\(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx\) [624]

Optimal result

Integrand size = 45, antiderivative size = 313 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx=-\frac {(283 A-163 B+75 C) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}-\frac {(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \sec (c+d x)}} \] Output:

-1/32*(283*A-163*B+75*C)*arctanh(1/2*a^(1/2)*sec(d*x+c)^(1/2)*sin(d*x+c)*2 
^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2)/a^(5/2)/d-1/4*(A-B+C)*sin(d*x+c)/d/ 
sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^(5/2)-1/16*(21*A-13*B+5*C)*sin(d*x+c)/a/ 
d/sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^(3/2)+1/80*(157*A-85*B+45*C)*sin(d*x+c 
)/a^2/d/sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^(1/2)-1/240*(787*A-475*B+195*C)* 
sin(d*x+c)/a^2/d/sec(d*x+c)^(1/2)/(a+a*sec(d*x+c))^(1/2)+1/240*(2671*A-149 
5*B+735*C)*sec(d*x+c)^(1/2)*sin(d*x+c)/a^2/d/(a+a*sec(d*x+c))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.75 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.71 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx=-\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (15 (283 A-163 B+75 C) \text {arctanh}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right )-\frac {1}{2} (3491 A-1895 B+975 C+5 (887 A-479 B+255 C) \cos (c+d x)+16 (52 A-25 B+15 C) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+40 B \cos (3 (c+d x))+12 A \cos (4 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{60 a d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sqrt {\sec (c+d x)} (a (1+\sec (c+d x)))^{3/2}} \] Input:

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

Output:

-1/60*(Sec[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(15*(283*A 
 - 163*B + 75*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 - ((3491*A - 
 1895*B + 975*C + 5*(887*A - 479*B + 255*C)*Cos[c + d*x] + 16*(52*A - 25*B 
 + 15*C)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 40*B*Cos[3*(c + d*x)] 
+ 12*A*Cos[4*(c + d*x)])*Sin[(c + d*x)/2])/2))/(a*d*(A + 2*C + 2*B*Cos[c + 
 d*x] + A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2 
))
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.08, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {3042, 4572, 27, 3042, 4508, 27, 3042, 4510, 27, 3042, 4510, 27, 3042, 4501, 3042, 4295, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4572

\(\displaystyle \frac {\int \frac {a (13 A-5 B+5 C)-8 a (A-B) \sec (c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) (\sec (c+d x) a+a)^{3/2}}dx}{4 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (13 A-5 B+5 C)-8 a (A-B) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (\sec (c+d x) a+a)^{3/2}}dx}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (13 A-5 B+5 C)-8 a (A-B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 4508

\(\displaystyle \frac {\frac {\int \frac {a^2 (157 A-85 B+45 C)-6 a^2 (21 A-13 B+5 C) \sec (c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {\sec (c+d x) a+a}}dx}{2 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {a^2 (157 A-85 B+45 C)-6 a^2 (21 A-13 B+5 C) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {\sec (c+d x) a+a}}dx}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {a^2 (157 A-85 B+45 C)-6 a^2 (21 A-13 B+5 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 4510

\(\displaystyle \frac {\frac {\frac {2 \int -\frac {a^3 (787 A-475 B+195 C)-4 a^3 (157 A-85 B+45 C) \sec (c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {\sec (c+d x) a+a}}dx}{5 a}+\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^3 (787 A-475 B+195 C)-4 a^3 (157 A-85 B+45 C) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {\sec (c+d x) a+a}}dx}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^3 (787 A-475 B+195 C)-4 a^3 (157 A-85 B+45 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 4510

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 \int -\frac {a^4 (2671 A-1495 B+735 C)-2 a^4 (787 A-475 B+195 C) \sec (c+d x)}{2 \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x) a+a}}dx}{3 a}+\frac {2 a^3 (787 A-475 B+195 C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^3 (787 A-475 B+195 C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (2671 A-1495 B+735 C)-2 a^4 (787 A-475 B+195 C) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {\sec (c+d x) a+a}}dx}{3 a}}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^3 (787 A-475 B+195 C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (2671 A-1495 B+735 C)-2 a^4 (787 A-475 B+195 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 4501

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^3 (787 A-475 B+195 C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^4 (2671 A-1495 B+735 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {a \sec (c+d x)+a}}-15 a^4 (283 A-163 B+75 C) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {\sec (c+d x) a+a}}dx}{3 a}}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^3 (787 A-475 B+195 C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^4 (2671 A-1495 B+735 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {a \sec (c+d x)+a}}-15 a^4 (283 A-163 B+75 C) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 4295

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^3 (787 A-475 B+195 C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {\frac {30 a^4 (283 A-163 B+75 C) \int \frac {1}{2 a-\frac {a^2 \sin (c+d x) \tan (c+d x)}{\sec (c+d x) a+a}}d\left (-\frac {a \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}+\frac {2 a^4 (2671 A-1495 B+735 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {a \sec (c+d x)+a}}}{3 a}}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {2 a^2 (157 A-85 B+45 C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^3 (787 A-475 B+195 C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^4 (2671 A-1495 B+735 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {a \sec (c+d x)+a}}-\frac {15 \sqrt {2} a^{7/2} (283 A-163 B+75 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x) \sqrt {\sec (c+d x)}}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{4 a^2}-\frac {a (21 A-13 B+5 C) \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}}{8 a^2}-\frac {(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}\)

