\(\int (a+b \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \sec ^7(c+d x) \, dx\) [557]

Optimal result

Integrand size = 33, antiderivative size = 307 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]

[Out]

1/16*(8*b^4*(A+2*C)+12*a^2*b^2*(3*A+4*C)+a^4*(5*A+6*C))*arctanh(sin(d*x+c))/d+4/15*a*b*(5*b^2*(2*A+3*C)+2*a^2*
(4*A+5*C))*tan(d*x+c)/d+1/240*(24*A*b^4+15*a^4*(5*A+6*C)+10*a^2*b^2*(49*A+66*C))*sec(d*x+c)*tan(d*x+c)/d+1/60*
a*b*(4*A*b^2+a^2*(39*A+50*C))*sec(d*x+c)^2*tan(d*x+c)/d+1/120*(12*A*b^2+5*a^2*(5*A+6*C))*(a+b*cos(d*x+c))^2*se
c(d*x+c)^3*tan(d*x+c)/d+2/15*A*b*(a+b*cos(d*x+c))^3*sec(d*x+c)^4*tan(d*x+c)/d+1/6*A*(a+b*cos(d*x+c))^4*sec(d*x
+c)^5*tan(d*x+c)/d

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3127, 3126, 3110, 3100, 2827, 3852, 8, 3855} \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {4 a b \left (2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{15 d}+\frac {a b \left (a^2 (39 A+50 C)+4 A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac {\left (5 a^2 (5 A+6 C)+12 A b^2\right ) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{120 d}+\frac {\left (a^4 (5 A+6 C)+12 a^2 b^2 (3 A+4 C)+8 b^4 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)+24 A b^4\right ) \tan (c+d x) \sec (c+d x)}{240 d}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}+\frac {2 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d} \]

[In]

Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]

[Out]

((8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a*b*(5*b^2*(2
*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((24*A*b^4 + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*
C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + (a*b*(4*A*b^2 + a^2*(39*A + 50*C))*Sec[c + d*x]^2*Tan[c + d*x])/(60*d
) + ((12*A*b^2 + 5*a^2*(5*A + 6*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (2*A*b*(a +
b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])
/(6*d)

Rule 8

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

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3100

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

Rule 3110

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)
*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)
), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d +
b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m
+ 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] &&
NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3126

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3127

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n
+ 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2
*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3852

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

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+b \cos (c+d x))^3 \left (4 A b+a (5 A+6 C) \cos (c+d x)+b (A+6 C) \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx \\ & = \frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{30} \int (a+b \cos (c+d x))^2 \left (12 A b^2+5 a^2 (5 A+6 C)+2 a b (23 A+30 C) \cos (c+d x)+3 b^2 (3 A+10 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx \\ & = \frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{120} \int (a+b \cos (c+d x)) \left (6 b \left (4 A b^2+a^2 (39 A+50 C)\right )+a \left (15 a^2 (5 A+6 C)+8 b^2 (32 A+45 C)\right ) \cos (c+d x)+b \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {1}{360} \int \left (-3 \left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right )-96 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \cos (c+d x)-3 b^2 \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {1}{720} \int \left (-192 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )-45 \left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{15} \left (4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )\right ) \int \sec ^2(c+d x) \, dx-\frac {1}{16} \left (-8 b^4 (A+2 C)-12 a^2 b^2 (3 A+4 C)-a^4 (5 A+6 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {\left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {\left (4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d} \\ & = \frac {\left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac {\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.95 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.66 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (8 A b^4+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \sec (c+d x)+10 a^2 \left (36 A b^2+a^2 (5 A+6 C)\right ) \sec ^3(c+d x)+40 a^4 A \sec ^5(c+d x)+64 a b \left (15 \left (a^2+b^2\right ) (A+C)+5 \left (A b^2+a^2 (2 A+C)\right ) \tan ^2(c+d x)+3 a^2 A \tan ^4(c+d x)\right )\right )}{240 d} \]

[In]

Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]

[Out]

