\(\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx\) [743]

Optimal result

Integrand size = 25, antiderivative size = 494 \[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (147 a^4+279 a^2 b^2-10 b^4\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (147 a^3-114 a^2 b+165 a b^2+10 b^3\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {38 a b \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \] Output:

2/315*(a-b)*(a+b)^(1/2)*(147*a^4+279*a^2*b^2-10*b^4)*cos(d*x+c)^(1/2)*csc( 
d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+ 
b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^( 
1/2)/a^3/d/sec(d*x+c)^(1/2)-2/315*(a-b)*(a+b)^(1/2)*(147*a^3-114*a^2*b+165 
*a*b^2+10*b^3)*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2 
)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+ 
b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d/sec(d*x+c)^(1/2)+2/315*b*(1 
63*a^2+5*b^2)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(3/2)*sin(d*x+c)/a/d+2/315 
*(49*a^2+75*b^2)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+38/6 
3*a*b*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(7/2)*sin(d*x+c)/d+2/9*a^2*(a+b*co 
s(d*x+c))^(1/2)*sec(d*x+c)^(9/2)*sin(d*x+c)/d
 

Mathematica [A] (warning: unable to verify)

Time = 13.33 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.05 \[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \left (-2 \left (147 a^5+147 a^4 b+279 a^3 b^2+279 a^2 b^3-10 a b^4-10 b^5\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right )+2 a \left (147 a^4+261 a^3 b+279 a^2 b^2+155 a b^3-10 b^4\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )-\left (147 a^4+279 a^2 b^2-10 b^4\right ) \cos (c+d x) (a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 a^2 d \sqrt {a+b \cos (c+d x)} \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )}}+\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) \sin (c+d x)}{315 a^2}+\frac {2}{315} \sec ^2(c+d x) \left (49 a^2 \sin (c+d x)+75 b^2 \sin (c+d x)\right )+\frac {2 \sec (c+d x) \left (163 a^2 b \sin (c+d x)+5 b^3 \sin (c+d x)\right )}{315 a}+\frac {38}{63} a b \sec ^2(c+d x) \tan (c+d x)+\frac {2}{9} a^2 \sec ^3(c+d x) \tan (c+d x)\right )}{d} \] Input:

Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2),x]
 

Output:

(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(147*a^5 + 147*a^4*b + 279*a^ 
3*b^2 + 279*a^2*b^3 - 10*a*b^4 - 10*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d* 
x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[Arc 
Sin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(147*a^4 + 261*a^3*b + 279* 
a^2*b^2 + 155*a*b^3 - 10*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[( 
a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c 
+ d*x)/2]], (-a + b)/(a + b)] - (147*a^4 + 279*a^2*b^2 - 10*b^4)*Cos[c + d 
*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*a^2*d* 
Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d 
*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Sin[c + d*x]) 
/(315*a^2) + (2*Sec[c + d*x]^2*(49*a^2*Sin[c + d*x] + 75*b^2*Sin[c + d*x]) 
)/315 + (2*Sec[c + d*x]*(163*a^2*b*Sin[c + d*x] + 5*b^3*Sin[c + d*x]))/(31 
5*a) + (38*a*b*Sec[c + d*x]^2*Tan[c + d*x])/63 + (2*a^2*Sec[c + d*x]^3*Tan 
[c + d*x])/9))/d
 

Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 489, normalized size of antiderivative = 0.99, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 4710, 3042, 3271, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}

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.

\(\displaystyle \int \sec ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{11/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx\)

\(\Big \downarrow \) 4710

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac {11}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\)

\(\Big \downarrow \) 3271

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {19 b a^2+\left (7 a^2+27 b^2\right ) \cos (c+d x) a+3 b \left (2 a^2+3 b^2\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {19 b a^2+\left (7 a^2+27 b^2\right ) \cos (c+d x) a+3 b \left (2 a^2+3 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {19 b a^2+\left (7 a^2+27 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+3 b \left (2 a^2+3 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3534

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 \int \frac {76 b^2 \cos ^2(c+d x) a^2+\left (49 a^2+75 b^2\right ) a^2+b \left (137 a^2+63 b^2\right ) \cos (c+d x) a}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\int \frac {76 b^2 \cos ^2(c+d x) a^2+\left (49 a^2+75 b^2\right ) a^2+b \left (137 a^2+63 b^2\right ) \cos (c+d x) a}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\int \frac {76 b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+\left (49 a^2+75 b^2\right ) a^2+b \left (137 a^2+63 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3534

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {2 \int \frac {\left (147 a^2+605 b^2\right ) \cos (c+d x) a^3+2 b \left (49 a^2+75 b^2\right ) \cos ^2(c+d x) a^2+3 b \left (163 a^2+5 b^2\right ) a^2}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (147 a^2+605 b^2\right ) \cos (c+d x) a^3+2 b \left (49 a^2+75 b^2\right ) \cos ^2(c+d x) a^2+3 b \left (163 a^2+5 b^2\right ) a^2}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (147 a^2+605 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3+2 b \left (49 a^2+75 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+3 b \left (163 a^2+5 b^2\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3534

