3.6.77 \(\int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)} \, dx\) [577]

3.6.77.1 Optimal result

Integrand size = 33, antiderivative size = 363 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (9 a^3 A b-12 a A b^3-15 a^4 B+16 a^2 b^2 B+2 b^4 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 b^4 \left (a^2-b^2\right ) d}+\frac {a^2 \left (3 a^2 A b-5 A b^3-5 a^3 B+7 a b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{(a-b) b^4 (a+b)^2 d}-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \]

output

-1/3*(3*A*a*b-5*B*a^2+2*B*b^2)*sin(d*x+c)/b^2/(a^2-b^2)/d/sec(d*x+c)^(1/2) 
+a*(A*b-B*a)*sin(d*x+c)/b/(a^2-b^2)/d/(b+a*sec(d*x+c))/sec(d*x+c)^(1/2)+(3 
*A*a^2*b-2*A*b^3-5*B*a^3+4*B*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d 
*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c 
)^(1/2)/b^3/(a^2-b^2)/d-1/3*(9*A*a^3*b-12*A*a*b^3-15*B*a^4+16*B*a^2*b^2+2* 
B*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d 
*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/b^4/(a^2-b^2)/d+a^2*( 
3*A*a^2*b-5*A*b^3-5*B*a^3+7*B*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2* 
d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*cos(d*x+c)^(1/ 
2)*sec(d*x+c)^(1/2)/(a-b)/b^4/(a+b)^2/d
 

3.6.77.2 Mathematica [A] (warning: unable to verify)

Time = 6.84 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.93 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {\frac {2 \left (-3 a^2 A b+6 A b^3+5 a^3 B-8 a b^2 B\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (-12 a A b^2+8 a^2 b B+4 b^3 B\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-9 a^2 A b+6 A b^3+15 a^3 B-12 a b^2 B\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{12 (a-b) b^2 (a+b) d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {a^2 (-A b+a B) \sin (c+d x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3 A b \sin (c+d x)-a^4 B \sin (c+d x)}{b^3 \left (-a^2+b^2\right ) (a+b \cos (c+d x))}+\frac {B \sin (2 (c+d x))}{3 b^2}\right )}{d} \]

input

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

output

-1/12*((2*(-3*a^2*A*b + 6*A*b^3 + 5*a^3*B - 8*a*b^2*B)*Cos[c + d*x]^2*(Ell 
ipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Se 
c[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d 
*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-12*a*A*b^2 + 8*a 
^2*b*B + 4*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d* 
x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*( 
a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-9*a^2*A*b + 6*A*b^3 + 15*a^ 
3*B - 12*a*b^2*B)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Se 
c[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c 
+ d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec 
[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*Ellip 
ticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - 
Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1] 
*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b* 
Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2) 
))/((a - b)*b^2*(a + b)*d) + (Sqrt[Sec[c + d*x]]*((a^2*(-(A*b) + a*B)*Sin[ 
c + d*x])/(b^3*(a^2 - b^2)) - (a^3*A*b*Sin[c + d*x] - a^4*B*Sin[c + d*x])/ 
(b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])) + (B*Sin[2*(c + d*x)])/(3*b^2)))/d
 

3.6.77.3 Rubi [A] (verified)

Time = 2.58 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 3439, 3042, 4518, 27, 3042, 4592, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 3284}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3439

\(\displaystyle \int \frac {A \sec (c+d x)+B}{\sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A \csc \left (c+d x+\frac {\pi }{2}\right )+B}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^2}dx\)

\(\Big \downarrow \) 4518

\(\displaystyle \frac {\int -\frac {-5 B a^2-3 (A b-a B) \sec ^2(c+d x) a+3 A b a+2 b^2 B+2 b (A b-a B) \sec (c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))}dx}{b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\int \frac {-5 B a^2-3 (A b-a B) \sec ^2(c+d x) a+3 A b a+2 b^2 B+2 b (A b-a B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))}dx}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\int \frac {-5 B a^2-3 (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a+3 A b a+2 b^2 B+2 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-a \left (-5 B a^2+3 A b a+2 b^2 B\right ) \sec ^2(c+d x)+2 b \left (-2 B a^2+3 A b a-b^2 B\right ) \sec (c+d x)+3 \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right )}{2 \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-a \left (-5 B a^2+3 A b a+2 b^2 B\right ) \sec ^2(c+d x)+2 b \left (-2 B a^2+3 A b a-b^2 B\right ) \sec (c+d x)+3 \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right )}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-a \left (-5 B a^2+3 A b a+2 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (-2 B a^2+3 A b a-b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 4594