(15*(8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(8*A
*b^4 + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*Sec[c + d*x] + 10*a^2*(36*A*b^2 + a^2*(5*A + 6*C))*Sec[c + d*
x]^3 + 40*a^4*A*Sec[c + d*x]^5 + 64*a*b*(15*(a^2 + b^2)*(A + C) + 5*(A*b^2 + a^2*(2*A + C))*Tan[c + d*x]^2 + 3
*a^2*A*Tan[c + d*x]^4)))/(240*d)

Maple [A] (verified)

Time = 13.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.95

method	result	size
parts	\(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 A a \,b^{3}+4 C \,a^{3} b \right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+C \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {4 A \,a^{3} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {4 C a \,b^{3} \tan \left (d x +c \right )}{d}\)	\(291\)
derivativedivides	\(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 A \,a^{3} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-4 C \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 C \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A a \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \tan \left (d x +c \right ) a \,b^{3}+A \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\)	\(360\)
default	\(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 A \,a^{3} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-4 C \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 C \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A a \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \tan \left (d x +c \right ) a \,b^{3}+A \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\)	\(360\)
parallelrisch	\(\frac {-75 \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {36 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+75 \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {36 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (850 A +1020 C \right ) a^{4}+6120 \left (A +\frac {12 C}{17}\right ) b^{2} a^{2}+720 A \,b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (\left (150 A +180 C \right ) a^{4}+1080 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}+240 A \,b^{4}\right ) \sin \left (5 d x +5 c \right )+7680 \left (a^{2} \left (A +\frac {3 C}{4}\right )+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{2}}{4}\right ) b a \sin \left (2 d x +2 c \right )+3072 b \left (a^{2} \left (A +\frac {5 C}{4}\right )+\frac {5 b^{2} \left (A +C \right )}{4}\right ) a \sin \left (4 d x +4 c \right )+512 b a \left (a^{2} \left (A +\frac {5 C}{4}\right )+\frac {5 b^{2} \left (A +\frac {3 C}{2}\right )}{4}\right ) \sin \left (6 d x +6 c \right )+1980 \sin \left (d x +c \right ) \left (\left (A +\frac {14 C}{33}\right ) a^{4}+\frac {28 b^{2} \left (A +\frac {4 C}{7}\right ) a^{2}}{11}+\frac {8 A \,b^{4}}{33}\right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(418\)
risch	\(\text {Expression too large to display}\)	\(1067\)

[In]

int((a+cos(d*x+c)*b)^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.97 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, {\left (A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, {\left (A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (192 \, A a^{3} b \cos \left (d x + c\right ) + 64 \, {\left (2 \, {\left (4 \, A + 5 \, C\right )} a^{3} b + 5 \, {\left (2 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 40 \, A a^{4} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 64 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{3} b + 5 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="fricas")

[Out]

1/480*(15*((5*A + 6*C)*a^4 + 12*(3*A + 4*C)*a^2*b^2 + 8*(A + 2*C)*b^4)*cos(d*x + c)^6*log(sin(d*x + c) + 1) -
15*((5*A + 6*C)*a^4 + 12*(3*A + 4*C)*a^2*b^2 + 8*(A + 2*C)*b^4)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(192
*A*a^3*b*cos(d*x + c) + 64*(2*(4*A + 5*C)*a^3*b + 5*(2*A + 3*C)*a*b^3)*cos(d*x + c)^5 + 40*A*a^4 + 15*((5*A +
6*C)*a^4 + 12*(3*A + 4*C)*a^2*b^2 + 8*A*b^4)*cos(d*x + c)^4 + 64*((4*A + 5*C)*a^3*b + 5*A*a*b^3)*cos(d*x + c)^
3 + 10*((5*A + 6*C)*a^4 + 36*A*a^2*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^6)

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**7,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.52 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {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 a^{3} b + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} b + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{3} - 5 \, A a^{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 )} - 30 \, C 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 )} - 180 \, A 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 )} - 720 \, C a^{2} b^{2} {\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 )} - 120 \, A b^{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 )} + 240 \, C b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1920 \, C a b^{3} \tan \left (d x + c\right )}{480 \, d} \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="maxima")

[Out]

1/480*(128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^3*b + 640*(tan(d*x + c)^3 + 3*tan(d*x
+ c))*C*a^3*b + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a*b^3 - 5*A*a^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)) - 30*C*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)) - 180*A*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)) - 720*C*a^2*
b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 120*A*b^4*(2*sin(d
*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 240*C*b^4*(log(sin(d*x + c) +
1) - log(sin(d*x + c) - 1)) + 1920*C*a*b^3*tan(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1100 vs. \(2 (293) = 586\).

