\(\int \frac {\sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) [1340]

Optimal result

Integrand size = 45, antiderivative size = 447 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (24 A b^2+18 a b B-a^2 C+16 b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 b^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}} \] Output:

1/24*(24*A*b^2+18*B*a*b-C*a^2+16*C*b^2)*((b+a*cos(d*x+c))/(a+b))^(1/2)*Inv 
erseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^(1/2))/b/d/cos(d*x+c)^(1/2)/( 
a+b*sec(d*x+c))^(1/2)-1/8*(2*B*a^2*b-8*B*b^3-a^3*C-4*a*b^2*(2*A+C))*((b+a* 
cos(d*x+c))/(a+b))^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b)) 
^(1/2))/b^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-1/24*(24*A*b^2+6*B*a 
*b-3*C*a^2+16*C*b^2)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2) 
*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/b^2/d/((b+a*cos(d*x+c))/(a+b))^(1 
/2)+1/3*C*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d/cos(d*x+c)^(5/2)+1/12*(6*B*b 
+C*a)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/b/d/cos(d*x+c)^(3/2)+1/24*(24*A*b^ 
2+6*B*a*b-3*C*a^2+16*C*b^2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/b^2/d/cos(d* 
x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 35.74 (sec) , antiderivative size = 132996, normalized size of antiderivative = 297.53 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4753

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4584

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left ((6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (3 A b+2 C b+3 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (2 A+C)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 4590

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right ) \sec ^2(c+d x)+2 b (12 a A+6 b B+7 a C) \sec (c+d x)+a (6 b B+a C)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right ) \sec ^2(c+d x)+2 b (12 a A+6 b B+7 a C) \sec (c+d x)+a (6 b B+a C)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b (12 a A+6 b B+7 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+a (6 b B+a C)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 4590

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\int -\frac {3 \left (-C a^3+2 b B a^2-4 b^2 (2 A+C) a-8 b^3 B\right ) \sec ^2(c+d x)-2 a b (6 b B+a C) \sec (c+d x)+a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{b}+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {3 \left (-C a^3+2 b B a^2-4 b^2 (2 A+C) a-8 b^3 B\right ) \sec ^2(c+d x)-2 a b (6 b B+a C) \sec (c+d x)+a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {3 \left (-C a^3+2 b B a^2-4 b^2 (2 A+C) a-8 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b (6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 4596

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )-2 a b (6 b B+a C) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx+3 \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )-2 a b (6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 4346

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )-2 a b (6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )-2 a b (6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3286

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )-2 a b (6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )-2 a b (6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3284

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right )-2 a b (6 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 4523

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 4343

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3134

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3132

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 4345

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {b \sqrt {\sec (c+d x)} \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {b \sqrt {\sec (c+d x)} \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3142

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {b \sqrt {\sec (c+d x)} \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {b \sqrt {\sec (c+d x)} \left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(a C+6 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\)

Input:

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

Output:

$Aborted
 

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 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

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 3286

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

rule 4343

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

rule 4345

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

rule 4346

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

rule 4523

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

rule 4584

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[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs 
c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(m + n + 1)   Int[(a + b*Csc[e 
+ f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a 
*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc 
[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 
 - b^2, 0] && GtQ[m, 0] &&  !LeQ[n, -1]
 

rule 4590

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[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 
)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) 
   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( 
A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*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] && GtQ[n, 0]
 

rule 4596

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

rule 4753

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 29.59 (sec) , antiderivative size = 2785, normalized size of antiderivative = 6.23

method	result	size
default	\(\text {Expression too large to display}\)	\(2785\)

Input:

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

Output:

1/24/d*(sin(d*x+c)*cos(d*x+c)*(12*cos(d*x+c)+12)*B*((a-b)/(a+b))^(1/2)*b^3 
+24*A*((a-b)/(a+b))^(1/2)*a*b^2*cos(d*x+c)^3*sin(d*x+c)+6*B*((a-b)/(a+b))^ 
(1/2)*a^2*b*cos(d*x+c)^3*sin(d*x+c)+C*(1/(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(((a-b)/(a+b))^(1/2)*(c 
sc(d*x+c)-cot(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(6*cos(d*x+c)^5+1 
2*cos(d*x+c)^4+6*cos(d*x+c)^3)+A*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*co 
s(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^3*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x 
+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(24*cos(d*x+c)^5+48*cos(d*x+c)^4+24* 
cos(d*x+c)^3)+C*(1/(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(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),( 
-(a+b)/(a-b))^(1/2))*(3*cos(d*x+c)^5+6*cos(d*x+c)^4+3*cos(d*x+c)^3)+C*(1/( 
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*E 
llipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2)) 
*(16*cos(d*x+c)^5+32*cos(d*x+c)^4+16*cos(d*x+c)^3)+B*(1/(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*EllipticF(((a-b)/( 
a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-24*cos(d*x+c)^ 
5-48*cos(d*x+c)^4-24*cos(d*x+c)^3)+C*(1/(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(((a-b)/(a+b))^(1/2)*(csc 
(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-6*cos(d*x+c)^5-12*cos(d*x+c)^4 
-6*cos(d*x+c)^3)+B*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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