3.5.45 \(\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [445]

Optimal. Leaf size=225 \[ \frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \]

[Out]

1/8*a^4*(28*A+35*B+48*C)*x+a^4*C*arctanh(sin(d*x+c))/d+1/8*a^4*(28*A+35*B+40*C)*sin(d*x+c)/d+1/20*a*(4*A+5*B)*
cos(d*x+c)^3*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+1/5*A*cos(d*x+c)^4*(a+a*sec(d*x+c))^4*sin(d*x+c)/d+1/60*(28*A+35*
B+20*C)*cos(d*x+c)^2*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/d+1/24*(28*A+35*B+32*C)*cos(d*x+c)*(a^4+a^4*sec(d*x+c))
*sin(d*x+c)/d

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Rubi [A]
time = 0.40, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {4171, 4102, 4081, 3855} \begin {gather*} \frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {(28 A+35 B+32 C) \sin (c+d x) \cos (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (28 A+35 B+48 C)+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(28 A+35 B+20 C) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{60 d}+\frac {a (4 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(28*A + 35*B + 48*C)*x)/8 + (a^4*C*ArcTanh[Sin[c + d*x]])/d + (a^4*(28*A + 35*B + 40*C)*Sin[c + d*x])/(8*
d) + (a*(4*A + 5*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[
c + d*x])^4*Sin[c + d*x])/(5*d) + ((28*A + 35*B + 20*C)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x]
)/(60*d) + ((28*A + 35*B + 32*C)*Cos[c + d*x]*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)

Rule 3855

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

Rule 4081

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(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) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B},
x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]

Rule 4102

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

Rule 4171

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

Rubi steps

\begin {align*} \int \cos ^5(c+d x) (a+a \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+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (a (4 A+5 B)+5 a C \sec (c+d x)) \, dx}{5 a}\\ &=\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (a^2 (28 A+35 B+20 C)+20 a^2 C \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left (5 a^3 (28 A+35 B+32 C)+60 a^3 C \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^4 (28 A+35 B+40 C)+120 a^4 C \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}-\frac {\int \left (-15 a^5 (28 A+35 B+48 C)-120 a^5 C \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}

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Mathematica [A]
time = 0.66, size = 182, normalized size = 0.81 \begin {gather*} \frac {a^4 \left (1680 A d x+2100 B d x+2880 C d x-480 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 (49 A+56 B+54 C) \sin (c+d x)+120 (8 A+7 B+4 C) \sin (2 (c+d x))+290 A \sin (3 (c+d x))+160 B \sin (3 (c+d x))+40 C \sin (3 (c+d x))+60 A \sin (4 (c+d x))+15 B \sin (4 (c+d x))+6 A \sin (5 (c+d x))\right )}{480 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(1680*A*d*x + 2100*B*d*x + 2880*C*d*x - 480*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*C*Log[Cos[(c
+ d*x)/2] + Sin[(c + d*x)/2]] + 60*(49*A + 56*B + 54*C)*Sin[c + d*x] + 120*(8*A + 7*B + 4*C)*Sin[2*(c + d*x)]
+ 290*A*Sin[3*(c + d*x)] + 160*B*Sin[3*(c + d*x)] + 40*C*Sin[3*(c + d*x)] + 60*A*Sin[4*(c + d*x)] + 15*B*Sin[
4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)

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Maple [A]
time = 0.96, size = 340, normalized size = 1.51