Time = 0.40 (sec) , antiderivative size = 1100, normalized size of antiderivative = 3.58 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Too large to display} \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="giac")

[Out]

1/240*(15*(5*A*a^4 + 6*C*a^4 + 36*A*a^2*b^2 + 48*C*a^2*b^2 + 8*A*b^4 + 16*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c)
+ 1)) - 15*(5*A*a^4 + 6*C*a^4 + 36*A*a^2*b^2 + 48*C*a^2*b^2 + 8*A*b^4 + 16*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c)
- 1)) + 2*(165*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 150*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 960*A*a^3*b*tan(1/2*d*x +
1/2*c)^11 - 960*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 900*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 720*C*a^2*b^2*tan(1/
2*d*x + 1/2*c)^11 - 960*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 - 960*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 120*A*b^4*tan(
1/2*d*x + 1/2*c)^11 + 25*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 210*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 2240*A*a^3*b*tan(1/
2*d*x + 1/2*c)^9 + 3520*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 1260*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 2160*C*a^2*b^
2*tan(1/2*d*x + 1/2*c)^9 + 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 4800*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 360*A*b
^4*tan(1/2*d*x + 1/2*c)^9 + 450*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 60*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 4992*A*a^3*b*
tan(1/2*d*x + 1/2*c)^7 - 5760*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 1440*C*a
^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 5760*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 9600*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 24
0*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 450*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 60*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 4992*A*a
^3*b*tan(1/2*d*x + 1/2*c)^5 + 5760*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 144
0*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 5760*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 9600*C*a*b^3*tan(1/2*d*x + 1/2*c)^5
+ 240*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 25*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 210*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 224
0*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 3520*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 1260*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3
- 2160*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 4800*C*a*b^3*tan(1/2*d*x + 1/
2*c)^3 - 360*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 165*A*a^4*tan(1/2*d*x + 1/2*c) + 150*C*a^4*tan(1/2*d*x + 1/2*c) +
960*A*a^3*b*tan(1/2*d*x + 1/2*c) + 960*C*a^3*b*tan(1/2*d*x + 1/2*c) + 900*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 720
*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 960*A*a*b^3*tan(1/2*d*x + 1/2*c) + 960*C*a*b^3*tan(1/2*d*x + 1/2*c) + 120*A*
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 = 4.37 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.25 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\left (\frac {11\,A\,a^4}{8}+A\,b^4+\frac {5\,C\,a^4}{4}+\frac {15\,A\,a^2\,b^2}{2}+6\,C\,a^2\,b^2-8\,A\,a\,b^3-8\,A\,a^3\,b-8\,C\,a\,b^3-8\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,A\,a^4}{24}-3\,A\,b^4-\frac {7\,C\,a^4}{4}-\frac {21\,A\,a^2\,b^2}{2}-18\,C\,a^2\,b^2+\frac {88\,A\,a\,b^3}{3}+\frac {56\,A\,a^3\,b}{3}+40\,C\,a\,b^3+\frac {88\,C\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {15\,A\,a^4}{4}+2\,A\,b^4+\frac {C\,a^4}{2}+3\,A\,a^2\,b^2+12\,C\,a^2\,b^2-48\,A\,a\,b^3-\frac {208\,A\,a^3\,b}{5}-80\,C\,a\,b^3-48\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,A\,a^4}{4}+2\,A\,b^4+\frac {C\,a^4}{2}+3\,A\,a^2\,b^2+12\,C\,a^2\,b^2+48\,A\,a\,b^3+\frac {208\,A\,a^3\,b}{5}+80\,C\,a\,b^3+48\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,A\,a^4}{24}-3\,A\,b^4-\frac {7\,C\,a^4}{4}-\frac {21\,A\,a^2\,b^2}{2}-18\,C\,a^2\,b^2-\frac {88\,A\,a\,b^3}{3}-\frac {56\,A\,a^3\,b}{3}-40\,C\,a\,b^3-\frac {88\,C\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,A\,a^4}{8}+A\,b^4+\frac {5\,C\,a^4}{4}+\frac {15\,A\,a^2\,b^2}{2}+6\,C\,a^2\,b^2+8\,A\,a\,b^3+8\,A\,a^3\,b+8\,C\,a\,b^3+8\,C\,a^3\,b\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 )}+\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,A\,a^4}{16}+\frac {A\,b^4}{2}+\frac {3\,C\,a^4}{8}+C\,b^4+\frac {9\,A\,a^2\,b^2}{4}+3\,C\,a^2\,b^2\right )}{\frac {5\,A\,a^4}{4}+2\,A\,b^4+\frac {3\,C\,a^4}{2}+4\,C\,b^4+9\,A\,a^2\,b^2+12\,C\,a^2\,b^2}\right )\,\left (\frac {5\,A\,a^4}{8}+A\,b^4+\frac {3\,C\,a^4}{4}+2\,C\,b^4+\frac {9\,A\,a^2\,b^2}{2}+6\,C\,a^2\,b^2\right )}{d} \]

