3.966 \(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec (c+d x) \, dx\)

Optimal. Leaf size=290 \[ \frac{\sin (c+d x) \left (2 a^2 b^2 (85 A+56 C)+95 a^3 b B+12 a^4 C+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{30 d}+\frac{\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{60 d}+\frac{b \sin (c+d x) \cos (c+d x) \left (130 a^2 b B+24 a^3 C+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{120 d}+\frac{1}{8} x \left (16 a^3 b (2 A+C)+24 a^2 b^2 B+8 a^4 B+4 a b^3 (4 A+3 C)+3 b^4 B\right )+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]

[Out]

((8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*x)/8 + (a^4*A*ArcTanh[Sin[c + d
*x]])/d + ((95*a^3*b*B + 80*a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Sin[c + d*x])/(3
0*d) + (b*(130*a^2*b*B + 45*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((2
0*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*b*B + 4*a*C)*(a +
b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.905955, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {3049, 3033, 3023, 2735, 3770} \[ \frac{\sin (c+d x) \left (2 a^2 b^2 (85 A+56 C)+95 a^3 b B+12 a^4 C+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{30 d}+\frac{\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{60 d}+\frac{b \sin (c+d x) \cos (c+d x) \left (130 a^2 b B+24 a^3 C+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{120 d}+\frac{1}{8} x \left (16 a^3 b (2 A+C)+24 a^2 b^2 B+8 a^4 B+4 a b^3 (4 A+3 C)+3 b^4 B\right )+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*x)/8 + (a^4*A*ArcTanh[Sin[c + d
*x]])/d + ((95*a^3*b*B + 80*a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Sin[c + d*x])/(3
0*d) + (b*(130*a^2*b*B + 45*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((2
0*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*b*B + 4*a*C)*(a +
b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)

Rule 3049

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

Rule 3033

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[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3023

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

Rule 2735

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

Rule 3770

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

Rubi steps

\begin{align*} \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} \int (a+b \cos (c+d x))^3 \left (5 a A+(5 A b+5 a B+4 b C) \cos (c+d x)+(5 b B+4 a C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{(5 b B+4 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{20} \int (a+b \cos (c+d x))^2 \left (20 a^2 A+\left (40 a A b+20 a^2 B+15 b^2 B+28 a b C\right ) \cos (c+d x)+\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{(5 b B+4 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{60} \int (a+b \cos (c+d x)) \left (60 a^3 A+\left (60 a^3 B+115 a b^2 B+36 a^2 b (5 A+3 C)+8 b^3 (5 A+4 C)\right ) \cos (c+d x)+\left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac{\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{(5 b B+4 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{120} \int \left (120 a^4 A+15 \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \cos (c+d x)+4 \left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \sin (c+d x)}{30 d}+\frac{b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac{\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{(5 b B+4 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{120} \int \left (120 a^4 A+15 \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{8} \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) x+\frac{\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \sin (c+d x)}{30 d}+\frac{b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac{\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{(5 b B+4 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{8} \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) x+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \sin (c+d x)}{30 d}+\frac{b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac{\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{(5 b B+4 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 1.22174, size = 382, normalized size = 1.32 \[ \frac{60 \sin (c+d x) \left (12 a^2 b^2 (4 A+3 C)+32 a^3 b B+8 a^4 C+24 a b^3 B+b^4 (6 A+5 C)\right )+120 b \sin (2 (c+d x)) \left (6 a^2 b B+4 a^3 C+4 a b^2 (A+C)+b^3 B\right )+1920 a^3 A b c+1920 a^3 A b d x-480 a^4 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+480 a^4 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1440 a^2 b^2 B c+1440 a^2 b^2 B d x+240 a^2 b^2 C \sin (3 (c+d x))+960 a^3 b c C+960 a^3 b C d x+480 a^4 B c+480 a^4 B d x+960 a A b^3 c+960 a A b^3 d x+160 a b^3 B \sin (3 (c+d x))+60 a b^3 C \sin (4 (c+d x))+720 a b^3 c C+720 a b^3 C d x+40 A b^4 \sin (3 (c+d x))+15 b^4 B \sin (4 (c+d x))+180 b^4 B c+180 b^4 B d x+50 b^4 C \sin (3 (c+d x))+6 b^4 C \sin (5 (c+d x))}{480 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1920*a^3*A*b*c + 960*a*A*b^3*c + 480*a^4*B*c + 1440*a^2*b^2*B*c + 180*b^4*B*c + 960*a^3*b*c*C + 720*a*b^3*c*C
+ 1920*a^3*A*b*d*x + 960*a*A*b^3*d*x + 480*a^4*B*d*x + 1440*a^2*b^2*B*d*x + 180*b^4*B*d*x + 960*a^3*b*C*d*x +
720*a*b^3*C*d*x - 480*a^4*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*a^4*A*Log[Cos[(c + d*x)/2] + Sin[(
c + d*x)/2]] + 60*(32*a^3*b*B + 24*a*b^3*B + 8*a^4*C + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Sin[c + d*x]
+ 120*b*(6*a^2*b*B + b^3*B + 4*a^3*C + 4*a*b^2*(A + C))*Sin[2*(c + d*x)] + 40*A*b^4*Sin[3*(c + d*x)] + 160*a*b
^3*B*Sin[3*(c + d*x)] + 240*a^2*b^2*C*Sin[3*(c + d*x)] + 50*b^4*C*Sin[3*(c + d*x)] + 15*b^4*B*Sin[4*(c + d*x)]
+ 60*a*b^3*C*Sin[4*(c + d*x)] + 6*b^4*C*Sin[5*(c + d*x)])/(480*d)