method	result	size
derivativedivides	\(\frac {A \,a^{4} \sin \left (d x +c \right )+a^{4} B \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} B \sin \left (d x +c \right )+4 a^{4} C \left (d x +c \right )+2 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \sin \left (d x +c \right )+4 A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(340\)
default	\(\frac {A \,a^{4} \sin \left (d x +c \right )+a^{4} B \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} B \sin \left (d x +c \right )+4 a^{4} C \left (d x +c \right )+2 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \sin \left (d x +c \right )+4 A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(340\)
risch	\(\frac {7 a^{4} A x}{2}+\frac {35 a^{4} x B}{8}+6 a^{4} x C -\frac {27 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{8 d}-\frac {49 i A \,a^{4} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} B}{2 d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} a^{4} B}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{8 d}+\frac {49 i A \,a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {A \,a^{4} \sin \left (5 d x +5 c \right )}{80 d}+\frac {A \,a^{4} \sin \left (4 d x +4 c \right )}{8 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} B}{32 d}+\frac {29 A \,a^{4} \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} B}{3 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{12 d}+\frac {2 \sin \left (2 d x +2 c \right ) A \,a^{4}}{d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} B}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} C}{d}\)	\(341\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 332, normalized size = 1.48 \begin {gather*} \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} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A 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^{4} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B 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} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 480 \, {\left (d x + c\right )} B a^{4} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1920 \, {\left (d x + c\right )} C a^{4} + 240 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{4} \sin \left (d x + c\right ) + 1920 \, B a^{4} \sin \left (d x + c\right ) + 2880 \, C a^{4} \sin \left (d x + c\right )}{480 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]
time = 3.01, size = 154, normalized size = 0.68 \begin {gather*} \frac {15 \, {\left (28 \, A + 35 \, B + 48 \, C\right )} a^{4} d x + 60 \, C a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, C a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (34 \, A + 20 \, B + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (28 \, A + 27 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (83 \, A + 100 \, B + 100 \, C\right )} a^{4}\right )} \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+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/120*(15*(28*A + 35*B + 48*C)*a^4*d*x + 60*C*a^4*log(sin(d*x + c) + 1) - 60*C*a^4*log(-sin(d*x + c) + 1) + (2
4*A*a^4*cos(d*x + c)^4 + 30*(4*A + B)*a^4*cos(d*x + c)^3 + 8*(34*A + 20*B + 5*C)*a^4*cos(d*x + c)^2 + 15*(28*A
+ 27*B + 16*C)*a^4*cos(d*x + c) + 8*(83*A + 100*B + 100*C)*a^4)*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+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

