3.9.84 \(\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [884]

Optimal. Leaf size=223 \[ \frac {1}{8} \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) x+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \sin (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(3 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d} \]

[Out]

1/8*(12*a^2*b*B+8*b^3*B+12*a*b^2*(A+2*C)+a^3*(3*A+4*C))*x+b^3*C*arctanh(sin(d*x+c))/d+1/6*(3*A*b^3+4*a^3*B+16*
a*b^2*B+6*a^2*b*(2*A+3*C))*sin(d*x+c)/d+1/24*a*(6*A*b^2+20*a*b*B+3*a^2*(3*A+4*C))*cos(d*x+c)*sin(d*x+c)/d+1/12
*(3*A*b+4*B*a)*cos(d*x+c)^2*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/4*A*cos(d*x+c)^3*(a+b*sec(d*x+c))^3*sin(d*x+c)/d

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Rubi [A]
time = 0.45, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4179, 4159, 4132, 8, 4130, 3855} \begin {gather*} \frac {a \sin (c+d x) \cos (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{24 d}+\frac {\sin (c+d x) \left (4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right )}{6 d}+\frac {1}{8} x \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right )+\frac {(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*x)/8 + (b^3*C*ArcTanh[Sin[c + d*x]])/d + ((3*A*
b^3 + 4*a^3*B + 16*a*b^2*B + 6*a^2*b*(2*A + 3*C))*Sin[c + d*x])/(6*d) + (a*(6*A*b^2 + 20*a*b*B + 3*a^2*(3*A +
4*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((3*A*b + 4*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])
/(12*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*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]

Rubi steps

\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (3 A b+4 a B+(3 a A+4 b B+4 a C) \sec (c+d x)+4 b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(3 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)+\left (15 a A b+8 a^2 B+12 b^2 B+24 a b C\right ) \sec (c+d x)+12 b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(3 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{24} \int \cos (c+d x) \left (-4 \left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right )-3 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) \sec (c+d x)-24 b^3 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(3 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{24} \int \cos (c+d x) \left (-4 \left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right )-24 b^3 C \sec ^2(c+d x)\right ) \, dx-\frac {1}{8} \left (-12 a^2 b B-8 b^3 B-12 a b^2 (A+2 C)-a^3 (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) x+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \sin (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(3 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\left (b^3 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) x+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \sin (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(3 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.98, size = 215, normalized size = 0.96 \begin {gather*} \frac {12 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) (c+d x)-96 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \left (4 A b^3+3 a^3 B+12 a b^2 B+3 a^2 b (3 A+4 C)\right ) \sin (c+d x)+24 a \left (3 A b^2+3 a b B+a^2 (A+C)\right ) \sin (2 (c+d x))+8 a^2 (3 A b+a B) \sin (3 (c+d x))+3 a^3 A \sin (4 (c+d x))}{96 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(12*(12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*(c + d*x) - 96*b^3*C*Log[Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]] + 96*b^3*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(4*A*b^3 + 3*a^3*B + 12*a*b^2*B + 3
*a^2*b*(3*A + 4*C))*Sin[c + d*x] + 24*a*(3*A*b^2 + 3*a*b*B + a^2*(A + C))*Sin[2*(c + d*x)] + 8*a^2*(3*A*b + a*
B)*Sin[3*(c + d*x)] + 3*a^3*A*Sin[4*(c + d*x)])/(96*d)

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Maple [A]
time = 0.15, size = 251, normalized size = 1.13 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 246, normalized size = 1.10 \begin {gather*} \frac {3 \, {\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} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 288 \, {\left (d x + c\right )} C a b^{2} + 96 \, {\left (d x + c\right )} B b^{3} + 48 \, C b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 288 \, C a^{2} b \sin \left (d x + c\right ) + 288 \, B a b^{2} \sin \left (d x + c\right ) + 96 \, A b^{3} \sin \left (d x + c\right )}{96 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 - 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B
*a^3 + 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3 - 96*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b + 72*(2*d*x +
2*c + sin(2*d*x + 2*c))*B*a^2*b + 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a*b^2 + 288*(d*x + c)*C*a*b^2 + 96*(d*
x + c)*B*b^3 + 48*C*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 288*C*a^2*b*sin(d*x + c) + 288*B*a*b
^2*sin(d*x + c) + 96*A*b^3*sin(d*x + c))/d

