3.319 \(\int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx\)

Optimal. Leaf size=240 \[ -\frac {2 b^4 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d \sqrt {a-b} \sqrt {a+b}}-\frac {(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A-4 a b B+4 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {x \left (3 a^4 A-4 a^3 b B+4 a^2 A b^2-8 a b^3 B+8 A b^4\right )}{8 a^5}+\frac {A \sin (c+d x) \cos ^3(c+d x)}{4 a d} \]

[Out]

1/8*(3*A*a^4+4*A*a^2*b^2+8*A*b^4-4*B*a^3*b-8*B*a*b^3)*x/a^5-1/3*(2*a^2+3*b^2)*(A*b-B*a)*sin(d*x+c)/a^4/d+1/8*(
3*A*a^2+4*A*b^2-4*B*a*b)*cos(d*x+c)*sin(d*x+c)/a^3/d-1/3*(A*b-B*a)*cos(d*x+c)^2*sin(d*x+c)/a^2/d+1/4*A*cos(d*x
+c)^3*sin(d*x+c)/a/d-2*b^4*(A*b-B*a)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/d/(a-b)^(1/2)/(a+
b)^(1/2)

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Rubi [A]  time = 0.98, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4034, 4104, 3919, 3831, 2659, 208} \[ -\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A-4 a b B+4 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac {2 b^4 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d \sqrt {a-b} \sqrt {a+b}}+\frac {x \left (4 a^2 A b^2+3 a^4 A-4 a^3 b B-8 a b^3 B+8 A b^4\right )}{8 a^5}-\frac {(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac {A \sin (c+d x) \cos ^3(c+d x)}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B)*x)/(8*a^5) - (2*b^4*(A*b - a*B)*ArcTanh[(Sqrt[a - b
]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - ((2*a^2 + 3*b^2)*(A*b - a*B)*Sin[c + d*x])
/(3*a^4*d) + ((3*a^2*A + 4*A*b^2 - 4*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Cos[c + d*x]^2
*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3919

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

Rule 4034

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

Rule 4104

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

Rubi steps

\begin {align*} \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx &=\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {\cos ^3(c+d x) \left (4 (A b-a B)-3 a A \sec (c+d x)-3 A b \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a}\\ &=-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {\cos ^2(c+d x) \left (3 \left (3 a^2 A+4 b (A b-a B)\right )+a (A b+8 a B) \sec (c+d x)-8 b (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^2}\\ &=\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {\cos (c+d x) \left (8 \left (2 a^2+3 b^2\right ) (A b-a B)-a \left (9 a^2 A-4 A b^2+4 a b B\right ) \sec (c+d x)-3 b \left (3 a^2 A+4 A b^2-4 a b B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{24 a^3}\\ &=-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {3 \left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right )+3 a b \left (3 a^2 A+4 A b^2-4 a b B\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{24 a^4}\\ &=\frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (b^4 (A b-a B)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5}\\ &=\frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (b^3 (A b-a B)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^5}\\ &=\frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (2 b^3 (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}\\ &=\frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {2 b^4 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}\\ \end {align*}

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Mathematica [A]  time = 0.67, size = 202, normalized size = 0.84 \[ \frac {3 a^4 A \sin (4 (c+d x))+8 a^3 (a B-A b) \sin (3 (c+d x))+24 a^2 \left (a^2 A-a b B+A b^2\right ) \sin (2 (c+d x))+24 a \left (3 a^2+4 b^2\right ) (a B-A b) \sin (c+d x)+\frac {192 b^4 (A b-a B) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+12 (c+d x) \left (3 a^4 A-4 a^3 b B+4 a^2 A b^2-8 a b^3 B+8 A b^4\right )}{96 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*(3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B)*(c + d*x) + (192*b^4*(A*b - a*B)*ArcTanh[((-a +
b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 24*a*(3*a^2 + 4*b^2)*(-(A*b) + a*B)*Sin[c + d*x] + 24
*a^2*(a^2*A + A*b^2 - a*b*B)*Sin[2*(c + d*x)] + 8*a^3*(-(A*b) + a*B)*Sin[3*(c + d*x)] + 3*a^4*A*Sin[4*(c + d*x
)])/(96*a^5*d)

