3.7.70 \(\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [670]

Optimal. Leaf size=250 \[ \frac {1}{2} a b \left (4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {b^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (6 A b^4+2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{15 d}+\frac {a b \left (6 A b^2+a^2 (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {\left (3 A b^2+a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{15 d}+\frac {A b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d} \]

[Out]

1/2*a*b*(4*b^2*(A+2*C)+a^2*(3*A+4*C))*x+b^4*C*arctanh(sin(d*x+c))/d+1/15*(6*A*b^4+2*a^4*(4*A+5*C)+a^2*b^2*(56*
A+85*C))*sin(d*x+c)/d+1/30*a*b*(6*A*b^2+a^2*(29*A+40*C))*cos(d*x+c)*sin(d*x+c)/d+1/15*(3*A*b^2+a^2*(4*A+5*C))*
cos(d*x+c)^2*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/5*A*b*cos(d*x+c)^3*(a+b*sec(d*x+c))^3*sin(d*x+c)/d+1/5*A*cos(d*
x+c)^4*(a+b*sec(d*x+c))^4*sin(d*x+c)/d

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Rubi [A]
time = 0.62, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4180, 4179, 4159, 4132, 8, 4130, 3855} \begin {gather*} \frac {a b \left (a^2 (29 A+40 C)+6 A b^2\right ) \sin (c+d x) \cos (c+d x)}{30 d}+\frac {\left (a^2 (4 A+5 C)+3 A b^2\right ) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac {1}{2} a b x \left (a^2 (3 A+4 C)+4 b^2 (A+2 C)\right )+\frac {\left (2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)+6 A b^4\right ) \sin (c+d x)}{15 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac {A b \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{5 d}+\frac {b^4 C \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x)/2 + (b^4*C*ArcTanh[Sin[c + d*x]])/d + ((6*A*b^4 + 2*a^4*(4*A + 5*C
) + a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(15*d) + (a*b*(6*A*b^2 + a^2*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x]
)/(30*d) + ((3*A*b^2 + a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + (A*b*Cos[
c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[c + d*x])
/(5*d)

Rule 8

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

Rule 3855

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

Rule 4130

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 4132

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 4159

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 4179

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 4180

Int[((A_.) + 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] - Dis
t[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - 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, C}, x] && NeQ[a^2 - b^2,
0] && GtQ[m, 0] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+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+a (4 A+5 C) \sec (c+d x)+5 b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 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 (4 \left (3 A b^2+a^2 (4 A+5 C)\right )+4 a b (7 A+10 C) \sec (c+d x)+20 b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (3 A b^2+a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{15 d}+\frac {A b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 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 (4 b \left (6 A b^2+a^2 (29 A+40 C)\right )+4 a \left (9 b^2 (3 A+5 C)+2 a^2 (4 A+5 C)\right ) \sec (c+d x)+60 b^3 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a b \left (6 A b^2+a^2 (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {\left (3 A b^2+a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{15 d}+\frac {A b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 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 (-8 \left (6 A b^4+2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)\right )-60 a b \left (4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) \sec (c+d x)-120 b^4 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a b \left (6 A b^2+a^2 (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {\left (3 A b^2+a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{15 d}+\frac {A b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 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 (-8 \left (6 A b^4+2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)\right )-120 b^4 C \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (a b \left (4 b^2 (A+2 C)+a^2 (3 A+4 C)\right )\right ) \int 1 \, dx\\ &=\frac {1}{2} a b \left (4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {\left (6 A b^4+2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{15 d}+\frac {a b \left (6 A b^2+a^2 (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {\left (3 A b^2+a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{15 d}+\frac {A b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 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}{2} a b \left (4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {b^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (6 A b^4+2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{15 d}+\frac {a b \left (6 A b^2+a^2 (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {\left (3 A b^2+a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{15 d}+\frac {A b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.84, size = 223, normalized size = 0.89 \begin {gather*} \frac {120 a b \left (4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) (c+d x)-240 b^4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 b^4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 \left (8 A b^4+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \sin (c+d x)+240 a b \left (A b^2+a^2 (A+C)\right ) \sin (2 (c+d x))+5 a^2 \left (24 A b^2+a^2 (5 A+4 C)\right ) \sin (3 (c+d x))+30 a^3 A b \sin (4 (c+d x))+3 a^4 A \sin (5 (c+d x))}{240 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(120*a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*(c + d*x) - 240*b^4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] +
240*b^4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 30*(8*A*b^4 + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*S
in[c + d*x] + 240*a*b*(A*b^2 + a^2*(A + C))*Sin[2*(c + d*x)] + 5*a^2*(24*A*b^2 + a^2*(5*A + 4*C))*Sin[3*(c + d
*x)] + 30*a^3*A*b*Sin[4*(c + d*x)] + 3*a^4*A*Sin[5*(c + d*x)])/(240*d)

