\(\int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx\) [476]

Optimal result

Integrand size = 21, antiderivative size = 244 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {\left (8 a^4+36 a^2 b^2+5 b^4\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \tan (c+d x)}{60 b d}-\frac {\left (8 a^4-178 a^2 b^2-75 b^4\right ) \sec (c+d x) \tan (c+d x)}{240 d}-\frac {a \left (4 a^2-53 b^2\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}-\frac {\left (4 a^2-25 b^2\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {(a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d} \] Output:

1/16*(8*a^4+36*a^2*b^2+5*b^4)*arctanh(sin(d*x+c))/d-1/60*a*(4*a^4-121*a^2* 
b^2-128*b^4)*tan(d*x+c)/b/d-1/240*(8*a^4-178*a^2*b^2-75*b^4)*sec(d*x+c)*ta 
n(d*x+c)/d-1/120*a*(4*a^2-53*b^2)*(a+b*sec(d*x+c))^2*tan(d*x+c)/b/d-1/120* 
(4*a^2-25*b^2)*(a+b*sec(d*x+c))^3*tan(d*x+c)/b/d-1/30*a*(a+b*sec(d*x+c))^4 
*tan(d*x+c)/b/d+1/6*(a+b*sec(d*x+c))^5*tan(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.63 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {15 \left (8 a^4+36 a^2 b^2+5 b^4\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (8 a^4+36 a^2 b^2+5 b^4\right ) \sec (c+d x)+10 b^2 \left (36 a^2+5 b^2\right ) \sec ^3(c+d x)+40 b^4 \sec ^5(c+d x)+64 a b \left (15 \left (a^2+b^2\right )+5 \left (a^2+2 b^2\right ) \tan ^2(c+d x)+3 b^2 \tan ^4(c+d x)\right )\right )}{240 d} \] Input:

Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^4,x]
 

Output:

(15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15* 
(8*a^4 + 36*a^2*b^2 + 5*b^4)*Sec[c + d*x] + 10*b^2*(36*a^2 + 5*b^2)*Sec[c 
+ d*x]^3 + 40*b^4*Sec[c + d*x]^5 + 64*a*b*(15*(a^2 + b^2) + 5*(a^2 + 2*b^2 
)*Tan[c + d*x]^2 + 3*b^2*Tan[c + d*x]^4)))/(240*d)
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 4327, 3042, 4490, 3042, 4490, 27, 3042, 4490, 3042, 4485, 3042, 4274, 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.

\(\displaystyle \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4dx\)

\(\Big \downarrow \) 4327

\(\displaystyle \frac {\int \sec (c+d x) (5 b-a \sec (c+d x)) (a+b \sec (c+d x))^4dx}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (5 b-a \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4dx}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {1}{5} \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (21 a b-\left (4 a^2-25 b^2\right ) \sec (c+d x)\right )dx-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (21 a b+\left (25 b^2-4 a^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int 3 \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (24 a^2+25 b^2\right )-a \left (4 a^2-53 b^2\right ) \sec (c+d x)\right )dx-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (24 a^2+25 b^2\right )-a \left (4 a^2-53 b^2\right ) \sec (c+d x)\right )dx-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b \left (24 a^2+25 b^2\right )-a \left (4 a^2-53 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \sec (c+d x) (a+b \sec (c+d x)) \left (a b \left (64 a^2+181 b^2\right )-\left (8 a^4-178 b^2 a^2-75 b^4\right ) \sec (c+d x)\right )dx-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a b \left (64 a^2+181 b^2\right )+\left (-8 a^4+178 b^2 a^2+75 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4485

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \sec (c+d x) \left (15 b \left (8 a^4+36 b^2 a^2+5 b^4\right )-4 a \left (4 a^4-121 b^2 a^2-128 b^4\right ) \sec (c+d x)\right )dx-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (15 b \left (8 a^4+36 b^2 a^2+5 b^4\right )-4 a \left (4 a^4-121 b^2 a^2-128 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (8 a^4+36 a^2 b^2+5 b^4\right ) \int \sec (c+d x)dx-4 a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \int \sec ^2(c+d x)dx\right )-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (8 a^4+36 a^2 b^2+5 b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-4 a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {4 a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \int 1d(-\tan (c+d x))}{d}+15 b \left (8 a^4+36 a^2 b^2+5 b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx\right )-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (8 a^4+36 a^2 b^2+5 b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {4 a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \tan (c+d x)}{d}\right )-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {15 b \left (8 a^4+36 a^2 b^2+5 b^4\right ) \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \tan (c+d x)}{d}\right )-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a \left (4 a^2-53 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )-\frac {a \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

