\(\int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx\) [377]

Optimal result

Integrand size = 33, antiderivative size = 525 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {(a-b) \sqrt {a+b} \left (16 a^2 A+15 A b^2-18 a b B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a^3 b d}+\frac {\sqrt {a+b} \left (16 a^2 A-10 a A b+15 A b^2+12 a^2 B-18 a b B\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a^3 d}+\frac {\sqrt {a+b} \left (4 a^2 A b+5 A b^3-8 a^3 B-6 a b^2 B\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a^4 d}+\frac {\left (16 a^2 A+15 A b^2-18 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a^3 d}-\frac {(5 A b-6 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 a^2 d}+\frac {A \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a d} \] Output:

1/24*(a-b)*(a+b)^(1/2)*(16*A*a^2+15*A*b^2-18*B*a*b)*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+sec(d*x+c))/(a-b))^(1/2)/a^3/b/d+1/24*(a+b)^(1/2)*(16*A 
*a^2-10*A*a*b+15*A*b^2+12*B*a^2-18*B*a*b)*cot(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^3/d+1/8*(a+b)^(1/2)*(4*A*a^2*b+5*A*b^3 
-8*B*a^3-6*B*a*b^2)*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^4/d+1/24*(16*A*a^2+15*A*b^2-18*B*a*b)*(a+b*sec(d*x+ 
c))^(1/2)*sin(d*x+c)/a^3/d-1/12*(5*A*b-6*B*a)*cos(d*x+c)*(a+b*sec(d*x+c))^ 
(1/2)*sin(d*x+c)/a^2/d+1/3*A*cos(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c 
)/a/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1569\) vs. \(2(525)=1050\).

Time = 18.85 (sec) , antiderivative size = 1569, normalized size of antiderivative = 2.99 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

((b + a*Cos[c + d*x])*Sec[c + d*x]*((A*Sin[c + d*x])/(12*a) + ((-5*A*b + 6 
*a*B)*Sin[2*(c + d*x)])/(24*a^2) + (A*Sin[3*(c + d*x)])/(12*a)))/(d*Sqrt[a 
 + b*Sec[c + d*x]]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 
 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c 
 + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^ 
2*A*b*Tan[(c + d*x)/2] + 15*a*A*b^2*Tan[(c + d*x)/2] + 15*A*b^3*Tan[(c + d 
*x)/2] - 18*a^2*b*B*Tan[(c + d*x)/2] - 18*a*b^2*B*Tan[(c + d*x)/2] - 32*a^ 
3*A*Tan[(c + d*x)/2]^3 - 30*a*A*b^2*Tan[(c + d*x)/2]^3 + 36*a^2*b*B*Tan[(c 
 + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 
 + 15*a*A*b^2*Tan[(c + d*x)/2]^5 - 15*A*b^3*Tan[(c + d*x)/2]^5 - 18*a^2*b* 
B*Tan[(c + d*x)/2]^5 + 18*a*b^2*B*Tan[(c + d*x)/2]^5 - 24*a^2*A*b*Elliptic 
Pi[-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)] - 
 30*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)] + 48*a^3*B*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)] + 36*a*b^2*B*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)] - 24*a^2*A*b*Ell...
 

Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 535, 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, 4522, 27, 3042, 4592, 27, 3042, 4592, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4522

