\(\int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx\) [74]

Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=a^4 A x+\frac {a^4 (48 A+35 B) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^4 (8 A+7 B) \tan (c+d x)}{8 d}+\frac {a B (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d} \] Output:

a^4*A*x+1/8*a^4*(48*A+35*B)*arctanh(sin(d*x+c))/d+5/8*a^4*(8*A+7*B)*tan(d* 
x+c)/d+1/4*a*B*(a+a*sec(d*x+c))^3*tan(d*x+c)/d+1/12*(4*A+7*B)*(a^2+a^2*sec 
(d*x+c))^2*tan(d*x+c)/d+1/24*(32*A+35*B)*(a^4+a^4*sec(d*x+c))*tan(d*x+c)/d
 

Mathematica [A] (verified)

Time = 4.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.87 \[ \int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^4 \left (24 A d x+24 (4 A+B) \coth ^{-1}(\sin (c+d x))+(48 A+81 B) \text {arctanh}(\sin (c+d x))+168 A \tan (c+d x)+192 B \tan (c+d x)+48 A \sec (c+d x) \tan (c+d x)+81 B \sec (c+d x) \tan (c+d x)+6 B \sec ^3(c+d x) \tan (c+d x)+8 A \tan ^3(c+d x)+32 B \tan ^3(c+d x)\right )}{24 d} \] Input:

Integrate[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]
 

Output:

(a^4*(24*A*d*x + 24*(4*A + B)*ArcCoth[Sin[c + d*x]] + (48*A + 81*B)*ArcTan 
h[Sin[c + d*x]] + 168*A*Tan[c + d*x] + 192*B*Tan[c + d*x] + 48*A*Sec[c + d 
*x]*Tan[c + d*x] + 81*B*Sec[c + d*x]*Tan[c + d*x] + 6*B*Sec[c + d*x]^3*Tan 
[c + d*x] + 8*A*Tan[c + d*x]^3 + 32*B*Tan[c + d*x]^3))/(24*d)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4405, 3042, 4405, 3042, 4405, 27, 3042, 4402, 3042, 4254, 24, 4257}

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.

\(\displaystyle \int (a \sec (c+d x)+a)^4 (A+B \sec (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 4405

\(\displaystyle \frac {1}{4} \int (\sec (c+d x) a+a)^3 (4 a A+a (4 A+7 B) \sec (c+d x))dx+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (4 a A+a (4 A+7 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 4405

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (\sec (c+d x) a+a)^2 \left (12 A a^2+(32 A+35 B) \sec (c+d x) a^2\right )dx+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (12 A a^2+(32 A+35 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 4405

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int 3 (\sec (c+d x) a+a) \left (8 A a^3+5 (8 A+7 B) \sec (c+d x) a^3\right )dx+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int (\sec (c+d x) a+a) \left (8 A a^3+5 (8 A+7 B) \sec (c+d x) a^3\right )dx+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (8 A a^3+5 (8 A+7 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 4402

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (5 a^4 (8 A+7 B) \int \sec ^2(c+d x)dx+a^4 (48 A+35 B) \int \sec (c+d x)dx+8 a^4 A x\right )+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (a^4 (48 A+35 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+5 a^4 (8 A+7 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+8 a^4 A x\right )+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (-\frac {5 a^4 (8 A+7 B) \int 1d(-\tan (c+d x))}{d}+a^4 (48 A+35 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+8 a^4 A x\right )+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (a^4 (48 A+35 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^4 (8 A+7 B) \tan (c+d x)}{d}+8 a^4 A x\right )+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\frac {a^4 (48 A+35 B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \tan (c+d x)}{d}+8 a^4 A x\right )+\frac {(32 A+35 B) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )+\frac {(4 A+7 B) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}\right )+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\)

Input:

Int[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]
 

Output:

(a*B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (((4*A + 7*B)*(a^2 + a^2 
*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + (((32*A + 35*B)*(a^4 + a^4*Sec[c + 
d*x])*Tan[c + d*x])/(2*d) + (3*(8*a^4*A*x + (a^4*(48*A + 35*B)*ArcTanh[Sin 
[c + d*x]])/d + (5*a^4*(8*A + 7*B)*Tan[c + d*x])/d))/2)/3)/4
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 4402