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Maple [A]
time = 0.10, size = 240, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {A \,b^{4} \sin \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 C \,b^{3} a \left (d x +c \right )+2 a^{2} A \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 C \,a^{2} b^{2} \sin \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{3} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(240\)
default	\(\frac {A \,b^{4} \sin \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 C \,b^{3} a \left (d x +c \right )+2 a^{2} A \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 C \,a^{2} b^{2} \sin \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{3} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(240\)
risch	\(\frac {3 a^{3} A b x}{2}+2 A a \,b^{3} x +2 C \,a^{3} b x +4 C a \,b^{3} x +\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{2} b^{2}}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{8 d}-\frac {9 i {\mathrm e}^{i \left (d x +c \right )} a^{2} A \,b^{2}}{4 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{2} b^{2}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{4}}{2 d}-\frac {5 i A \,a^{4} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {9 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} A \,b^{2}}{4 d}+\frac {5 i A \,a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{4}}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{4}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{4}}{d}+\frac {A \,a^{4} \sin \left (5 d x +5 c \right )}{80 d}+\frac {A \,a^{3} b \sin \left (4 d x +4 c \right )}{8 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{4}}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} A \,b^{2}}{2 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3} b}{d}+\frac {\sin \left (2 d x +2 c \right ) a A \,b^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b C}{d}\)	\(427\)

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+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 239, normalized size = 0.96 \begin {gather*} \frac {8 \, {\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} - 40 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C 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 )} A a^{3} b + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 480 \, {\left (d x + c\right )} C a b^{3} + 60 \, 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 )} + 720 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 120 \, A b^{4} \sin \left (d x + c\right )}{120 \, d} \end {gather*}

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+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/120*(8*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 40*(sin(d*x + c)^3 - 3*sin(d*x + c))
*C*a^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3*b + 120*(2*d*x + 2*c + sin(2*d*x + 2
*c))*C*a^3*b - 240*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b^2 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a*b^3
+ 480*(d*x + c)*C*a*b^3 + 60*C*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 720*C*a^2*b^2*sin(d*x + c
) + 120*A*b^4*sin(d*x + c))/d

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Fricas [A]
time = 2.82, size = 195, normalized size = 0.78 \begin {gather*} \frac {15 \, C b^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, C b^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} b + 4 \, {\left (A + 2 \, C\right )} a b^{3}\right )} d x + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 30 \, A a^{3} b \cos \left (d x + c\right )^{3} + 4 \, {\left (4 \, A + 5 \, C\right )} a^{4} + 60 \, {\left (2 \, A + 3 \, C\right )} a^{2} b^{2} + 30 \, A b^{4} + 2 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{4} + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} b + 4 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, d} \end {gather*}

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+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/30*(15*C*b^4*log(sin(d*x + c) + 1) - 15*C*b^4*log(-sin(d*x + c) + 1) + 15*((3*A + 4*C)*a^3*b + 4*(A + 2*C)*a
*b^3)*d*x + (6*A*a^4*cos(d*x + c)^4 + 30*A*a^3*b*cos(d*x + c)^3 + 4*(4*A + 5*C)*a^4 + 60*(2*A + 3*C)*a^2*b^2 +
30*A*b^4 + 2*((4*A + 5*C)*a^4 + 30*A*a^2*b^2)*cos(d*x + c)^2 + 15*((3*A + 4*C)*a^3*b + 4*A*a*b^3)*cos(d*x + c
))*sin(d*x + c))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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+C*sec(d*x+c)**2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (238) = 476\).
time = 0.52, size = 753, normalized size = 3.01 \begin {gather*} \frac {30 \, C b^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 30 \, C b^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (3 \, A a^{3} b + 4 \, C a^{3} b + 4 \, A a b^{3} + 8 \, C a b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (30 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 30 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 180 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 180 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 30 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 40 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 80 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 120 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 480 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 120 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 116 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 100 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 600 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1080 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 180 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 480 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 720 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 30 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 180 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 180 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 30 \, A b^{4} \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 )}^{5}}}{30 \, d} \end {gather*}

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+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/30*(30*C*b^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 30*C*b^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 15*(3*A*a^3*
b + 4*C*a^3*b + 4*A*a*b^3 + 8*C*a*b^3)*(d*x + c) + 2*(30*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 30*C*a^4*tan(1/2*d*x +
1/2*c)^9 - 75*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 60*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 180*A*a^2*b^2*tan(1/2*d*x
+ 1/2*c)^9 + 180*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 60*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 30*A*b^4*tan(1/2*d*x +
1/2*c)^9 + 40*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 80*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 30*A*a^3*b*tan(1/2*d*x + 1/2*c
)^7 - 120*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 480*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 720*C*a^2*b^2*tan(1/2*d*x +
1/2*c)^7 - 120*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 116*A*a^4*tan(1/2*d*x + 1/2
*c)^5 + 100*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 600*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 1080*C*a^2*b^2*tan(1/2*d*x +
1/2*c)^5 + 180*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 40*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 80*C*a^4*tan(1/2*d*x + 1/2*c)
^3 + 30*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 120*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 480*A*a^2*b^2*tan(1/2*d*x + 1/2*
c)^3 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 120*A*b^4*tan(1/2*d*x + 1/2
*c)^3 + 30*A*a^4*tan(1/2*d*x + 1/2*c) + 30*C*a^4*tan(1/2*d*x + 1/2*c) + 75*A*a^3*b*tan(1/2*d*x + 1/2*c) + 60*C
*a^3*b*tan(1/2*d*x + 1/2*c) + 180*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 180*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 60*A*a
*b^3*tan(1/2*d*x + 1/2*c) + 30*A*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d