Input:

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

Output:

-1/4*((A - B + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]) 
^(5/2)) + (-1/2*(a*(21*A - 13*B + 5*C)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2) 
*(a + a*Sec[c + d*x])^(3/2)) + ((2*a^2*(157*A - 85*B + 45*C)*Sin[c + d*x]) 
/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((2*a^3*(787*A - 475* 
B + 195*C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) 
 - ((-15*Sqrt[2]*a^(7/2)*(283*A - 163*B + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[ 
c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/d + (2*a^4*(2 
671*A - 1495*B + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec 
[c + d*x]]))/(3*a))/(5*a))/(4*a^2))/(8*a^2)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4295

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

rule 4501

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

rule 4508

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

rule 4510

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*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d 
*n)   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B* 
n - A*b*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, 
 m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
 

rule 4572

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

Maple [A] (warning: unable to verify)

Time = 3.65 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.33

method	result	size
default	\(\frac {\left (\left (4245 \cos \left (d x +c \right )^{3}+12735 \cos \left (d x +c \right )^{2}+12735 \cos \left (d x +c \right )+4245\right ) A \sqrt {-\frac {2}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )+\left (-2445 \cos \left (d x +c \right )^{3}-7335 \cos \left (d x +c \right )^{2}-7335 \cos \left (d x +c \right )-2445\right ) B \sqrt {-\frac {2}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )+\left (1125 \cos \left (d x +c \right )^{3}+3375 \cos \left (d x +c \right )^{2}+3375 \cos \left (d x +c \right )+1125\right ) C \sqrt {-\frac {2}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )+\left (192 \cos \left (d x +c \right )^{4}-320 \cos \left (d x +c \right )^{3}+3136 \cos \left (d x +c \right )^{2}+9110 \cos \left (d x +c \right )+5342\right ) \sin \left (d x +c \right ) A +\left (320 \cos \left (d x +c \right )^{3}-1600 \cos \left (d x +c \right )^{2}-5030 \cos \left (d x +c \right )-2990\right ) \sin \left (d x +c \right ) B +\left (960 \cos \left (d x +c \right )^{2}+2550 \cos \left (d x +c \right )+1470\right ) \sin \left (d x +c \right ) C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{480 d \,a^{3} \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right ) \sqrt {\sec \left (d x +c \right )}}\)	\(417\)
parts	\(\frac {A \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (192 \cos \left (d x +c \right )^{4}-320 \cos \left (d x +c \right )^{3}+3136 \cos \left (d x +c \right )^{2}+9110 \cos \left (d x +c \right )+5342\right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )+\arctan \left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {2}{\cos \left (d x +c \right )+1}}\, \left (4245 \cos \left (d x +c \right )+12735+12735 \sec \left (d x +c \right )+4245 \sec \left (d x +c \right )^{2}\right )\right )}{480 d \,a^{3} \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {5}{2}}}-\frac {B \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (-64 \cos \left (d x +c \right )^{3}+320 \cos \left (d x +c \right )^{2}+1006 \cos \left (d x +c \right )+598\right ) \tan \left (d x +c \right )+\arctan \left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {2}{\cos \left (d x +c \right )+1}}\, \left (489 \cos \left (d x +c \right )^{2}+1467 \cos \left (d x +c \right )+1467+489 \sec \left (d x +c \right )\right )\right )}{96 d \,a^{3} \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {C \left (-\frac {\left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}}{16}+\frac {17 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}}{32}+\frac {75 \arctan \left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {2}{\cos \left (d x +c \right )+1}}}{32}+\frac {83 \csc \left (d x +c \right )}{32}-\frac {83 \cot \left (d x +c \right )}{32}\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{d \,a^{3} \sqrt {\sec \left (d x +c \right )}}\)	\(508\)

Input:

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(5/2 
),x,method=_RETURNVERBOSE)
 

Output:

1/480/d/a^3*((4245*cos(d*x+c)^3+12735*cos(d*x+c)^2+12735*cos(d*x+c)+4245)* 
A*(-2/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)/(-1/(cos(d*x+c)+1))^(1/2)*( 
-csc(d*x+c)+cot(d*x+c)))+(-2445*cos(d*x+c)^3-7335*cos(d*x+c)^2-7335*cos(d* 
x+c)-2445)*B*(-2/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)/(-1/(cos(d*x+c)+ 
1))^(1/2)*(-csc(d*x+c)+cot(d*x+c)))+(1125*cos(d*x+c)^3+3375*cos(d*x+c)^2+3 
375*cos(d*x+c)+1125)*C*(-2/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)/(-1/(c 
os(d*x+c)+1))^(1/2)*(-csc(d*x+c)+cot(d*x+c)))+(192*cos(d*x+c)^4-320*cos(d* 
x+c)^3+3136*cos(d*x+c)^2+9110*cos(d*x+c)+5342)*sin(d*x+c)*A+(320*cos(d*x+c 
)^3-1600*cos(d*x+c)^2-5030*cos(d*x+c)-2990)*sin(d*x+c)*B+(960*cos(d*x+c)^2 
+2550*cos(d*x+c)+1470)*sin(d*x+c)*C)*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)^ 
3+3*cos(d*x+c)^2+3*cos(d*x+c)+1)/sec(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.03 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c) 
)^(5/2),x, algorithm="fricas")
 

