\(\int (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \sec ^7(c+d x) \, dx\) [27]

Optimal result

Integrand size = 33, antiderivative size = 225 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3 (23 A+30 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac {a^3 (23 A+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]

[Out]

1/16*a^3*(23*A+30*C)*arctanh(sin(d*x+c))/d+1/15*a^3*(34*A+45*C)*tan(d*x+c)/d+1/16*a^3*(23*A+30*C)*sec(d*x+c)*t
an(d*x+c)/d+1/120*a^3*(73*A+90*C)*sec(d*x+c)^2*tan(d*x+c)/d+1/120*(31*A+30*C)*(a^3+a^3*cos(d*x+c))*sec(d*x+c)^
3*tan(d*x+c)/d+1/10*A*(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)^4*tan(d*x+c)/a/d+1/6*A*(a+a*cos(d*x+c))^3*sec(d*x+c)^5
*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3123, 3054, 3047, 3100, 2827, 3853, 3855, 3852, 8} \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3 (23 A+30 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac {a^3 (73 A+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac {a^3 (23 A+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {(31 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{120 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 a d}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]

[In]

Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]

[Out]

(a^3*(23*A + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 30*C
)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((31*A + 30*C)
*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*
Tan[c + d*x])/(10*a*d) + (A*(a + a*Cos[c + d*x])^3*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 3047

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

Rule 3054

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

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 3123

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

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 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

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+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x))^3 (3 a A+2 a (A+3 C) \cos (c+d x)) \sec ^6(c+d x) \, dx}{6 a} \\ & = \frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x))^2 \left (a^2 (31 A+30 C)+2 a^2 (8 A+15 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{30 a} \\ & = \frac {(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x)) \left (3 a^3 (73 A+90 C)+18 a^3 (7 A+10 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a} \\ & = \frac {(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int \left (3 a^4 (73 A+90 C)+\left (18 a^4 (7 A+10 C)+3 a^4 (73 A+90 C)\right ) \cos (c+d x)+18 a^4 (7 A+10 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a} \\ & = \frac {a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int \left (45 a^4 (23 A+30 C)+24 a^4 (34 A+45 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{360 a} \\ & = \frac {a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{8} \left (a^3 (23 A+30 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{15} \left (a^3 (34 A+45 C)\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {a^3 (23 A+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{16} \left (a^3 (23 A+30 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (34 A+45 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d} \\ & = \frac {a^3 (23 A+30 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac {a^3 (23 A+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.60 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.08 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {23 a^3 A \text {arctanh}(\sin (c+d x))}{16 d}+\frac {15 a^3 C \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^3 A \tan (c+d x)}{d}+\frac {4 a^3 C \tan (c+d x)}{d}+\frac {23 a^3 A \sec (c+d x) \tan (c+d x)}{16 d}+\frac {15 a^3 C \sec (c+d x) \tan (c+d x)}{8 d}+\frac {23 a^3 A \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^3 C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^3 A \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {7 a^3 A \tan ^3(c+d x)}{3 d}+\frac {a^3 C \tan ^3(c+d x)}{d}+\frac {3 a^3 A \tan ^5(c+d x)}{5 d} \]

[In]

Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]

[Out]

(23*a^3*A*ArcTanh[Sin[c + d*x]])/(16*d) + (15*a^3*C*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*A*Tan[c + d*x])/d +
(4*a^3*C*Tan[c + d*x])/d + (23*a^3*A*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (15*a^3*C*Sec[c + d*x]*Tan[c + d*x])/
(8*d) + (23*a^3*A*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a^3*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*A*Sec
[c + d*x]^5*Tan[c + d*x])/(6*d) + (7*a^3*A*Tan[c + d*x]^3)/(3*d) + (a^3*C*Tan[c + d*x]^3)/d + (3*a^3*A*Tan[c +
d*x]^5)/(5*d)

Maple [A] (verified)