Input:

Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^4,x]
 

Output:

((a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d) + (-1/5*(a*(a + b*Sec[c + d* 
x])^4*Tan[c + d*x])/d + (-1/4*((4*a^2 - 25*b^2)*(a + b*Sec[c + d*x])^3*Tan 
[c + d*x])/d + (3*(-1/3*(a*(4*a^2 - 53*b^2)*(a + b*Sec[c + d*x])^2*Tan[c + 
 d*x])/d + (-1/2*(b*(8*a^4 - 178*a^2*b^2 - 75*b^4)*Sec[c + d*x]*Tan[c + d* 
x])/d + ((15*b*(8*a^4 + 36*a^2*b^2 + 5*b^4)*ArcTanh[Sin[c + d*x]])/d - (4* 
a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*Tan[c + d*x])/d)/2)/3))/4)/5)/(6*b)
 

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 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4327

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

rule 4485

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

rule 4490

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

Maple [A] (verified)

Time = 3.17 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 b \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(209\)
default	\(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 b \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(209\)
parts	\(\frac {a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {4 a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {4 b \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(220\)
parallelrisch	\(\frac {-1800 \left (a^{4}+\frac {9}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+1800 \left (a^{4}+\frac {9}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (720 a^{4}+6120 a^{2} b^{2}+850 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (240 a^{4}+1080 a^{2} b^{2}+150 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (5760 b \,a^{3}+7680 a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (3840 b \,a^{3}+3072 a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+\left (640 b \,a^{3}+512 a \,b^{3}\right ) \sin \left (6 d x +6 c \right )+480 \left (a^{4}+\frac {21}{2} a^{2} b^{2}+\frac {33}{8} b^{4}\right ) \sin \left (d x +c \right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(322\)
norman	\(\frac {\frac {\left (8 a^{4}-64 b \,a^{3}+60 a^{2} b^{2}-64 a \,b^{3}+11 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (8 a^{4}+64 b \,a^{3}+60 a^{2} b^{2}+64 a \,b^{3}+11 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (40 a^{4}-960 b \,a^{3}+60 a^{2} b^{2}-832 a \,b^{3}+75 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (40 a^{4}+960 b \,a^{3}+60 a^{2} b^{2}+832 a \,b^{3}+75 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {\left (72 a^{4}-704 b \,a^{3}+252 a^{2} b^{2}-448 a \,b^{3}-5 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (72 a^{4}+704 b \,a^{3}+252 a^{2} b^{2}+448 a \,b^{3}-5 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {\left (8 a^{4}+36 a^{2} b^{2}+5 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (8 a^{4}+36 a^{2} b^{2}+5 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(371\)
risch	\(-\frac {i \left (-640 b \,a^{3}+360 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}+120 a^{4} {\mathrm e}^{11 i \left (d x +c \right )}+425 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}-425 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-360 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}-75 b^{4} {\mathrm e}^{i \left (d x +c \right )}-120 a^{4} {\mathrm e}^{i \left (d x +c \right )}+75 b^{4} {\mathrm e}^{11 i \left (d x +c \right )}+240 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}-1920 a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-3840 a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3060 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-7680 a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-3060 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-540 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}-6400 a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-3072 a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+540 a^{2} b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+2520 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-5120 a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-2520 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-7680 b \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-240 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-990 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+990 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-512 a \,b^{3}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} b^{2}}{4 d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{4}}{16 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} b^{2}}{4 d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{4}}{16 d}\)	\(541\)

Input:

int(sec(d*x+c)^3*(a+b*sec(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^4*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))-4*b*a^3 
*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+6*a^2*b^2*(-(-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)))-4*a*b^3*(-8/15-1/5*sec( 
d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+b^4*(-(-1/6*sec(d*x+c)^5-5/24*sec(d 
*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c)+5/16*ln(sec(d*x+c)+tan(d*x+c))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {15 \, {\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (128 \, {\left (5 \, a^{3} b + 4 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + 192 \, a b^{3} \cos \left (d x + c\right ) + 15 \, {\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 40 \, b^{4} + 64 \, {\left (5 \, a^{3} b + 4 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \] Input:

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^4,x, algorithm="fricas")
 

Output:

1/480*(15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*cos(d*x + c)^6*log(sin(d*x + c) + 1 
) - 15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) 
+ 2*(128*(5*a^3*b + 4*a*b^3)*cos(d*x + c)^5 + 192*a*b^3*cos(d*x + c) + 15* 
(8*a^4 + 36*a^2*b^2 + 5*b^4)*cos(d*x + c)^4 + 40*b^4 + 64*(5*a^3*b + 4*a*b 
^3)*cos(d*x + c)^3 + 10*(36*a^2*b^2 + 5*b^4)*cos(d*x + c)^2)*sin(d*x + c)) 
/(d*cos(d*x + c)^6)
 

Sympy [F]

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

integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**4,x)
 

Output:

Integral((a + b*sec(c + d*x))**4*sec(c + d*x)**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.13 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} b + 128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a b^{3} - 5 \, b^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \] Input:

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^4,x, algorithm="maxima")
 

Output:

1/480*(640*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^3*b + 128*(3*tan(d*x + c)^5 
 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*a*b^3 - 5*b^4*(2*(15*sin(d*x + c)^ 
5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^ 
4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) 
 - 1)) - 180*a^2*b^2*(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)) - 120*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)))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (230) = 460\).

Time = 0.19 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.43 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx =\text {Too large to display} \] Input:

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^4,x, algorithm="giac")
 

Output:

1/240*(15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) 
- 15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*( 
120*a^4*tan(1/2*d*x + 1/2*c)^11 - 960*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 900* 
a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 960*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 165* 
b^4*tan(1/2*d*x + 1/2*c)^11 - 360*a^4*tan(1/2*d*x + 1/2*c)^9 + 3520*a^3*b* 
tan(1/2*d*x + 1/2*c)^9 - 1260*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 2240*a*b^3* 
tan(1/2*d*x + 1/2*c)^9 + 25*b^4*tan(1/2*d*x + 1/2*c)^9 + 240*a^4*tan(1/2*d 
*x + 1/2*c)^7 - 5760*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 360*a^2*b^2*tan(1/2*d* 
x + 1/2*c)^7 - 4992*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 450*b^4*tan(1/2*d*x + 1 
/2*c)^7 + 240*a^4*tan(1/2*d*x + 1/2*c)^5 + 5760*a^3*b*tan(1/2*d*x + 1/2*c) 
^5 + 360*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 4992*a*b^3*tan(1/2*d*x + 1/2*c)^ 
5 + 450*b^4*tan(1/2*d*x + 1/2*c)^5 - 360*a^4*tan(1/2*d*x + 1/2*c)^3 - 3520 
*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 1260*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 2240 
*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 25*b^4*tan(1/2*d*x + 1/2*c)^3 + 120*a^4*ta 
n(1/2*d*x + 1/2*c) + 960*a^3*b*tan(1/2*d*x + 1/2*c) + 900*a^2*b^2*tan(1/2* 
d*x + 1/2*c) + 960*a*b^3*tan(1/2*d*x + 1/2*c) + 165*b^4*tan(1/2*d*x + 1/2* 
c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d
 

Mupad [B] (verification not implemented)

Time = 14.87 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.52 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4+\frac {9\,a^2\,b^2}{2}+\frac {5\,b^4}{8}\right )}{d}+\frac {\left (a^4-8\,a^3\,b+\frac {15\,a^2\,b^2}{2}-8\,a\,b^3+\frac {11\,b^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (-3\,a^4+\frac {88\,a^3\,b}{3}-\frac {21\,a^2\,b^2}{2}+\frac {56\,a\,b^3}{3}+\frac {5\,b^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (2\,a^4-48\,a^3\,b+3\,a^2\,b^2-\frac {208\,a\,b^3}{5}+\frac {15\,b^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (2\,a^4+48\,a^3\,b+3\,a^2\,b^2+\frac {208\,a\,b^3}{5}+\frac {15\,b^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-3\,a^4-\frac {88\,a^3\,b}{3}-\frac {21\,a^2\,b^2}{2}-\frac {56\,a\,b^3}{3}+\frac {5\,b^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (a^4+8\,a^3\,b+\frac {15\,a^2\,b^2}{2}+8\,a\,b^3+\frac {11\,b^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:

int((a + b/cos(c + d*x))^4/cos(c + d*x)^3,x)
 