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

rule 4405

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

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.30

method	result	size
parts	\(a^{4} A x -\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(196\)
parallelrisch	\(\frac {44 a^{4} \left (-\frac {18 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {35 B}{48}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{11}+\frac {18 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {35 B}{48}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{11}+\frac {3 A x d \cos \left (2 d x +2 c \right )}{11}+\frac {3 A x d \cos \left (4 d x +4 c \right )}{44}+\left (A +\frac {14 B}{11}\right ) \sin \left (2 d x +2 c \right )+\frac {3 \left (A +\frac {27 B}{16}\right ) \sin \left (3 d x +3 c \right )}{11}+\frac {5 \left (A +B \right ) \sin \left (4 d x +4 c \right )}{11}+\frac {3 \left (A +\frac {35 B}{16}\right ) \sin \left (d x +c \right )}{11}+\frac {9 A x d}{44}\right )}{3 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\)	\(207\)
derivativedivides	\(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \tan \left (d x +c \right )+6 a^{4} A \tan \left (d x +c \right )+6 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(250\)
default	\(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \tan \left (d x +c \right )+6 a^{4} A \tan \left (d x +c \right )+6 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(250\)
norman	\(\frac {a^{4} A x +a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-4 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {5 a^{4} \left (8 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {3 a^{4} \left (24 A +31 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{4} \left (424 A +385 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {a^{4} \left (520 A +511 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {a^{4} \left (48 A +35 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{4} \left (48 A +35 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\)	\(252\)
risch	\(a^{4} A x -\frac {i a^{4} \left (48 A \,{\mathrm e}^{7 i \left (d x +c \right )}+81 B \,{\mathrm e}^{7 i \left (d x +c \right )}-144 A \,{\mathrm e}^{6 i \left (d x +c \right )}-96 B \,{\mathrm e}^{6 i \left (d x +c \right )}+48 A \,{\mathrm e}^{5 i \left (d x +c \right )}+105 B \,{\mathrm e}^{5 i \left (d x +c \right )}-480 A \,{\mathrm e}^{4 i \left (d x +c \right )}-480 B \,{\mathrm e}^{4 i \left (d x +c \right )}-48 A \,{\mathrm e}^{3 i \left (d x +c \right )}-105 B \,{\mathrm e}^{3 i \left (d x +c \right )}-496 A \,{\mathrm e}^{2 i \left (d x +c \right )}-544 B \,{\mathrm e}^{2 i \left (d x +c \right )}-48 \,{\mathrm e}^{i \left (d x +c \right )} A -81 B \,{\mathrm e}^{i \left (d x +c \right )}-160 A -160 B \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{8 d}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{8 d}\)	\(293\)

Input:

int((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

a^4*A*x-(A*a^4+4*B*a^4)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(4*A*a^4+B*a^ 
4)/d*ln(sec(d*x+c)+tan(d*x+c))+(4*A*a^4+6*B*a^4)/d*(1/2*sec(d*x+c)*tan(d*x 
+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(6*A*a^4+4*B*a^4)/d*tan(d*x+c)+B*a^4/d* 
(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+ 
c)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04 \[ \int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {48 \, A a^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (48 \, A + 35 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (48 \, A + 35 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (160 \, {\left (A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (16 \, A + 27 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right ) + 6 \, B a^{4}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \] Input:

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="fricas")
 

Output:

1/48*(48*A*a^4*d*x*cos(d*x + c)^4 + 3*(48*A + 35*B)*a^4*cos(d*x + c)^4*log 
(sin(d*x + c) + 1) - 3*(48*A + 35*B)*a^4*cos(d*x + c)^4*log(-sin(d*x + c) 
+ 1) + 2*(160*(A + B)*a^4*cos(d*x + c)^3 + 3*(16*A + 27*B)*a^4*cos(d*x + c 
)^2 + 8*(A + 4*B)*a^4*cos(d*x + c) + 6*B*a^4)*sin(d*x + c))/(d*cos(d*x + c 
)^4)
 

Sympy [F]

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

integrate((a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)),x)
 

Output:

a**4*(Integral(A, x) + Integral(4*A*sec(c + d*x), x) + Integral(6*A*sec(c 
+ d*x)**2, x) + Integral(4*A*sec(c + d*x)**3, x) + Integral(A*sec(c + d*x) 
**4, x) + Integral(B*sec(c + d*x), x) + Integral(4*B*sec(c + d*x)**2, x) + 
 Integral(6*B*sec(c + d*x)**3, x) + Integral(4*B*sec(c + d*x)**4, x) + Int 
egral(B*sec(c + d*x)**5, x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (141) = 282\).