Time = 11.86 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.10

method	result	size
parallelrisch	\(\frac {75 \left (-\frac {23 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {30 C}{23}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{20}+\frac {23 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {30 C}{23}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{20}+\frac {4 \left (2 A +\frac {23 C}{15}\right ) \sin \left (2 d x +2 c \right )}{5}+\frac {\left (53 C +\frac {391 A}{6}\right ) \sin \left (3 d x +3 c \right )}{75}+\frac {16 \left (\frac {17 A}{5}+4 C \right ) \sin \left (4 d x +4 c \right )}{75}+\frac {\left (\frac {23 A}{30}+C \right ) \sin \left (5 d x +5 c \right )}{5}+\frac {4 \left (\frac {34 A}{45}+C \right ) \sin \left (6 d x +6 c \right )}{25}+\sin \left (d x +c \right ) \left (\frac {38 C}{75}+A \right )\right ) a^{3}}{4 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(247\)
parts	\(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (A \,a^{3}+3 C \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{3}+C \,a^{3}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \,a^{3} \tan \left (d x +c \right )}{d}-\frac {3 A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {3 C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(250\)
derivativedivides	\(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \,a^{3} \tan \left (d x +c \right )+3 A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-3 C \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(294\)
default	\(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \,a^{3} \tan \left (d x +c \right )+3 A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-3 C \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\)	\(294\)
risch	\(-\frac {i a^{3} \left (345 A \,{\mathrm e}^{11 i \left (d x +c \right )}+450 C \,{\mathrm e}^{11 i \left (d x +c \right )}-240 C \,{\mathrm e}^{10 i \left (d x +c \right )}+1955 A \,{\mathrm e}^{9 i \left (d x +c \right )}+1590 C \,{\mathrm e}^{9 i \left (d x +c \right )}-480 A \,{\mathrm e}^{8 i \left (d x +c \right )}-2640 C \,{\mathrm e}^{8 i \left (d x +c \right )}+2250 A \,{\mathrm e}^{7 i \left (d x +c \right )}+1140 C \,{\mathrm e}^{7 i \left (d x +c \right )}-5440 A \,{\mathrm e}^{6 i \left (d x +c \right )}-7200 C \,{\mathrm e}^{6 i \left (d x +c \right )}-2250 A \,{\mathrm e}^{5 i \left (d x +c \right )}-1140 C \,{\mathrm e}^{5 i \left (d x +c \right )}-7680 A \,{\mathrm e}^{4 i \left (d x +c \right )}-8160 C \,{\mathrm e}^{4 i \left (d x +c \right )}-1955 A \,{\mathrm e}^{3 i \left (d x +c \right )}-1590 C \,{\mathrm e}^{3 i \left (d x +c \right )}-3264 A \,{\mathrm e}^{2 i \left (d x +c \right )}-4080 C \,{\mathrm e}^{2 i \left (d x +c \right )}-345 A \,{\mathrm e}^{i \left (d x +c \right )}-450 C \,{\mathrm e}^{i \left (d x +c \right )}-544 A -720 C \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {23 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {23 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\)	\(371\)

[In]

int((a+cos(d*x+c)*a)^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

75/4*(-23/20*(1/15*cos(6*d*x+6*c)+2/5*cos(4*d*x+4*c)+cos(2*d*x+2*c)+2/3)*(A+30/23*C)*ln(tan(1/2*d*x+1/2*c)-1)+
23/20*(1/15*cos(6*d*x+6*c)+2/5*cos(4*d*x+4*c)+cos(2*d*x+2*c)+2/3)*(A+30/23*C)*ln(tan(1/2*d*x+1/2*c)+1)+4/5*(2*
A+23/15*C)*sin(2*d*x+2*c)+1/75*(53*C+391/6*A)*sin(3*d*x+3*c)+16/75*(17/5*A+4*C)*sin(4*d*x+4*c)+1/5*(23/30*A+C)
*sin(5*d*x+5*c)+4/25*(34/45*A+C)*sin(6*d*x+6*c)+sin(d*x+c)*(38/75*C+A))*a^3/d/(cos(6*d*x+6*c)+6*cos(4*d*x+4*c)
+15*cos(2*d*x+2*c)+10)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.80 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (34 \, A + 45 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \, {\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \, {\left (17 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \, {\left (23 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 144 \, A a^{3} \cos \left (d x + c\right ) + 40 \, A a^{3}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="fricas")