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Mupad [B]
time = 5.60, size = 2241, normalized size = 8.96 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 + 4*A*a*b^3 + 5*A*a^3*b + 4*C*a
^3*b) + tan(c/2 + (d*x)/2)^5*((116*A*a^4)/15 + 12*A*b^4 + (20*C*a^4)/3 + 40*A*a^2*b^2 + 72*C*a^2*b^2) + tan(c/
2 + (d*x)/2)^9*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 - 4*A*a*b^3 - 5*A*a^3*b - 4*C*a^3*b)
+ tan(c/2 + (d*x)/2)^3*((8*A*a^4)/3 + 8*A*b^4 + (16*C*a^4)/3 + 32*A*a^2*b^2 + 48*C*a^2*b^2 + 8*A*a*b^3 + 2*A*
a^3*b + 8*C*a^3*b) + tan(c/2 + (d*x)/2)^7*((8*A*a^4)/3 + 8*A*b^4 + (16*C*a^4)/3 + 32*A*a^2*b^2 + 48*C*a^2*b^2
- 8*A*a*b^3 - 2*A*a^3*b - 8*C*a^3*b))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x
)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) - (C*b^4*atan((C*b^4*(tan(c/2 + (d*x)/2)*(32*C^2
*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^
2 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2) + C*b^4*(32*C*b^4 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a
*b^3 + 64*C*a^3*b))*1i + C*b^4*(tan(c/2 + (d*x)/2)*(32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^
6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*
b^2) - C*b^4*(32*C*b^4 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*1i)/(1024*C^3*a^2*b^10 - 256*C^3
*a*b^11 - 128*C^3*a^3*b^9 + 1024*C^3*a^4*b^8 + 256*C^3*a^6*b^6 + C*b^4*(tan(c/2 + (d*x)/2)*(32*C^2*b^8 + 128*A
^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 512*A*C*
a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2) + C*b^4*(32*C*b^4 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*
a^3*b)) - C*b^4*(tan(c/2 + (d*x)/2)*(32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*C^2
*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2) - C*b^4*(3
2*C*b^4 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b)) - 128*A*C^2*a*b^11 + 1024*A*C^2*a^2*b^10 - 96*A
*C^2*a^3*b^9 + 1280*A*C^2*a^4*b^8 + 384*A*C^2*a^6*b^6 + 256*A^2*C*a^2*b^10 + 384*A^2*C*a^4*b^8 + 144*A^2*C*a^6
*b^6))*2i)/d - (a*b*atan(((a*b*(tan(c/2 + (d*x)/2)*(32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^
6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*
b^2) - (a*b*(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2)*(32*C*b^4 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a
^3*b)*1i)/2)*(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2))/2 + (a*b*(tan(c/2 + (d*x)/2)*(32*C^2*b^8 + 128*A^2*a^2*b
^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 512*A*C*a^2*b^6
+ 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2) + (a*b*(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2)*(32*C*b^4 + 64*A*a*b^3 + 4
8*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b)*1i)/2)*(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2))/2)/(1024*C^3*a^2*b^10 -
256*C^3*a*b^11 - 128*C^3*a^3*b^9 + 1024*C^3*a^4*b^8 + 256*C^3*a^6*b^6 - 128*A*C^2*a*b^11 + 1024*A*C^2*a^2*b^10
- 96*A*C^2*a^3*b^9 + 1280*A*C^2*a^4*b^8 + 384*A*C^2*a^6*b^6 + 256*A^2*C*a^2*b^10 + 384*A^2*C*a^4*b^8 + 144*A^
2*C*a^6*b^6 - (a*b*(tan(c/2 + (d*x)/2)*(32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*
C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2) - (a*b*
(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2)*(32*C*b^4 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b)*1i)/2)
*(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2)*1i)/2 + (a*b*(tan(c/2 + (d*x)/2)*(32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*
A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 512*A*C*a^2*b^6 + 640*A*C
*a^4*b^4 + 192*A*C*a^6*b^2) + (a*b*(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2)*(32*C*b^4 + 64*A*a*b^3 + 48*A*a^3*b
+ 128*C*a*b^3 + 64*C*a^3*b)*1i)/2)*(3*A*a^2 + 4*A*b^2 + 4*C*a^2 + 8*C*b^2)*1i)/2))*(3*A*a^2 + 4*A*b^2 + 4*C*a
^2 + 8*C*b^2))/d

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