[In]

int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/cos(c + d*x)^7,x)

[Out]

(tan(c/2 + (d*x)/2)*((11*A*a^4)/8 + A*b^4 + (5*C*a^4)/4 + (15*A*a^2*b^2)/2 + 6*C*a^2*b^2 + 8*A*a*b^3 + 8*A*a^3
*b + 8*C*a*b^3 + 8*C*a^3*b) + tan(c/2 + (d*x)/2)^11*((11*A*a^4)/8 + A*b^4 + (5*C*a^4)/4 + (15*A*a^2*b^2)/2 + 6
*C*a^2*b^2 - 8*A*a*b^3 - 8*A*a^3*b - 8*C*a*b^3 - 8*C*a^3*b) - tan(c/2 + (d*x)/2)^3*(3*A*b^4 - (5*A*a^4)/24 + (
7*C*a^4)/4 + (21*A*a^2*b^2)/2 + 18*C*a^2*b^2 + (88*A*a*b^3)/3 + (56*A*a^3*b)/3 + 40*C*a*b^3 + (88*C*a^3*b)/3)
+ tan(c/2 + (d*x)/2)^9*((5*A*a^4)/24 - 3*A*b^4 - (7*C*a^4)/4 - (21*A*a^2*b^2)/2 - 18*C*a^2*b^2 + (88*A*a*b^3)/
3 + (56*A*a^3*b)/3 + 40*C*a*b^3 + (88*C*a^3*b)/3) + tan(c/2 + (d*x)/2)^5*((15*A*a^4)/4 + 2*A*b^4 + (C*a^4)/2 +
3*A*a^2*b^2 + 12*C*a^2*b^2 + 48*A*a*b^3 + (208*A*a^3*b)/5 + 80*C*a*b^3 + 48*C*a^3*b) + tan(c/2 + (d*x)/2)^7*(
(15*A*a^4)/4 + 2*A*b^4 + (C*a^4)/2 + 3*A*a^2*b^2 + 12*C*a^2*b^2 - 48*A*a*b^3 - (208*A*a^3*b)/5 - 80*C*a*b^3 -
48*C*a^3*b))/(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)) + (atanh((4*tan(c/2 + (d*x)/2)*((5*A*a^4)/16 +
(A*b^4)/2 + (3*C*a^4)/8 + C*b^4 + (9*A*a^2*b^2)/4 + 3*C*a^2*b^2))/((5*A*a^4)/4 + 2*A*b^4 + (3*C*a^4)/2 + 4*C*
b^4 + 9*A*a^2*b^2 + 12*C*a^2*b^2))*((5*A*a^4)/8 + A*b^4 + (3*C*a^4)/4 + 2*C*b^4 + (9*A*a^2*b^2)/2 + 6*C*a^2*b^
2))/d