[Out]

1/480*(15*(23*A + 30*C)*a^3*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 15*(23*A + 30*C)*a^3*cos(d*x + c)^6*log(-si
n(d*x + c) + 1) + 2*(16*(34*A + 45*C)*a^3*cos(d*x + c)^5 + 15*(23*A + 30*C)*a^3*cos(d*x + c)^4 + 16*(17*A + 15
*C)*a^3*cos(d*x + c)^3 + 10*(23*A + 6*C)*a^3*cos(d*x + c)^2 + 144*A*a^3*cos(d*x + c) + 40*A*a^3)*sin(d*x + c))
/(d*cos(d*x + c)^6)

Sympy [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)*sec(d*x+c)**7,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.70 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{3} + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 5 \, A a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, C a^{3} \tan \left (d x + c\right )}{480 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="maxima")

[Out]

1/480*(96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^3 + 160*(tan(d*x + c)^3 + 3*tan(d*x + c
))*A*a^3 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 - 5*A*a^3*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 3
3*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)) - 90*A*a^3*(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)) - 30*C*a^3*(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)) - 360*C*a^3*(2*sin(d*x +
c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*C*a^3*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.24 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left (23 \, A a^{3} + 30 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (23 \, A a^{3} + 30 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (345 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 450 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1955 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2550 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5940 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5814 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 7500 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5130 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1470 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="giac")

[Out]

1/240*(15*(23*A*a^3 + 30*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(23*A*a^3 + 30*C*a^3)*log(abs(tan(1/2*
d*x + 1/2*c) - 1)) - 2*(345*A*a^3*tan(1/2*d*x + 1/2*c)^11 + 450*C*a^3*tan(1/2*d*x + 1/2*c)^11 - 1955*A*a^3*tan
(1/2*d*x + 1/2*c)^9 - 2550*C*a^3*tan(1/2*d*x + 1/2*c)^9 + 4554*A*a^3*tan(1/2*d*x + 1/2*c)^7 + 5940*C*a^3*tan(1
/2*d*x + 1/2*c)^7 - 5814*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 7500*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 3165*A*a^3*tan(1/2
*d*x + 1/2*c)^3 + 5130*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 1575*A*a^3*tan(1/2*d*x + 1/2*c) - 1470*C*a^3*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 = 3.08 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.16 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\left (-\frac {23\,A\,a^3}{8}-\frac {15\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {391\,A\,a^3}{24}+\frac {85\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {759\,A\,a^3}{20}-\frac {99\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {969\,A\,a^3}{20}+\frac {125\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {211\,A\,a^3}{8}-\frac {171\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {105\,A\,a^3}{8}+\frac {49\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (23\,A+30\,C\right )}{8\,d} \]

[In]

int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3)/cos(c + d*x)^7,x)

[Out]

(tan(c/2 + (d*x)/2)*((105*A*a^3)/8 + (49*C*a^3)/4) - tan(c/2 + (d*x)/2)^11*((23*A*a^3)/8 + (15*C*a^3)/4) - tan
(c/2 + (d*x)/2)^3*((211*A*a^3)/8 + (171*C*a^3)/4) + tan(c/2 + (d*x)/2)^9*((391*A*a^3)/24 + (85*C*a^3)/4) - tan
(c/2 + (d*x)/2)^7*((759*A*a^3)/20 + (99*C*a^3)/2) + tan(c/2 + (d*x)/2)^5*((969*A*a^3)/20 + (125*C*a^3)/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)) + (a^3*atanh(tan(c/2 + (d*x)/2))*(23*A + 30*C))/(8*d)