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

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Giac [A]
time = 0.52, size = 337, normalized size = 1.50 \begin {gather*} \frac {120 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 120 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (28 \, A a^{4} + 35 \, B a^{4} + 48 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (420 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 600 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1960 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2450 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3584 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3160 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3950 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3680 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1500 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1080 \, C a^{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}}}{120 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*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*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 120*C*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 15*(28*A*
a^4 + 35*B*a^4 + 48*C*a^4)*(d*x + c) + 2*(420*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 525*B*a^4*tan(1/2*d*x + 1/2*c)^9
+ 600*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 1960*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 2450*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 2
720*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 3584*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 4480*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 472
0*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 3160*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 3950*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 3680*
C*a^4*tan(1/2*d*x + 1/2*c)^3 + 1500*A*a^4*tan(1/2*d*x + 1/2*c) + 1395*B*a^4*tan(1/2*d*x + 1/2*c) + 1080*C*a^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 = 4.45, size = 1151, normalized size = 5.12 \begin {gather*} \frac {\left (7\,A\,a^4+\frac {35\,B\,a^4}{4}+10\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {98\,A\,a^4}{3}+\frac {245\,B\,a^4}{6}+\frac {136\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {896\,A\,a^4}{15}+\frac {224\,B\,a^4}{3}+\frac {236\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {158\,A\,a^4}{3}+\frac {395\,B\,a^4}{6}+\frac {184\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {93\,B\,a^4}{4}+18\,C\,a^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 )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {C\,a^4\,\mathrm {atan}\left (\frac {C\,a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )+C\,a^4\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\right )\,1{}\mathrm {i}+C\,a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )-C\,a^4\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\right )\,1{}\mathrm {i}}{1920\,C^3\,a^{12}+2464\,A\,C^2\,a^{12}+784\,A^2\,C\,a^{12}+3080\,B\,C^2\,a^{12}+1225\,B^2\,C\,a^{12}+C\,a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )+C\,a^4\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\right )-C\,a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )-C\,a^4\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\right )+1960\,A\,B\,C\,a^{12}}\right )\,2{}\mathrm {i}}{d}-\frac {a^4\,\mathrm {atan}\left (\frac {\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )-\frac {a^4\,\left (28\,A+35\,B+48\,C\right )\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (28\,A+35\,B+48\,C\right )}{8}+\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )+\frac {a^4\,\left (28\,A+35\,B+48\,C\right )\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (28\,A+35\,B+48\,C\right )}{8}}{1920\,C^3\,a^{12}+2464\,A\,C^2\,a^{12}+784\,A^2\,C\,a^{12}+3080\,B\,C^2\,a^{12}+1225\,B^2\,C\,a^{12}+1960\,A\,B\,C\,a^{12}-\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )-\frac {a^4\,\left (28\,A+35\,B+48\,C\right )\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (28\,A+35\,B+48\,C\right )\,1{}\mathrm {i}}{8}+\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A^2\,a^8+980\,A\,B\,a^8+1344\,A\,C\,a^8+\frac {1225\,B^2\,a^8}{2}+1680\,B\,C\,a^8+1184\,C^2\,a^8\right )+\frac {a^4\,\left (28\,A+35\,B+48\,C\right )\,\left (112\,A\,a^4+140\,B\,a^4+224\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (28\,A+35\,B+48\,C\right )\,1{}\mathrm {i}}{8}}\right )\,\left (28\,A+35\,B+48\,C\right )}{4\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)^9*(7*A*a^4 + (35*B*a^4)/4 + 10*C*a^4) + tan(c/2 + (d*x)/2)^7*((98*A*a^4)/3 + (245*B*a^4)/6
+ (136*C*a^4)/3) + tan(c/2 + (d*x)/2)^3*((158*A*a^4)/3 + (395*B*a^4)/6 + (184*C*a^4)/3) + tan(c/2 + (d*x)/2)^
5*((896*A*a^4)/15 + (224*B*a^4)/3 + (236*C*a^4)/3) + tan(c/2 + (d*x)/2)*(25*A*a^4 + (93*B*a^4)/4 + 18*C*a^4))/
(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*a^4*atan((C*a^4*(tan(c/2 + (d*x)/2)*(392*A^2*a^8 + (1225*B^2*a^8)/2 + 1184*C^2*a^
8 + 980*A*B*a^8 + 1344*A*C*a^8 + 1680*B*C*a^8) + C*a^4*(112*A*a^4 + 140*B*a^4 + 224*C*a^4))*1i + C*a^4*(tan(c/
2 + (d*x)/2)*(392*A^2*a^8 + (1225*B^2*a^8)/2 + 1184*C^2*a^8 + 980*A*B*a^8 + 1344*A*C*a^8 + 1680*B*C*a^8) - C*a
^4*(112*A*a^4 + 140*B*a^4 + 224*C*a^4))*1i)/(1920*C^3*a^12 + 2464*A*C^2*a^12 + 784*A^2*C*a^12 + 3080*B*C^2*a^1
2 + 1225*B^2*C*a^12 + C*a^4*(tan(c/2 + (d*x)/2)*(392*A^2*a^8 + (1225*B^2*a^8)/2 + 1184*C^2*a^8 + 980*A*B*a^8 +
1344*A*C*a^8 + 1680*B*C*a^8) + C*a^4*(112*A*a^4 + 140*B*a^4 + 224*C*a^4)) - C*a^4*(tan(c/2 + (d*x)/2)*(392*A^
2*a^8 + (1225*B^2*a^8)/2 + 1184*C^2*a^8 + 980*A*B*a^8 + 1344*A*C*a^8 + 1680*B*C*a^8) - C*a^4*(112*A*a^4 + 140*
B*a^4 + 224*C*a^4)) + 1960*A*B*C*a^12))*2i)/d - (a^4*atan(((a^4*(tan(c/2 + (d*x)/2)*(392*A^2*a^8 + (1225*B^2*a
^8)/2 + 1184*C^2*a^8 + 980*A*B*a^8 + 1344*A*C*a^8 + 1680*B*C*a^8) - (a^4*(28*A + 35*B + 48*C)*(112*A*a^4 + 140
*B*a^4 + 224*C*a^4)*1i)/8)*(28*A + 35*B + 48*C))/8 + (a^4*(tan(c/2 + (d*x)/2)*(392*A^2*a^8 + (1225*B^2*a^8)/2
+ 1184*C^2*a^8 + 980*A*B*a^8 + 1344*A*C*a^8 + 1680*B*C*a^8) + (a^4*(28*A + 35*B + 48*C)*(112*A*a^4 + 140*B*a^4
+ 224*C*a^4)*1i)/8)*(28*A + 35*B + 48*C))/8)/(1920*C^3*a^12 + 2464*A*C^2*a^12 + 784*A^2*C*a^12 + 3080*B*C^2*a
^12 + 1225*B^2*C*a^12 - (a^4*(tan(c/2 + (d*x)/2)*(392*A^2*a^8 + (1225*B^2*a^8)/2 + 1184*C^2*a^8 + 980*A*B*a^8
+ 1344*A*C*a^8 + 1680*B*C*a^8) - (a^4*(28*A + 35*B + 48*C)*(112*A*a^4 + 140*B*a^4 + 224*C*a^4)*1i)/8)*(28*A +
35*B + 48*C)*1i)/8 + (a^4*(tan(c/2 + (d*x)/2)*(392*A^2*a^8 + (1225*B^2*a^8)/2 + 1184*C^2*a^8 + 980*A*B*a^8 + 1
344*A*C*a^8 + 1680*B*C*a^8) + (a^4*(28*A + 35*B + 48*C)*(112*A*a^4 + 140*B*a^4 + 224*C*a^4)*1i)/8)*(28*A + 35*
B + 48*C)*1i)/8 + 1960*A*B*C*a^12))*(28*A + 35*B + 48*C))/(4*d)

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