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 A b \sec ^2(c+d x)-4 a A \sec (c+d x)+5 A b-6 a B\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 A b \sec ^2(c+d x)-4 a A \sec (c+d x)+5 A b-6 a B\right )}{\sqrt {a+b \sec (c+d x)}}dx}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\int \frac {-3 A b \csc \left (c+d x+\frac {\pi }{2}\right )^2-4 a A \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b-6 a B}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{6 a}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (16 A a^2-18 b B a+2 (A b+6 a B) \sec (c+d x) a+15 A b^2-b (5 A b-6 a B) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}}{6 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (16 A a^2-18 b B a+2 (A b+6 a B) \sec (c+d x) a+15 A b^2-b (5 A b-6 a B) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {16 A a^2-18 b B a+2 (A b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+15 A b^2-b (5 A b-6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}}{6 a}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (16 A a^2-18 b B a+15 A b^2\right ) \sec ^2(c+d x)+2 a b (5 A b-6 a B) \sec (c+d x)+3 \left (-8 B a^3+4 A b a^2-6 b^2 B a+5 A b^3\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (16 A a^2-18 b B a+15 A b^2\right ) \sec ^2(c+d x)+2 a b (5 A b-6 a B) \sec (c+d x)+3 \left (-8 B a^3+4 A b a^2-6 b^2 B a+5 A b^3\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (16 A a^2-18 b B a+15 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (5 A b-6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-8 B a^3+4 A b a^2-6 b^2 B a+5 A b^3\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (16 a^2 A-18 a b B+15 A b^2\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 \left (-8 B a^3+4 A b a^2-6 b^2 B a+5 A b^3\right )+\left (2 a b (5 A b-6 a B)-b \left (16 A a^2-18 b B a+15 A b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (16 a^2 A-18 a b B+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 \left (-8 B a^3+4 A b a^2-6 b^2 B a+5 A b^3\right )+\left (2 a b (5 A b-6 a B)-b \left (16 A a^2-18 b B a+15 A b^2\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (16 a^2 A-18 a b B+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (4 a^2 (4 A+3 B)-2 a b (5 A+9 B)+15 A b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 \left (-8 a^3 B+4 a^2 A b-6 a b^2 B+5 A b^3\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (4 a^2 (4 A+3 B)-2 a b (5 A+9 B)+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (16 a^2 A-18 a b B+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-8 a^3 B+4 a^2 A b-6 a b^2 B+5 A b^3\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (4 a^2 (4 A+3 B)-2 a b (5 A+9 B)+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (16 a^2 A-18 a b B+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (-8 a^3 B+4 a^2 A b-6 a b^2 B+5 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (16 a^2 A-18 a b B+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \left (4 a^2 (4 A+3 B)-2 a b (5 A+9 B)+15 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \left (-8 a^3 B+4 a^2 A b-6 a b^2 B+5 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 a d}-\frac {\frac {(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\left (16 a^2 A-18 a b B+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-\frac {2 \sqrt {a+b} \left (4 a^2 (4 A+3 B)-2 a b (5 A+9 B)+15 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 (a-b) \sqrt {a+b} \left (16 a^2 A-18 a b B+15 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \left (-8 a^3 B+4 a^2 A b-6 a b^2 B+5 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}}{6 a}\)

Input:

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

Output:

(A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d) - (((5*A* 
b - 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d) - ( 
-1/2*((-2*(a - b)*Sqrt[a + b]*(16*a^2*A + 15*A*b^2 - 18*a*b*B)*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]*(15*A*b^2 + 4*a^2*(4*A + 3*B) - 2*a*b*(5*A + 9 
*B))*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))])/d - (6*Sqrt[a + b]*(4*a^2*A*b + 5*A*b^3 - 8*a^3*B - 6 
*a*b^2*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)]*Sqr 
t[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))/a + ((16*a^2*A + 15*A*b^2 - 1 
8*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d))/(4*a))/(6*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 4522

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), 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] + Sim 
p[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*(n + 1)*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, m}, x] && NeQ[A*b - a*B, 0] 
 && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

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 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. \(1677\) vs. \(2(480)=960\).

Time = 13.52 (sec) , antiderivative size = 1678, normalized size of antiderivative = 3.20

method	result	size
default	\(\text {Expression too large to display}\)	\(1678\)

Input:

int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

1/24/d/a^3*((24*cos(d*x+c)^2+48*cos(d*x+c)+24)*A*(1/(a+b)*(b+a*cos(d*x+c)) 
/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b*EllipticPi( 
-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(30*cos(d*x+c)^2+60*cos(d*x 
+c)+30)*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+c 
os(d*x+c)))^(1/2)*b^3*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^( 
1/2))+(-48*cos(d*x+c)^2-96*cos(d*x+c)-48)*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1 
/2)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*EllipticPi(-csc(d* 
x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(-36*cos(d*x+c)^2-72*cos(d*x+c)-36 
)*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x 
+c)))^(1/2)*a*b^2*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2) 
)+(-16*cos(d*x+c)^2-32*cos(d*x+c)-16)*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)* 
(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*EllipticE(-csc(d*x+c)+ 
cot(d*x+c),((a-b)/(a+b))^(1/2))+(-16*cos(d*x+c)^2-32*cos(d*x+c)-16)*A*(cos 
(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1 
/2)*a^2*b*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(-15*cos(d 
*x+c)^2-30*cos(d*x+c)-15)*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2 
)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^2*EllipticE(-csc(d*x+c)+cot(d*x+c) 
,((a-b)/(a+b))^(1/2))+(-15*cos(d*x+c)^2-30*cos(d*x+c)-15)*A*(1/(a+b)*(b+a* 
cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^3*El 
lipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(18*cos(d*x+c)^2+36...
 

Fricas [F]

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

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/2),x, algorith 
m="fricas")
 

Output:

integral((B*cos(d*x + c)^3*sec(d*x + c) + A*cos(d*x + c)^3)/sqrt(b*sec(d*x 
 + c) + a), x)
 

Sympy [F]

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

integrate(cos(d*x+c)**3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/2),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

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

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x))^(1/2),x)
 

Output:

int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x))^(1/2), x)
 

Reduce [F]

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

int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(1/2),x)
 

Output:

int(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3,x)