3.8.29 \(\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\) [729]

3.8.29.1 Optimal result

Integrand size = 33, antiderivative size = 478 \[ \int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 b d}+\frac {\sqrt {a+b} \left (a^2 b (15 A-46 C)+30 a^3 C-6 b^3 (5 A+3 C)+2 a b^2 (45 A+17 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 b d}-\frac {5 a A b \sqrt {a+b} \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{d}+\frac {A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{d}-\frac {a b (15 A-16 C) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{15 d}-\frac {b (5 A-2 C) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{5 d} \]

output

A*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d+1/15*(a-b)*(a^2*(15*A-46*C)-6*b^2*(5 
*A+3*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a 
-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/ 
(a-b))^(1/2)/b/d+1/15*(a^2*b*(15*A-46*C)+30*a^3*C-6*b^3*(5*A+3*C)+2*a*b^2* 
(45*A+17*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b 
)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+ 
c))/(a-b))^(1/2)/b/d-5*a*A*b*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^(1/2)/ 
(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+ 
b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/d-1/5*b*(5*A-2*C)*(a+b*sec(d*x+c 
))^(3/2)*tan(d*x+c)/d-1/15*a*b*(15*A-16*C)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+ 
c)/d
 

3.8.29.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(6785\) vs. \(2(478)=956\).

Time = 29.00 (sec) , antiderivative size = 6785, normalized size of antiderivative = 14.19 \[ \int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.8.29.3 Rubi [A] (verified)

Time = 2.05 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 4583, 27, 3042, 4544, 27, 3042, 4544, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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.

\(\displaystyle \int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4583

\(\displaystyle \int \frac {1}{2} (a+b \sec (c+d x))^{3/2} \left (-b (5 A-2 C) \sec ^2(c+d x)+2 a C \sec (c+d x)+5 A b\right )dx+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int (a+b \sec (c+d x))^{3/2} \left (-b (5 A-2 C) \sec ^2(c+d x)+2 a C \sec (c+d x)+5 A b\right )dx+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-b (5 A-2 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a C \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b\right )dx+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 4544

\(\displaystyle \frac {1}{2} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \sec (c+d x)} \left (-a b (15 A-16 C) \sec ^2(c+d x)+2 \left (5 C a^2+5 A b^2+3 b^2 C\right ) \sec (c+d x)+25 a A b\right )dx-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int \sqrt {a+b \sec (c+d x)} \left (-a b (15 A-16 C) \sec ^2(c+d x)+2 \left (5 C a^2+5 A b^2+3 b^2 C\right ) \sec (c+d x)+25 a A b\right )dx-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (-a b (15 A-16 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (5 C a^2+5 A b^2+3 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+25 a A b\right )dx-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 4544

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {75 A b a^2+2 \left (15 C a^2+45 A b^2+17 b^2 C\right ) \sec (c+d x) a-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \sec (c+d x)}}dx-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {75 A b a^2+2 \left (15 C a^2+45 A b^2+17 b^2 C\right ) \sec (c+d x) a-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {75 A b a^2+2 \left (15 C a^2+45 A b^2+17 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {75 A b a^2+\left (2 a \left (15 C a^2+45 A b^2+17 b^2 C\right )+b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {75 A b a^2+\left (2 a \left (15 C a^2+45 A b^2+17 b^2 C\right )+b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+75 a^2 A b \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+\left (30 a^3 C+a^2 b (15 A-46 C)+2 a b^2 (45 A+17 C)-6 b^3 (5 A+3 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+75 a^2 A b \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (30 a^3 C+a^2 b (15 A-46 C)+2 a b^2 (45 A+17 C)-6 b^3 (5 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (30 a^3 C+a^2 b (15 A-46 C)+2 a b^2 (45 A+17 C)-6 b^3 (5 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {150 a A b \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (-b \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (30 a^3 C+a^2 b (15 A-46 C)+2 a b^2 (45 A+17 C)-6 b^3 (5 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\frac {150 a A b \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (a-b) \sqrt {a+b} \left (a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}+\frac {2 \sqrt {a+b} \left (30 a^3 C+a^2 b (15 A-46 C)+2 a b^2 (45 A+17 C)-6 b^3 (5 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\frac {150 a A b \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )-\frac {2 a b (15 A-16 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}\)

input

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

output

(A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d + ((-2*b*(5*A - 2*C)*(a + b* 
Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d) + (((2*(a - b)*Sqrt[a + b]*(a^2*(1 
5*A - 46*C) - 6*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b* 
Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/( 
a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(a 
^2*b*(15*A - 46*C) + 30*a^3*C - 6*b^3*(5*A + 3*C) + 2*a*b^2*(45*A + 17*C)) 
*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + 
 b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d 
*x]))/(a - b))])/(b*d) - (150*a*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a 
 + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sq 
rt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))] 
)/d)/3 - (2*a*b*(15*A - 16*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d) 
)/5)/2
 

3.8.29.3.1 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 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

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

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4544

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

rule 4546

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

rule 4583

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

3.8.29.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4437\) vs. \(2(437)=874\).

Time = 63.89 (sec) , antiderivative size = 4438, normalized size of antiderivative = 9.28

method	result	size
default	\(\text {Expression too large to display}\)	\(4438\)

input

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

output

1/15/d*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+c))/(cos(d*x+c)+1)*(6*C*b^3*tan 
(d*x+c)+18*C*sin(d*x+c)*b^3+30*A*b^3*sin(d*x+c)-92*C*EllipticF(cot(d*x+c)- 
csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b) 
*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)+60*A*EllipticE(co 
t(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2) 
*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^2*cos(d*x+c)-180*A*El 
lipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c) 
+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^2*cos(d*x+c 
)+92*C*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(c 
os(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b* 
cos(d*x+c)+30*A*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos( 
d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/ 
2)*a*b^2*cos(d*x+c)^2-90*A*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^( 
1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x 
+c)+1))^(1/2)*a*b^2*cos(d*x+c)^2+46*C*EllipticE(cot(d*x+c)-csc(d*x+c),((a- 
b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c 
))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)^2-46*C*EllipticF(cot(d*x+c)-csc( 
d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+ 
a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)^2+22*C*a^2*b*cos(d*x+ 
c)*sin(d*x+c)+30*A*a*b^2*cos(d*x+c)*sin(d*x+c)+30*A*(cos(d*x+c)/(cos(d*...
 

3.8.29.5 Fricas [F]

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

input

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

output

integral((C*b^2*cos(d*x + c)*sec(d*x + c)^4 + 2*C*a*b*cos(d*x + c)*sec(d*x 
 + c)^3 + 2*A*a*b*cos(d*x + c)*sec(d*x + c) + A*a^2*cos(d*x + c) + (C*a^2 
+ A*b^2)*cos(d*x + c)*sec(d*x + c)^2)*sqrt(b*sec(d*x + c) + a), x)
 

3.8.29.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.8.29.7 Maxima [F]

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

input

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

output

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

3.8.29.8 Giac [F]

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

input

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

output

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

3.8.29.9 Mupad [F(-1)]

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

input

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

output

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