Output:

(atanh(tan(c/2 + (d*x)/2))*(a^4 + (5*b^4)/8 + (9*a^2*b^2)/2))/d + (tan(c/2 
 + (d*x)/2)^9*((56*a*b^3)/3 + (88*a^3*b)/3 - 3*a^4 + (5*b^4)/24 - (21*a^2* 
b^2)/2) - tan(c/2 + (d*x)/2)^3*((56*a*b^3)/3 + (88*a^3*b)/3 + 3*a^4 - (5*b 
^4)/24 + (21*a^2*b^2)/2) + tan(c/2 + (d*x)/2)^5*((208*a*b^3)/5 + 48*a^3*b 
+ 2*a^4 + (15*b^4)/4 + 3*a^2*b^2) + tan(c/2 + (d*x)/2)^7*(2*a^4 - 48*a^3*b 
 - (208*a*b^3)/5 + (15*b^4)/4 + 3*a^2*b^2) + tan(c/2 + (d*x)/2)*(8*a*b^3 + 
 8*a^3*b + a^4 + (11*b^4)/8 + (15*a^2*b^2)/2) + tan(c/2 + (d*x)/2)^11*(a^4 
 - 8*a^3*b - 8*a*b^3 + (11*b^4)/8 + (15*a^2*b^2)/2))/(d*(15*tan(c/2 + (d*x 
)/2)^4 - 6*tan(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + ( 
d*x)/2)^8 - 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.48 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^3*(a+b*sec(d*x+c))^4,x)
 

Output:

( - 640*cos(c + d*x)*sin(c + d*x)**5*a**3*b - 512*cos(c + d*x)*sin(c + d*x 
)**5*a*b**3 + 1600*cos(c + d*x)*sin(c + d*x)**3*a**3*b + 1280*cos(c + d*x) 
*sin(c + d*x)**3*a*b**3 - 960*cos(c + d*x)*sin(c + d*x)*a**3*b - 960*cos(c 
 + d*x)*sin(c + d*x)*a*b**3 - 120*log(tan((c + d*x)/2) - 1)*sin(c + d*x)** 
6*a**4 - 540*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**2*b**2 - 75*log( 
tan((c + d*x)/2) - 1)*sin(c + d*x)**6*b**4 + 360*log(tan((c + d*x)/2) - 1) 
*sin(c + d*x)**4*a**4 + 1620*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a** 
2*b**2 + 225*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*b**4 - 360*log(tan( 
(c + d*x)/2) - 1)*sin(c + d*x)**2*a**4 - 1620*log(tan((c + d*x)/2) - 1)*si 
n(c + d*x)**2*a**2*b**2 - 225*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b* 
*4 + 120*log(tan((c + d*x)/2) - 1)*a**4 + 540*log(tan((c + d*x)/2) - 1)*a* 
*2*b**2 + 75*log(tan((c + d*x)/2) - 1)*b**4 + 120*log(tan((c + d*x)/2) + 1 
)*sin(c + d*x)**6*a**4 + 540*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a** 
2*b**2 + 75*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*b**4 - 360*log(tan(( 
c + d*x)/2) + 1)*sin(c + d*x)**4*a**4 - 1620*log(tan((c + d*x)/2) + 1)*sin 
(c + d*x)**4*a**2*b**2 - 225*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b** 
4 + 360*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**4 + 1620*log(tan((c + 
 d*x)/2) + 1)*sin(c + d*x)**2*a**2*b**2 + 225*log(tan((c + d*x)/2) + 1)*si 
n(c + d*x)**2*b**4 - 120*log(tan((c + d*x)/2) + 1)*a**4 - 540*log(tan((c + 
 d*x)/2) + 1)*a**2*b**2 - 75*log(tan((c + d*x)/2) + 1)*b**4 - 120*sin(c...