))^(1/2)/(b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arcta...
 

3.5.15.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.21 (sec) , antiderivative size = 2805, normalized size of antiderivative = 7.34 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\text {Too large to display} \]

input

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(5/2),x, algorithm="fricas")
 

output

1/9*((sqrt(2)*(I*a^2*b^3 + 3*I*b^5 - I*a^2*b*c^2 - a^2*c^3 - 3*I*b*c^4 - 3 
*c^5 + (a^2*b^2 + 3*b^4)*c)*cos(e*x + d)^2 - 2*sqrt(2)*(-I*a^3*b^2 - 3*I*a 
*b^4 - 3*I*a*b^2*c^2 - 3*a*b*c^3 - (a^3*b + 3*a*b^3)*c)*cos(e*x + d) - 2*( 
sqrt(2)*(-3*I*b^2*c^3 - 3*b*c^4 - (a^2*b + 3*b^3)*c^2 - I*(a^2*b^2 + 3*b^4 
)*c)*cos(e*x + d) + sqrt(2)*(-3*I*a*b*c^3 - 3*a*c^4 - (a^3 + 3*a*b^2)*c^2 
- I*(a^3*b + 3*a*b^3)*c))*sin(e*x + d) + sqrt(2)*(I*a^4*b + 3*I*a^2*b^3 + 
3*I*b*c^4 + 3*c^5 + (4*a^2 + 3*b^2)*c^3 + I*(4*a^2*b + 3*b^3)*c^2 + (a^4 + 
 3*a^2*b^2)*c))*sqrt(b + 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)) + (sqrt(2)*(-I 
*a^2*b^3 - 3*I*b^5 + I*a^2*b*c^2 - a^2*c^3 + 3*I*b*c^4 - 3*c^5 + (a^2*b^2 
+ 3*b^4)*c)*cos(e*x + d)^2 - 2*sqrt(2)*(I*a^3*b^2 + 3*I*a*b^4 + 3*I*a*b^2* 
c^2 - 3*a*b*c^3 - (a^3*b + 3*a*b^3)*c)*cos(e*x + d) - 2*(sqrt(2)*(3*I*b^2* 
c^3 - 3*b*c^4 - (a^2*b + 3*b^3)*c^2 + I*(a^2*b^2 + 3*b^4)*c)*cos(e*x + d) 
+ sqrt(2)*(3*I*a*b*c^3 - 3*a*c^4 - (a^3 + 3*a*b^2)*c^2 + I*(a^3*b + 3*a*b^ 
3)*c))*sin(e*x + d) + sqrt(2)*(-I*a^4*b - 3*I*a^2*b^3 - 3*I*b*c^4 + 3*c^5 
+ (4*a^2 + 3*b^2)*c^3 - I*(4*a^2*b + 3*b^3)*c^2 + (a^4 + 3*a^2*b^2)*c))...
 

3.5.15.6 Sympy [F]

\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )}\right )^{\frac {5}{2}}}\, dx \]

input

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))**(5/2),x)
 

output

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

3.5.15.7 Maxima [F]

\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(1/(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)
 

3.5.15.8 Giac [F]

\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(1/(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)
 

3.5.15.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )\right )}^{5/2}} \,d x \]

input

int(1/(a + b*cos(d + e*x) + c*sin(d + e*x))^(5/2),x)
 

output

int(1/(a + b*cos(d + e*x) + c*sin(d + e*x))^(5/2), x)