3.894 \(\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=314 \[ \frac{\sin (c+d x) \left (2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right )}{30 d}+\frac{a \sin (c+d x) \cos (c+d x) \left (4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right )}{120 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{60 d}+\frac{1}{8} x \left (4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right )+\frac{(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac{b^4 C \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*x)/8 + (b^4*C*ArcTanh[Sin[c + d
*x]])/d + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(3
0*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((1
2*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(60*d) + ((4*A*b +
5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*
Sin[c + d*x])/(5*d)

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Rubi [A]  time = 1.04871, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {4094, 4074, 4047, 8, 4045, 3770} \[ \frac{\sin (c+d x) \left (2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right )}{30 d}+\frac{a \sin (c+d x) \cos (c+d x) \left (4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right )}{120 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{60 d}+\frac{1}{8} x \left (4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right )+\frac{(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac{b^4 C \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*x)/8 + (b^4*C*ArcTanh[Sin[c + d
*x]])/d + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(3
0*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((1
2*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(60*d) + ((4*A*b +
5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*
Sin[c + d*x])/(5*d)

Rule 4094

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

Rule 4074

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 8

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

Rule 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

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

Mathematica [A]  time = 1.24771, size = 382, normalized size = 1.22 \[ \frac{120 a \sin (2 (c+d x)) \left (4 a^2 b (A+C)+a^3 B+6 a b^2 B+4 A b^3\right )+60 \sin (c+d x) \left (12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+24 a^3 b B+32 a b^3 B+8 A b^4\right )+240 a^2 A b^2 \sin (3 (c+d x))+60 a^3 A b \sin (4 (c+d x))+720 a^3 A b c+720 a^3 A b d x+50 a^4 A \sin (3 (c+d x))+6 a^4 A \sin (5 (c+d x))+1440 a^2 b^2 B c+1440 a^2 b^2 B d x+160 a^3 b B \sin (3 (c+d x))+960 a^3 b c C+960 a^3 b C d x+15 a^4 B \sin (4 (c+d x))+180 a^4 B c+180 a^4 B d x+40 a^4 C \sin (3 (c+d x))+960 a A b^3 c+960 a A b^3 d x+1920 a b^3 c C+1920 a b^3 C d x+480 b^4 B c+480 b^4 B d x-480 b^4 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+480 b^4 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

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

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Maple [A]  time = 0.092, size = 543, normalized size = 1.7 \begin{align*}{\frac{2\,{a}^{4}C\sin \left ( dx+c \right ) }{3\,d}}+2\,{a}^{3}bCx+3\,{a}^{2}{b}^{2}Bx+2\,Aa{b}^{3}x+{\frac{C{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,A{a}^{3}b\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2\,d}}+2\,{\frac{C\cos \left ( dx+c \right ){a}^{3}b\sin \left ( dx+c \right ) }{d}}+{\frac{4\,B\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{3}b}{3\,d}}+2\,{\frac{A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{2}{b}^{2}}{d}}+3\,{\frac{B\cos \left ( dx+c \right ){a}^{2}{b}^{2}\sin \left ( dx+c \right ) }{d}}+2\,{\frac{Aa{b}^{3}c}{d}}+2\,{\frac{{a}^{3}bCc}{d}}+3\,{\frac{{a}^{2}{b}^{2}Bc}{d}}+{\frac{A{b}^{4}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{Ca{b}^{3}c}{d}}+B{b}^{4}x+2\,{\frac{A\cos \left ( dx+c \right ) a{b}^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{A{a}^{3}b\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{3\,A{a}^{3}bc}{2\,d}}+{\frac{4\,A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{15\,d}}+{\frac{3\,B{a}^{4}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{8\,d}}+{\frac{8\,B{a}^{3}b\sin \left ( dx+c \right ) }{3\,d}}+4\,{\frac{A{a}^{2}{b}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{3\,B{a}^{4}c}{8\,d}}+{\frac{8\,A{a}^{4}\sin \left ( dx+c \right ) }{15\,d}}+{\frac{B{b}^{4}c}{d}}+4\,Ca{b}^{3}x+{\frac{C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{3\,d}}+{\frac{B{a}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{A{a}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+6\,{\frac{C{a}^{2}{b}^{2}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{a{b}^{3}B\sin \left ( dx+c \right ) }{d}}+{\frac{3\,B{a}^{4}x}{8}}+{\frac{3\,{a}^{3}Abx}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.06217, size = 468, normalized size = 1.49 \begin{align*} \frac{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 )} A a^{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 a^{4} - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 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 )} A a^{3} b - 640 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b + 480 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 480 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 1920 \,{\left (d x + c\right )} C a b^{3} + 480 \,{\left (d x + c\right )} B b^{4} + 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 )} + 2880 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 1920 \, B a b^{3} \sin \left (d x + c\right ) + 480 \, A b^{4} \sin \left (d x + c\right )}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c
) + 8*sin(2*d*x + 2*c))*B*a^4 - 160*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 60*(12*d*x + 12*c + sin(4*d*x +
4*c) + 8*sin(2*d*x + 2*c))*A*a^3*b - 640*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3*b + 480*(2*d*x + 2*c + sin(2*
d*x + 2*c))*C*a^3*b - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b^2 + 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*B
*a^2*b^2 + 480*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a*b^3 + 1920*(d*x + c)*C*a*b^3 + 480*(d*x + c)*B*b^4 + 240*C
*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 2880*C*a^2*b^2*sin(d*x + c) + 1920*B*a*b^3*sin(d*x + c)
+ 480*A*b^4*sin(d*x + c))/d

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Fricas [A]  time = 0.612538, size = 640, normalized size = 2.04 \begin{align*} \frac{60 \, C b^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, C b^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (3 \, B a^{4} + 4 \,{\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \,{\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} d x +{\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 16 \,{\left (4 \, A + 5 \, C\right )} a^{4} + 320 \, B a^{3} b + 240 \,{\left (2 \, A + 3 \, C\right )} a^{2} b^{2} + 480 \, B a b^{3} + 120 \, A b^{4} + 30 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left ({\left (4 \, A + 5 \, C\right )} a^{4} + 20 \, B a^{3} b + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (3 \, B a^{4} + 4 \,{\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3}\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(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.41457, size = 1477, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/120*(120*C*b^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 120*C*b^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 15*(3*B*a
^4 + 12*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3 + 32*C*a*b^3 + 8*B*b^4)*(d*x + c) + 2*(120*A*a^4*tan(
1/2*d*x + 1/2*c)^9 - 75*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 120*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 300*A*a^3*b*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*tan(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 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 + 160*A*a^4*tan
(1/2*d*x + 1/2*c)^7 - 30*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 320*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 120*A*a^3*b*tan(1/2
*d*x + 1/2*c)^7 + 1280*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 + 1920*A*a^2*b^2*ta
n(1/2*d*x + 1/2*c)^7 - 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 2880*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 + 1920*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 480*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 464*A*a^
4*tan(1/2*d*x + 1/2*c)^5 + 400*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 1600*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 2400*A*a^2
*b^2*tan(1/2*d*x + 1/2*c)^5 + 4320*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 2880*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 72
0*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 160*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 30*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 320*C*a^
4*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 1280*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 + 1920*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 + 28
80*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 + 1920*B*a*b^3*tan(1/2*d*x + 1/2*c)^3
+ 480*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^4*tan(1/2*d*x + 1/2*c) + 75*B*a^4*tan(1/2*d*x + 1/2*c) + 120*C*a
^4*tan(1/2*d*x + 1/2*c) + 300*A*a^3*b*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*ta
n(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*t
an(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) + 120*A*b^4*tan(1/2*
d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d