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)}dx}{b^2}+\frac {\int \frac {3 b \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right )-\left (-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\int \frac {3 b \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right )+\left (15 B a^4-9 A b a^3-16 b^2 B a^2+12 A b^3 a-2 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-\left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \int \sqrt {\sec (c+d x)}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx-\left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {6 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {6 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 4336

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b^2}+\frac {\frac {6 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b^2}+\frac {\frac {6 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {6 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^2 d (a+b)}+\frac {\frac {6 b \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}\)

input

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

output

(a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(b + a*Se 
c[c + d*x])) - (-1/3*(((6*b*(3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sq 
rt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - (2*(9*a 
^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*Sqrt[Cos[c + d*x] 
]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/b^2 + (6*a^2*(3*a^2*A*b 
 - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + 
 b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d))/b + (2*(3*a*A*b 
- 5*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]]))/(2*b*(a^2 - 
 b^2))
 

3.6.77.3.1 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 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], 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[c + d, 0]
 

rule 3439

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* 
(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim 
p[g^(m + n)   Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + 
c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c 
- a*d, 0] &&  !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4336

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]]   Int[ 
1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4518

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[b*(A*b 
- a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*( 
m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2))   Int[(a + b*Csc[ 
e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[A*(a^2*(m + 1) - b^2*(m + n + 1)) 
 + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2) 
*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A* 
b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] && IL 
tQ[n, 0])
 

rule 4592

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d 
*Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n)   Int[(a + b*Csc[e + f*x])^m 
*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* 
Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d 
, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

rule 4594

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2)   Int[(d*Csc[e + 
f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2   Int[(a*A - (A*b - a 
*B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, 
B, C}, x] && NeQ[a^2 - b^2, 0]
 

3.6.77.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1065\) vs. \(2(421)=842\).

Time = 11.30 (sec) , antiderivative size = 1066, normalized size of antiderivative = 2.94

method	result	size
default	\(\text {Expression too large to display}\)	\(1066\)

input

int((A+B*cos(d*x+c))/(a+cos(d*x+c)*b)^2/sec(d*x+c)^(5/2),x,method=_RETURNV 
ERBOSE)
 

output

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4*a^2/b^3*(3* 
A*b-4*B*a)/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2 
*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipt 
icPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))-2*a^3*(A*b-B*a)/b^4*(-1/a*b^2/ 
(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2 
)^(1/2)/(2*b*cos(1/2*d*x+1/2*c)^2+a-b)-1/2/a/(a+b)*(sin(1/2*d*x+1/2*c)^2)^ 
(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d 
*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/(a^2-b^2)*b/a 
*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/ 
2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^ 
(1/2))+1/2/(a^2-b^2)*b/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2* 
c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipti 
cE(cos(1/2*d*x+1/2*c),2^(1/2))-3*a/(a^2-b^2)/(-2*a*b+2*b^2)*b*(sin(1/2*d*x 
+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^ 
4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^( 
1/2))+1/a/(a^2-b^2)/(-2*a*b+2*b^2)*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*co 
s(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2) 
^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2)))-2/3/b^4/(-2*sin( 
1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4*B*cos(1/2*d*x+1/2*c)*sin( 
1/2*d*x+1/2*c)^4*b^2+6*A*a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*...
 

3.6.77.5 Fricas [F(-1)]

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

input

integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^2/sec(d*x+c)^(5/2),x, algorith 
m="fricas")
 

output

Timed out
 

3.6.77.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.77.7 Maxima [F]

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

input

integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^2/sec(d*x+c)^(5/2),x, algorith 
m="maxima")
 

output

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

3.6.77.8 Giac [F]

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

input

integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^2/sec(d*x+c)^(5/2),x, algorith 
m="giac")
 

output

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

3.6.77.9 Mupad [F(-1)]

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

input

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

output

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