Time = 0.04 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.94 \[ \int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 48 \, {\left (d x + c\right )} A a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} - 3 \, B a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 192 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, B a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 288 \, A a^{4} \tan \left (d x + c\right ) + 192 \, B a^{4} \tan \left (d x + c\right )}{48 \, d} \] Input:

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="maxima")
 

Output:

1/48*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 48*(d*x + c)*A*a^4 + 64 
*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^4 - 3*B*a^4*(2*(3*sin(d*x + c)^3 - 
5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + 
c) + 1) + 3*log(sin(d*x + c) - 1)) - 48*A*a^4*(2*sin(d*x + c)/(sin(d*x + c 
)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 72*B*a^4*(2*si 
n(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) 
 - 1)) + 192*A*a^4*log(sec(d*x + c) + tan(d*x + c)) + 48*B*a^4*log(sec(d*x 
 + c) + tan(d*x + c)) + 288*A*a^4*tan(d*x + c) + 192*B*a^4*tan(d*x + c))/d
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.48 \[ \int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {24 \, {\left (d x + c\right )} A a^{4} + 3 \, {\left (48 \, A a^{4} + 35 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (48 \, A a^{4} + 35 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (120 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 424 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 385 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 520 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 216 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 279 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \] Input:

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="giac")
 

Output:

1/24*(24*(d*x + c)*A*a^4 + 3*(48*A*a^4 + 35*B*a^4)*log(abs(tan(1/2*d*x + 1 
/2*c) + 1)) - 3*(48*A*a^4 + 35*B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 
 2*(120*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 105*B*a^4*tan(1/2*d*x + 1/2*c)^7 - 
424*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 385*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 520* 
A*a^4*tan(1/2*d*x + 1/2*c)^3 + 511*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 216*A*a^ 
4*tan(1/2*d*x + 1/2*c) - 279*B*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/ 
2*c)^2 - 1)^4)/d
 

Mupad [B] (verification not implemented)

Time = 11.07 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.69 \[ \int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {2\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {12\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {35\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {20\,B\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {27\,B\,a^4\,\sin \left (c+d\,x\right )}{8\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {4\,B\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {B\,a^4\,\sin \left (c+d\,x\right )}{4\,d\,{\cos \left (c+d\,x\right )}^4} \] Input:

int((A + B/cos(c + d*x))*(a + a/cos(c + d*x))^4,x)
 

Output:

(2*A*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (12*A*a^4*atanh( 
sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (35*B*a^4*atanh(sin(c/2 + (d*x 
)/2)/cos(c/2 + (d*x)/2)))/(4*d) + (20*A*a^4*sin(c + d*x))/(3*d*cos(c + d*x 
)) + (2*A*a^4*sin(c + d*x))/(d*cos(c + d*x)^2) + (A*a^4*sin(c + d*x))/(3*d 
*cos(c + d*x)^3) + (20*B*a^4*sin(c + d*x))/(3*d*cos(c + d*x)) + (27*B*a^4* 
sin(c + d*x))/(8*d*cos(c + d*x)^2) + (4*B*a^4*sin(c + d*x))/(3*d*cos(c + d 
*x)^3) + (B*a^4*sin(c + d*x))/(4*d*cos(c + d*x)^4)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.72 \[ \int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^{4} \left (-160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -160 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +168 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +192 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} a -105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} b +288 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a +210 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} b -144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a -105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b +144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} a +105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} b -288 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a -210 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} b +144 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a +105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b +24 \sin \left (d x +c \right )^{4} a d x -48 \sin \left (d x +c \right )^{3} a -81 \sin \left (d x +c \right )^{3} b -48 \sin \left (d x +c \right )^{2} a d x +48 \sin \left (d x +c \right ) a +87 \sin \left (d x +c \right ) b +24 a d x \right )}{24 d \left (\sin \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{2}+1\right )} \] Input:

int((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)),x)
 

Output:

(a**4*( - 160*cos(c + d*x)*sin(c + d*x)**3*a - 160*cos(c + d*x)*sin(c + d* 
x)**3*b + 168*cos(c + d*x)*sin(c + d*x)*a + 192*cos(c + d*x)*sin(c + d*x)* 
b - 144*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a - 105*log(tan((c + d*x 
)/2) - 1)*sin(c + d*x)**4*b + 288*log(tan((c + d*x)/2) - 1)*sin(c + d*x)** 
2*a + 210*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b - 144*log(tan((c + d 
*x)/2) - 1)*a - 105*log(tan((c + d*x)/2) - 1)*b + 144*log(tan((c + d*x)/2) 
 + 1)*sin(c + d*x)**4*a + 105*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b 
- 288*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a - 210*log(tan((c + d*x)/ 
2) + 1)*sin(c + d*x)**2*b + 144*log(tan((c + d*x)/2) + 1)*a + 105*log(tan( 
(c + d*x)/2) + 1)*b + 24*sin(c + d*x)**4*a*d*x - 48*sin(c + d*x)**3*a - 81 
*sin(c + d*x)**3*b - 48*sin(c + d*x)**2*a*d*x + 48*sin(c + d*x)*a + 87*sin 
(c + d*x)*b + 24*a*d*x))/(24*d*(sin(c + d*x)**4 - 2*sin(c + d*x)**2 + 1))