\(\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} (A+C \sec ^2(c+d x)) \, dx\) [721]

Optimal result

Integrand size = 33, antiderivative size = 408 \[ \int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a (a-b) \sqrt {a+b} (3 A-8 C) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 b d}+\frac {\sqrt {a+b} \left (a b (3 A-8 C)+6 a^2 C+2 b^2 (3 A+C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 b d}-\frac {3 A b \sqrt {a+b} \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{d}+\frac {A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {b (3 A-2 C) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3 d} \] Output:

1/3*a*(a-b)*(a+b)^(1/2)*(3*A-8*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 
+sec(d*x+c))/(a-b))^(1/2)/b/d+1/3*(a+b)^(1/2)*(a*b*(3*A-8*C)+6*C*a^2+2*b^2 
*(3*A+C))*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 
)/b/d-3*A*b*(a+b)^(1/2)*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)/d+A*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d-1/3*b*(3* 
A-2*C)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d
 

Mathematica [A] (warning: unable to verify)

Time = 19.22 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.38 \[ \int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \left (2 a (a+b) (3 A-8 C) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )+4 \left (3 a^2 C+b^2 (3 A+C)+a (-6 A b+4 b C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )+a \left (36 A b \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )+(3 A-8 C) \cos (c+d x) (b+a \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}{3 d (b+a \cos (c+d x))^2 (A+2 C+A \cos (2 c+2 d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} \sec ^{\frac {7}{2}}(c+d x)}+\frac {\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {16}{3} a C \sin (c+d x)+\frac {4}{3} b C \tan (c+d x)\right )}{d (b+a \cos (c+d x)) (A+2 C+A \cos (2 c+2 d x))} \] Input:

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

Output:

(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + C 
*Sec[c + d*x]^2)*(2*a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d 
*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[Ar 
cSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*a^2*C + 
 b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] 
*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[ 
Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(36*A*b*Sqrt[Co 
s[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos 
[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec 
[(c + d*x)/2]^2 + (3*A - 8*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d 
*x)/2]^4*Tan[(c + d*x)/2])))/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[ 
2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(7/2)) + (Cos[c + d* 
x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((16*a*C*Sin[c + d* 
x])/3 + (4*b*C*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[ 
2*c + 2*d*x]))
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3042, 4583, 27, 3042, 4544, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4583

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4544

\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \int \frac {-a b (3 A-8 C) \sec ^2(c+d x)+2 \left (3 A b^2+\left (3 a^2+b^2\right ) C\right ) \sec (c+d x)+9 a A b}{2 \sqrt {a+b \sec (c+d x)}}dx-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {-a b (3 A-8 C) \sec ^2(c+d x)+2 \left (3 A b^2+\left (3 a^2+b^2\right ) C\right ) \sec (c+d x)+9 a A b}{\sqrt {a+b \sec (c+d x)}}dx-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {-a b (3 A-8 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (3 A b^2+\left (3 a^2+b^2\right ) C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+9 a A b}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \frac {9 a A b+\left (a b (3 A-8 C)+2 \left (3 A b^2+\left (3 a^2+b^2\right ) C\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-a b (3 A-8 C) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \frac {9 a A b+\left (a b (3 A-8 C)+2 \left (3 A b^2+\left (3 a^2+b^2\right ) C\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b (3 A-8 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\left (6 a^2 C+a b (3 A-8 C)+2 b^2 (3 A+C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-a b (3 A-8 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+9 a A b \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\left (6 a^2 C+a b (3 A-8 C)+2 b^2 (3 A+C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b (3 A-8 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+9 a A b \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\left (6 a^2 C+a b (3 A-8 C)+2 b^2 (3 A+C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b (3 A-8 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {18 A b \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-a b (3 A-8 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (6 a^2 C+a b (3 A-8 C)+2 b^2 (3 A+C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\frac {18 A b \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {2 \sqrt {a+b} \left (6 a^2 C+a b (3 A-8 C)+2 b^2 (3 A+C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 a (a-b) \sqrt {a+b} (3 A-8 C) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {18 A b \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )-\frac {2 b (3 A-2 C) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}\)

Input:

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

Output:

(A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d + (((2*a*(a - b)*Sqrt[a + b] 
*(3*A - 8*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a 
 + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 
 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(a*b*(3*A - 8*C) + 6*a 
^2*C + 2*b^2*(3*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d 
*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*S 
qrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (18*A*b*Sqrt[a + b]*Cot[c 
+ d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], 
 (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[ 
c + d*x]))/(a - b))])/d)/3 - (2*b*(3*A - 2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan 
[c + d*x])/(3*d))/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 3042

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

rule 4271

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

rule 4319

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

rule 4409

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

rule 4492

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

rule 4544

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

rule 4546

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

rule 4583

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1149\) vs. \(2(371)=742\).

Time = 20.84 (sec) , antiderivative size = 1150, normalized size of antiderivative = 2.82

method	result	size
default	\(\text {Expression too large to display}\)	\(1150\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**3,x)*b*c + int(sqr 
t(sec(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**2,x)*a*c + int(sqrt(sec(c 
 + d*x)*b + a)*cos(c + d*x)*sec(c + d*x),x)*a*b + int(sqrt(sec(c + d*x)*b 
+ a)*cos(c + d*x),x)*a**2