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Maple [A]  time = 0.063, size = 543, normalized size = 1.9 \begin{align*}{\frac{3\,a{b}^{3}Cx}{2}}+2\,{\frac{A\cos \left ( dx+c \right ) a{b}^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{Ca{b}^{3}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{3\,C\cos \left ( dx+c \right ) a{b}^{3}\sin \left ( dx+c \right ) }{2\,d}}+2\,{\frac{C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{2}{b}^{2}}{d}}+2\,{\frac{C\cos \left ( dx+c \right ){a}^{3}b\sin \left ( dx+c \right ) }{d}}+{\frac{3\,{b}^{4}Bc}{8\,d}}+3\,{a}^{2}{b}^{2}Bx+{\frac{2\,A{b}^{4}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{8\,C{b}^{4}\sin \left ( dx+c \right ) }{15\,d}}+2\,aA{b}^{3}x+4\,A{a}^{3}bx+2\,{a}^{3}bCx+{\frac{B{a}^{4}c}{d}}+{\frac{{a}^{4}C\sin \left ( dx+c \right ) }{d}}+{\frac{A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{aA{b}^{3}c}{d}}+{\frac{3\,Ca{b}^{3}c}{2\,d}}+4\,{\frac{A{a}^{3}bc}{d}}+2\,{\frac{{a}^{3}bCc}{d}}+{\frac{C{b}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{4\,C{b}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+6\,{\frac{{a}^{2}A{b}^{2}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{2}{b}^{2}C\sin \left ( dx+c \right ) }{d}}+{\frac{A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ){b}^{4}}{3\,d}}+{a}^{4}Bx+{\frac{3\,{b}^{4}Bx}{8}}+4\,{\frac{{a}^{3}bB\sin \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{2}{b}^{2}Bc}{d}}+{\frac{{b}^{4}B\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{b}^{4}B\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{8\,a{b}^{3}B\sin \left ( dx+c \right ) }{3\,d}}+{\frac{4\,B\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}a{b}^{3}}{3\,d}}+3\,{\frac{B\cos \left ( dx+c \right ){a}^{2}{b}^{2}\sin \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01442, size = 459, normalized size = 1.58 \begin{align*} \frac{480 \,{\left (d x + c\right )} B a^{4} + 1920 \,{\left (d x + c\right )} A a^{3} b + 480 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b + 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} + 480 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} - 640 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{3} + 60 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} + 32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{4} + 480 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 480 \, C a^{4} \sin \left (d x + c\right ) + 1920 \, B a^{3} b \sin \left (d x + c\right ) + 2880 \, A a^{2} b^{2} \sin \left (d x + c\right )}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(480*(d*x + c)*B*a^4 + 1920*(d*x + c)*A*a^3*b + 480*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3*b + 720*(2*d*
x + 2*c + sin(2*d*x + 2*c))*B*a^2*b^2 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2*b^2 + 480*(2*d*x + 2*c + s
in(2*d*x + 2*c))*A*a*b^3 - 640*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a*b^3 + 60*(12*d*x + 12*c + sin(4*d*x + 4*c
) + 8*sin(2*d*x + 2*c))*C*a*b^3 - 160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*b^4 + 15*(12*d*x + 12*c + sin(4*d*x
+ 4*c) + 8*sin(2*d*x + 2*c))*B*b^4 + 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*b^4 + 480*A
*a^4*log(sec(d*x + c) + tan(d*x + c)) + 480*C*a^4*sin(d*x + c) + 1920*B*a^3*b*sin(d*x + c) + 2880*A*a^2*b^2*si
n(d*x + c))/d