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {2 \int \frac {3 \left (b \left (261 a^2+155 b^2\right ) \cos (c+d x) a^3+\left (147 a^4+279 b^2 a^2-10 b^4\right ) a^2\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 a b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {b \left (261 a^2+155 b^2\right ) \cos (c+d x) a^3+\left (147 a^4+279 b^2 a^2-10 b^4\right ) a^2}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {2 a b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {b \left (261 a^2+155 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3+\left (147 a^4+279 b^2 a^2-10 b^4\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 a b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3477

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {a^2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-a^2 (a-b) \left (147 a^3-114 a^2 b+165 a b^2+10 b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {2 a b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {a^2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a^2 (a-b) \left (147 a^3-114 a^2 b+165 a b^2+10 b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 a b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3295

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {a^2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a (a-b) \sqrt {a+b} \left (147 a^3-114 a^2 b+165 a b^2+10 b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{a}+\frac {2 a b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3473

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (\frac {\frac {2 a \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {\frac {2 a b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\frac {2 (a-b) \sqrt {a+b} \left (147 a^4+279 a^2 b^2-10 b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{d}-\frac {2 a (a-b) \sqrt {a+b} \left (147 a^3-114 a^2 b+165 a b^2+10 b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{a}}{5 a}}{7 a}+\frac {38 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\right )\)

Input:

Int[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2),x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin 
[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((38*a*b*Sqrt[a + b*Cos[c + d*x]]*Si 
n[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*a*(49*a^2 + 75*b^2)*Sqrt[a + b* 
Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (((2*(a - b)*Sqrt[a 
 + b]*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[ 
a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))] 
*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)] 
)/d - (2*a*(a - b)*Sqrt[a + b]*(147*a^3 - 114*a^2*b + 165*a*b^2 + 10*b^3)* 
Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[C 
os[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*S 
qrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d)/a + (2*a*b*(163*a^2 + 5*b^2)*Sqrt[ 
a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)))/(5*a))/(7*a))/9)
 

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 3271

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co 
s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* 
(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin 
[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 
2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 
+ b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - 
 d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] 
, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - 
b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || 
IntegersQ[2*m, 2*n])
 

rule 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3473

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* 
(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] 
)/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + 
d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], 
x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && 
PosQ[(c + d)/b]
 

rule 3477

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S 
imp[(A - B)/(a - b)   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* 
x]]), x], x] - Simp[(A*b - a*B)/(a - b)   Int[(1 + Sin[e + f*x])/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e 
, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && NeQ[A, B]
 

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 4710

Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Csc[a 
+ b*x])^m*(c*Sin[a + b*x])^m   Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1562\) vs. \(2(436)=872\).

Time = 2216.85 (sec) , antiderivative size = 1563, normalized size of antiderivative = 3.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1563\)

Input:

int((a+cos(d*x+c)*b)^(5/2)*sec(d*x+c)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

-2/315/d*(a+cos(d*x+c)*b)^(1/2)*sec(d*x+c)^(11/2)/(b*cos(d*x+c)^2+a*cos(d* 
x+c)+cos(d*x+c)*b+a)*(((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d 
*x+c)/(cos(d*x+c)+1))^(1/2)*a^5*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a 
+b))^(1/2))*(-147*cos(d*x+c)^7-294*cos(d*x+c)^6-147*cos(d*x+c)^5)+((a+cos( 
d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^ 
4*b*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-147*cos(d*x+c) 
^7-294*cos(d*x+c)^6-147*cos(d*x+c)^5)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+ 
b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^3*b^2*EllipticE(cot(d*x+c)-c 
sc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-279*cos(d*x+c)^7-558*cos(d*x+c)^6-279*co 
s(d*x+c)^5)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos 
(d*x+c)+1))^(1/2)*a^2*b^3*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^( 
1/2))*(-279*cos(d*x+c)^7-558*cos(d*x+c)^6-279*cos(d*x+c)^5)+((a+cos(d*x+c) 
*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a*b^4*El 
lipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(10*cos(d*x+c)^7+20*co 
s(d*x+c)^6+10*cos(d*x+c)^5)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)* 
(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^5*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a 
-b)/(a+b))^(1/2))*(10*cos(d*x+c)^7+20*cos(d*x+c)^6+10*cos(d*x+c)^5)+((a+co 
s(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)* 
a^5*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(147*cos(d*x+c)^ 
7+294*cos(d*x+c)^6+147*cos(d*x+c)^5)+((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(...
 

Fricas [F]

\[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^(11/2),x, algorithm="fricas")
 

Output:

integral((b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)*sqrt(b*cos(d*x + 
c) + a)*sec(d*x + c)^(11/2), x)
 

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))**(5/2)*sec(d*x+c)**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^(11/2),x, algorithm="maxima")
 

Output:

integrate((b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^(11/2), x)
 

Giac [F]

\[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^(11/2),x, algorithm="giac")
 

Output:

integrate((b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:

int((1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int((1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \, dx=2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) a b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a^{2} \] Input:

int((a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^(11/2),x)
 

Output:

2*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x 
)**5,x)*a*b + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x) 
**2*sec(c + d*x)**5,x)*b**2 + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + 
 a)*sec(c + d*x)**5,x)*a**2