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Fricas [A]
time = 2.82, size = 189, normalized size = 0.85 \begin {gather*} \frac {12 \, C b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, C b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, {\left (A + 2 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} d x + {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 16 \, B a^{3} + 24 \, {\left (2 \, A + 3 \, C\right )} a^{2} b + 72 \, B a b^{2} + 24 \, A b^{3} + 8 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/24*(12*C*b^3*log(sin(d*x + c) + 1) - 12*C*b^3*log(-sin(d*x + c) + 1) + 3*((3*A + 4*C)*a^3 + 12*B*a^2*b + 12*
(A + 2*C)*a*b^2 + 8*B*b^3)*d*x + (6*A*a^3*cos(d*x + c)^3 + 16*B*a^3 + 24*(2*A + 3*C)*a^2*b + 72*B*a*b^2 + 24*A
*b^3 + 8*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^2 + 3*((3*A + 4*C)*a^3 + 12*B*a^2*b + 12*A*a*b^2)*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)**4*(a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 723 vs. \(2 (213) = 426\).
time = 0.56, size = 723, normalized size = 3.24 \begin {gather*} \frac {24 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (3 \, A a^{3} + 4 \, C a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2} + 24 \, C a b^{2} + 8 \, B b^{3}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 120 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 216 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 216 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 216 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 216 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/24*(24*C*b^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 24*C*b^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 3*(3*A*a^3 +
4*C*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 24*C*a*b^2 + 8*B*b^3)*(d*x + c) - 2*(15*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 24
*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 72*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 36*B*a^2
*b*tan(1/2*d*x + 1/2*c)^7 - 72*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 72*B*a*b^2
*tan(1/2*d*x + 1/2*c)^7 - 24*A*b^3*tan(1/2*d*x + 1/2*c)^7 - 9*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 40*B*a^3*tan(1/2*
d*x + 1/2*c)^5 + 12*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 120*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^2*b*tan(1/2*d*x
+ 1/2*c)^5 - 216*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 216*B*a*b^2*tan(1/2*d*x
+ 1/2*c)^5 - 72*A*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 40*B*a^3*tan(1/2*d*x + 1/2*c)
^3 - 12*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 120*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^3
- 216*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 216*B*a*b^2*tan(1/2*d*x + 1/2*c)^3
- 72*A*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*A*a^3*tan(1/2*d*x + 1/2*c) - 24*B*a^3*tan(1/2*d*x + 1/2*c) - 12*C*a^3*t
an(1/2*d*x + 1/2*c) - 72*A*a^2*b*tan(1/2*d*x + 1/2*c) - 36*B*a^2*b*tan(1/2*d*x + 1/2*c) - 72*C*a^2*b*tan(1/2*d
*x + 1/2*c) - 36*A*a*b^2*tan(1/2*d*x + 1/2*c) - 72*B*a*b^2*tan(1/2*d*x + 1/2*c) - 24*A*b^3*tan(1/2*d*x + 1/2*c
))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d

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Mupad [B]
time = 7.31, size = 2500, normalized size = 11.21 \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)^4*(a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(atan(((((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i)*(12*A*a^3 + 32
*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) + tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*
B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2
*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*
a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a
*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i)*1i - (((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*
a^2*b*3i)/2 + C*a*b^2*3i)*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) -
tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 9
6*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b +
192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((A*a^
3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i)*1i)/((((A*a^3*3i)/8 + B*b^3*
1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i)*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 +
48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) + tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b
^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12
*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 12
0*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 +
C*a*b^2*3i) + (((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2 + (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i)*(12*A*a
^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) - tan(c/2 + (d*x)/2)*((9*A^2*a^6)/
2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 +
288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 1
68*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((A*a^3*3i)/8 + B*b^3*1i + (C*a^3*1i)/2
+ (A*a*b^2*3i)/2 + (B*a^2*b*3i)/2 + C*a*b^2*3i) - 64*B*C^2*b^9 + 64*B^2*C*b^9 - 192*C^3*a*b^8 + 576*C^3*a^2*b
^7 - 32*C^3*a^3*b^6 + 192*C^3*a^4*b^5 + 16*C^3*a^6*b^3 - 96*A*C^2*a*b^8 + 384*B*C^2*a*b^8 + 576*A*C^2*a^2*b^7
- 24*A*C^2*a^3*b^6 + 240*A*C^2*a^4*b^5 + 24*A*C^2*a^6*b^3 + 144*A^2*C*a^2*b^7 + 72*A^2*C*a^4*b^5 + 9*A^2*C*a^6
*b^3 - 96*B*C^2*a^2*b^7 + 640*B*C^2*a^3*b^6 + 96*B*C^2*a^5*b^4 + 192*B^2*C*a^2*b^7 + 144*B^2*C*a^4*b^5 + 192*A
*B*C*a*b^8 + 336*A*B*C*a^3*b^6 + 72*A*B*C*a^5*b^4))*((3*A*a^3)/4 + 2*B*b^3 + C*a^3 + 3*A*a*b^2 + 3*B*a^2*b + 6
*C*a*b^2))/d + (tan(c/2 + (d*x)/2)^7*(2*A*b^3 - (5*A*a^3)/4 + 2*B*a^3 - C*a^3 - 3*A*a*b^2 + 6*A*a^2*b + 6*B*a*
b^2 - 3*B*a^2*b + 6*C*a^2*b) + tan(c/2 + (d*x)/2)^3*(6*A*b^3 - (3*A*a^3)/4 + (10*B*a^3)/3 + C*a^3 + 3*A*a*b^2
+ 10*A*a^2*b + 18*B*a*b^2 + 3*B*a^2*b + 18*C*a^2*b) + tan(c/2 + (d*x)/2)^5*((3*A*a^3)/4 + 6*A*b^3 + (10*B*a^3)
/3 - C*a^3 - 3*A*a*b^2 + 10*A*a^2*b + 18*B*a*b^2 - 3*B*a^2*b + 18*C*a^2*b) + tan(c/2 + (d*x)/2)*((5*A*a^3)/4 +
2*A*b^3 + 2*B*a^3 + C*a^3 + 3*A*a*b^2 + 6*A*a^2*b + 6*B*a*b^2 + 3*B*a^2*b + 6*C*a^2*b))/(d*(4*tan(c/2 + (d*x)
/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) - (C*b^3*atan((C*b^3*(tan
(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^
2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192
*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3) + C*b^3*(12
*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2))*1i + C*b^3*(tan(c/2 + (d*x)/2
)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72
*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 4
8*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3) - C*b^3*(12*A*a^3 + 32*B*
b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2))*1i)/(64*B^2*C*b^9 - 64*B*C^2*b^9 - 192*C^3*
a*b^8 + C*b^3*(tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A
^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 +
36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^
3*b^3) + C*b^3*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2)) - C*b^3*(ta
n(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B
^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 +...

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