\(\int \frac {\cos (c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx\) [968]

Optimal result

Integrand size = 41, antiderivative size = 451 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^2 b \sqrt {a+b} d}+\frac {\left (3 A b^2+a b (A-2 B)+2 a^2 C\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^2 b \sqrt {a+b} d}+\frac {\sqrt {a+b} (3 A b-2 a B) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^3 d}+\frac {A \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}} \] Output:

-(3*A*b^2-2*B*a*b-a^2*(A-2*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2) 
/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+se 
c(d*x+c))/(a-b))^(1/2)/a^2/b/(a+b)^(1/2)/d+(3*A*b^2+a*b*(A-2*B)+2*C*a^2)*c 
ot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2) 
)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/b/(a+ 
b)^(1/2)/d+(a+b)^(1/2)*(3*A*b-2*B*a)*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))*(b*(1-sec(d*x+c))/(a+b))^ 
(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/d+A*sin(d*x+c)/a/d/(a+b*sec(d*x+ 
c))^(1/2)-b*(3*A*b^2-2*B*a*b-a^2*(A-2*C))*tan(d*x+c)/a^2/(a^2-b^2)/d/(a+b* 
sec(d*x+c))^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1798\) vs. \(2(451)=902\).

Time = 20.27 (sec) , antiderivative size = 1798, normalized size of antiderivative = 3.99 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Se 
c[c + d*x])^(3/2),x]
 

Output:

((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(A*b^2 
 - a*b*B + a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) - (4*(A*b^3*Sin[c + d*x] 
 - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*C 
os[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + 
b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])^(3/2)*(A + B*Sec[c + d*x] 
 + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*T 
an[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a^3*A 
*Tan[(c + d*x)/2] + a^2*A*b*Tan[(c + d*x)/2] - 3*a*A*b^2*Tan[(c + d*x)/2] 
- 3*A*b^3*Tan[(c + d*x)/2] + 2*a^2*b*B*Tan[(c + d*x)/2] + 2*a*b^2*B*Tan[(c 
 + d*x)/2] - 2*a^3*C*Tan[(c + d*x)/2] - 2*a^2*b*C*Tan[(c + d*x)/2] - 2*a^3 
*A*Tan[(c + d*x)/2]^3 + 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 4*a^2*b*B*Tan[(c + 
d*x)/2]^3 + 4*a^3*C*Tan[(c + d*x)/2]^3 + a^3*A*Tan[(c + d*x)/2]^5 - a^2*A* 
b*Tan[(c + d*x)/2]^5 - 3*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*A*b^3*Tan[(c + d*x 
)/2]^5 + 2*a^2*b*B*Tan[(c + d*x)/2]^5 - 2*a*b^2*B*Tan[(c + d*x)/2]^5 - 2*a 
^3*C*Tan[(c + d*x)/2]^5 + 2*a^2*b*C*Tan[(c + d*x)/2]^5 - 6*a^2*A*b*Ellipti 
cPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/ 
2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] 
+ 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 
 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x 
)/2]^2)/(a + b)] + 4*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a ...
 

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 4592, 27, 3042, 4548, 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.

\(\displaystyle \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4592

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4546

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4409

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4271

\(\displaystyle \frac {A \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {-(a-b) \left (2 a^2 C+a b (A-2 B)+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) (3 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {A \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {-b \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (2 a^2 C+a b (A-2 B)+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) (3 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {A \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (2 a^2 C+a b (A-2 B)+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) (3 A b-2 a B) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}+\frac {2 b \tan (c+d x) \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}\)

Input:

Int[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + 
d*x])^(3/2),x]
 

Output:

(A*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) - (((2*(a - b)*Sqrt[a + b] 
*(3*A*b^2 - 2*a*b*B - a^2*(A - 2*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*(a - b)*Sq 
rt[a + b]*(3*A*b^2 + a*b*(A - 2*B) + 2*a^2*C)*Cot[c + d*x]*EllipticF[ArcSi 
n[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*S 
qrt[a + b]*(a^2 - b^2)*(3*A*b - 2*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)]*Sqrt[(b*(1 
- Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))/ 
(a*(a^2 - b^2)) + (2*b*(3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Tan[c + d*x])/( 
a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]))/(2*a)
 

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 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 4548

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2053\) vs. \(2(420)=840\).

Time = 13.55 (sec) , antiderivative size = 2054, normalized size of antiderivative = 4.55

method	result	size
default	\(\text {Expression too large to display}\)	\(2054\)

Input:

int(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(3/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

1/d/a^2/(a+b)/(a-b)*(A*a^2*b*cos(d*x+c)*sin(d*x+c)+(3*cos(d*x+c)^2+6*cos(d 
*x+c)+3)*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(co 
s(d*x+c)+1))^(1/2)*b^3*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2 
))+(-2*cos(d*x+c)^2-4*cos(d*x+c)-2)*C*(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^3*EllipticF(-csc(d*x+c)+co 
t(d*x+c),((a-b)/(a+b))^(1/2))+(2*cos(d*x+c)^2+4*cos(d*x+c)+2)*B*(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^ 
3*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(2*cos(d*x+c)^2+4* 
cos(d*x+c)+2)*C*(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^3*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b)) 
^(1/2))+(-cos(d*x+c)^2-2*cos(d*x+c)-1)*A*(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^3*EllipticE(-csc(d*x+c) 
+cot(d*x+c),((a-b)/(a+b))^(1/2))+sin(d*x+c)*cos(d*x+c)*(-cos(d*x+c)+2)*a*A 
*b^2+(-6*cos(d*x+c)^2-12*cos(d*x+c)-6)*A*(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)*b^3*EllipticPi(-csc(d*x+c 
)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(-4*cos(d*x+c)^2-8*cos(d*x+c)-4)*B*(c 
os(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^3*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(-2*co 
s(d*x+c)^2-4*cos(d*x+c)-2)*A*(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*EllipticF(-csc(d*x+c)+cot(d*...
 

Fricas [F]

\[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(3/2 
),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**(3 
/2),x)
 

Output:

Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos(c + d*x)/(a + b*sec( 
c + d*x))**(3/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(3/2 
),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(3/2 
),x, algorithm="giac")
 

Output:

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*cos(d*x + c)/(b*sec(d*x 
+ c) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:

int((cos(c + d*x)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + b/cos(c + 
d*x))^(3/2),x)
 

Output:

int((cos(c + d*x)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + b/cos(c + 
d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2} b^{2}+2 \sec \left (d x +c \right ) a b +a^{2}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )}{\sec \left (d x +c \right )^{2} b^{2}+2 \sec \left (d x +c \right ) a b +a^{2}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )^{2} b^{2}+2 \sec \left (d x +c \right ) a b +a^{2}}d x \right ) a \] Input:

int(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(3/2),x)
 

Output:

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