Output:

[1/960*(15*sqrt(2)*((283*A - 163*B + 75*C)*cos(d*x + c)^3 + 3*(283*A - 163 
*B + 75*C)*cos(d*x + c)^2 + 3*(283*A - 163*B + 75*C)*cos(d*x + c) + 283*A 
- 163*B + 75*C)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a)*sqrt((a 
*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - 2*a*cos 
(d*x + c) - 3*a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 4*(96*A*cos(d*x 
+ c)^5 - 160*(A - B)*cos(d*x + c)^4 + 32*(49*A - 25*B + 15*C)*cos(d*x + c) 
^3 + 5*(911*A - 503*B + 255*C)*cos(d*x + c)^2 + (2671*A - 1495*B + 735*C)* 
cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/sqrt(co 
s(d*x + c)))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos( 
d*x + c) + a^3*d), 1/480*(15*sqrt(2)*((283*A - 163*B + 75*C)*cos(d*x + c)^ 
3 + 3*(283*A - 163*B + 75*C)*cos(d*x + c)^2 + 3*(283*A - 163*B + 75*C)*cos 
(d*x + c) + 283*A - 163*B + 75*C)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a 
*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))/(a*sin(d*x + c))) + 2* 
(96*A*cos(d*x + c)^5 - 160*(A - B)*cos(d*x + c)^4 + 32*(49*A - 25*B + 15*C 
)*cos(d*x + c)^3 + 5*(911*A - 503*B + 255*C)*cos(d*x + c)^2 + (2671*A - 14 
95*B + 735*C)*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d* 
x + c)/sqrt(cos(d*x + c)))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 
+ 3*a^3*d*cos(d*x + c) + a^3*d)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 645572 vs. \(2 (270) = 540\).

Time = 9.07 (sec) , antiderivative size = 645572, normalized size of antiderivative = 2062.53 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c) 
)^(5/2),x, algorithm="maxima")
 

Output:

1/480*((16*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x + 5*c) 
^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin 
(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos(5*d*x 
 + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c), cos( 
5/2*d*x + 5/2*c))))*cos(13/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5 
/2*c)))^4 + 4096*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin(5*d*x 
+ 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^ 
2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 300*(cos 
(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 5/2*c) 
, cos(5/2*d*x + 5/2*c))))*cos(11/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d 
*x + 5/2*c)))^4 + 20736*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 3*sin 
(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5*d*x 
+ 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 3 
00*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2*d*x + 
 5/2*c), cos(5/2*d*x + 5/2*c))))*cos(9/5*arctan2(sin(5/2*d*x + 5/2*c), cos 
(5/2*d*x + 5/2*c)))^4 + 4096*(3*cos(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) + 
3*sin(5*d*x + 5*c)^2*sin(5/2*d*x + 5/2*c) - 25*(cos(5*d*x + 5*c)^2 + sin(5 
*d*x + 5*c)^2)*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c)) 
) + 300*(cos(5*d*x + 5*c)^2 + sin(5*d*x + 5*c)^2)*sin(1/5*arctan2(sin(5/2* 
d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*cos(7/5*arctan2(sin(5/2*d*x + 5/2...
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c) 
)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:index.cc index_m i_lex_is_greater Error: Bad Argument V 
alue
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )+1}}{\sec \left (d x +c \right )^{6}+3 \sec \left (d x +c \right )^{5}+3 \sec \left (d x +c \right )^{4}+\sec \left (d x +c \right )^{3}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )+1}}{\sec \left (d x +c \right )^{5}+3 \sec \left (d x +c \right )^{4}+3 \sec \left (d x +c \right )^{3}+\sec \left (d x +c \right )^{2}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )+1}}{\sec \left (d x +c \right )^{4}+3 \sec \left (d x +c \right )^{3}+3 \sec \left (d x +c \right )^{2}+\sec \left (d x +c \right )}d x \right ) c \right )}{a^{3}} \] Input:

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

Output:

(sqrt(a)*(int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x) + 1))/(sec(c + d*x)**6 
 + 3*sec(c + d*x)**5 + 3*sec(c + d*x)**4 + sec(c + d*x)**3),x)*a + int((sq 
rt(sec(c + d*x))*sqrt(sec(c + d*x) + 1))/(sec(c + d*x)**5 + 3*sec(c + d*x) 
**4 + 3*sec(c + d*x)**3 + sec(c + d*x)**2),x)*b + int((sqrt(sec(c + d*x))* 
sqrt(sec(c + d*x) + 1))/(sec(c + d*x)**4 + 3*sec(c + d*x)**3 + 3*sec(c + d 
*x)**2 + sec(c + d*x)),x)*c))/a**3