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Fricas [A]  time = 1.96246, size = 640, normalized size = 2.21 \begin{align*} \frac{60 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (8 \, B a^{4} + 16 \,{\left (2 \, A + C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 4 \,{\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} d x +{\left (24 \, C b^{4} \cos \left (d x + c\right )^{4} + 120 \, C a^{4} + 480 \, B a^{3} b + 240 \,{\left (3 \, A + 2 \, C\right )} a^{2} b^{2} + 320 \, B a b^{3} + 16 \,{\left (5 \, A + 4 \, C\right )} b^{4} + 30 \,{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (30 \, C a^{2} b^{2} + 20 \, B a b^{3} +{\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (16 \, C a^{3} b + 24 \, B a^{2} b^{2} + 4 \,{\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(60*A*a^4*log(sin(d*x + c) + 1) - 60*A*a^4*log(-sin(d*x + c) + 1) + 15*(8*B*a^4 + 16*(2*A + C)*a^3*b + 2
4*B*a^2*b^2 + 4*(4*A + 3*C)*a*b^3 + 3*B*b^4)*d*x + (24*C*b^4*cos(d*x + c)^4 + 120*C*a^4 + 480*B*a^3*b + 240*(3
*A + 2*C)*a^2*b^2 + 320*B*a*b^3 + 16*(5*A + 4*C)*b^4 + 30*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^3 + 8*(30*C*a^2*b^2
+ 20*B*a*b^3 + (5*A + 4*C)*b^4)*cos(d*x + c)^2 + 15*(16*C*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C)*a*b^3 + 3*B*b^
4)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.33528, size = 1477, normalized size = 5.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(120*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 120*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 15*(8*B*a
^4 + 32*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3 + 12*C*a*b^3 + 3*B*b^4)*(d*x + c) + 2*(120*C*a^4*tan(
1/2*d*x + 1/2*c)^9 + 480*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 240*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*t
an(1/2*d*x + 1/2*c)^9 - 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 240*A*a*
b^3*tan(1/2*d*x + 1/2*c)^9 + 480*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 300*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*A*b
^4*tan(1/2*d*x + 1/2*c)^9 - 75*B*b^4*tan(1/2*d*x + 1/2*c)^9 + 120*C*b^4*tan(1/2*d*x + 1/2*c)^9 + 480*C*a^4*tan
(1/2*d*x + 1/2*c)^7 + 1920*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 2880*A*a^2*b^
2*tan(1/2*d*x + 1/2*c)^7 - 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 1920*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 480*
A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 1280*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 120*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 32
0*A*b^4*tan(1/2*d*x + 1/2*c)^7 - 30*B*b^4*tan(1/2*d*x + 1/2*c)^7 + 160*C*b^4*tan(1/2*d*x + 1/2*c)^7 + 720*C*a^
4*tan(1/2*d*x + 1/2*c)^5 + 2880*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 4320*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 2400*
C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 1600*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 400*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 46
4*C*b^4*tan(1/2*d*x + 1/2*c)^5 + 480*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 1920*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 480*
C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 2880*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3
+ 1920*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 1280*B*a*b^3*tan(1/2*d*x + 1/2*
c)^3 + 120*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 320*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 30*B*b^4*tan(1/2*d*x + 1/2*c)^3
+ 160*C*b^4*tan(1/2*d*x + 1/2*c)^3 + 120*C*a^4*tan(1/2*d*x + 1/2*c) + 480*B*a^3*b*tan(1/2*d*x + 1/2*c) + 240*
C*a^3*b*tan(1/2*d*x + 1/2*c) + 720*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 360*B*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) + 240*A*a*b^3*tan(1/2*d*x + 1/2*c) + 480*B*a*b^3*tan(1/2*d*x + 1/2*c) + 300*C*a*
b^3*tan(1/2*d*x + 1/2*c) + 120*A*b^4*tan(1/2*d*x + 1/2*c) + 75*B*b^4*tan(1/2*d*x + 1/2*c) + 120*C*b^4*tan(1/2*
d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d