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fricas [A]  time = 0.55, size = 685, normalized size = 2.85 \[ \left [\frac {3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} - 4 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} + 8 \, B a b^{5} - 8 \, A b^{6}\right )} d x - 12 \, {\left (B a b^{4} - A b^{5}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + {\left (16 \, B a^{6} - 16 \, A a^{5} b + 8 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3} - 24 \, B a^{2} b^{4} + 24 \, A a b^{5} + 6 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (B a^{6} - A a^{5} b - B a^{4} b^{2} + A a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} + 4 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}, \frac {3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} - 4 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} + 8 \, B a b^{5} - 8 \, A b^{6}\right )} d x + 24 \, {\left (B a b^{4} - A b^{5}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (16 \, B a^{6} - 16 \, A a^{5} b + 8 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3} - 24 \, B a^{2} b^{4} + 24 \, A a b^{5} + 6 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (B a^{6} - A a^{5} b - B a^{4} b^{2} + A a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} + 4 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 - 4*B*a^3*b^3 + 4*A*a^2*b^4 + 8*B*a*b^5 - 8*A*b^6)*d*x - 12*(B*a*b^4
- A*b^5)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*
x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + (16*B*a^6 - 16*A*a^
5*b + 8*B*a^4*b^2 - 8*A*a^3*b^3 - 24*B*a^2*b^4 + 24*A*a*b^5 + 6*(A*a^6 - A*a^4*b^2)*cos(d*x + c)^3 + 8*(B*a^6
- A*a^5*b - B*a^4*b^2 + A*a^3*b^3)*cos(d*x + c)^2 + 3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 + 4*B*a^3*b^3 - 4*A*a^2
*b^4)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5*b^2)*d), 1/24*(3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 - 4*B*a^3*b^3
+ 4*A*a^2*b^4 + 8*B*a*b^5 - 8*A*b^6)*d*x + 24*(B*a*b^4 - A*b^5)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*c
os(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (16*B*a^6 - 16*A*a^5*b + 8*B*a^4*b^2 - 8*A*a^3*b^3 - 24*B*a^2*b
^4 + 24*A*a*b^5 + 6*(A*a^6 - A*a^4*b^2)*cos(d*x + c)^3 + 8*(B*a^6 - A*a^5*b - B*a^4*b^2 + A*a^3*b^3)*cos(d*x +
c)^2 + 3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 + 4*B*a^3*b^3 - 4*A*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^
5*b^2)*d)]

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giac [B]  time = 0.30, size = 642, normalized size = 2.68 \[ \frac {\frac {3 \, {\left (3 \, A a^{4} - 4 \, B a^{3} b + 4 \, A a^{2} b^{2} - 8 \, B a b^{3} + 8 \, A b^{4}\right )} {\left (d x + c\right )}}{a^{5}} + \frac {48 \, {\left (B a b^{4} - A b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{5}} - \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} + 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, 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} + 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, 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} + 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, 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 ) + 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, 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} a^{4}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(3*A*a^4 - 4*B*a^3*b + 4*A*a^2*b^2 - 8*B*a*b^3 + 8*A*b^4)*(d*x + c)/a^5 + 48*(B*a*b^4 - A*b^5)*(pi*flo
or(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a
^2 + b^2)))/(sqrt(-a^2 + b^2)*a^5) - 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 + 24
*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 12*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 24*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*ta
n(1/2*d*x + 1/2*c)^5 + 40*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 12*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 12*A*a*b^2*tan(
1/2*d*x + 1/2*c)^5 - 72*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 + 40*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 12*B*a^2*b*tan(1/2*d*x + 1
/2*c)^3 - 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 72*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) + 24*A*a^2*b*tan(1/2*d*x + 1/2*c) + 12*B*a
^2*b*tan(1/2*d*x + 1/2*c) - 12*A*a*b^2*tan(1/2*d*x + 1/2*c) - 24*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*a^4))/d

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maple [B]  time = 1.18, size = 1212, normalized size = 5.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*b^2*B+10/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/
2*c)^3*B+5/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*A-2/d/a^4*arctan(tan(1/2*d*x+1/2*c))*B*b^3+2/d/
a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*B+3/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*A+
2/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*B+1/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*
B*b+2/d*b^4/a^4/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+3/4/d*A/a*arctan(t
an(1/2*d*x+1/2*c))-2/d*b^5/a^5/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+10/
3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*B-3/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^
3*A+2/d/a^5*arctan(tan(1/2*d*x+1/2*c))*A*b^4+1/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*A*b^2-2/d/a
^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*A*b-10/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^
5*A*b-6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*A*b^3+2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2
*d*x+1/2*c)^7*b^2*B-1/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*A*b^2-2/d/a^4/(1+tan(1/2*d*x+1/2*c
)^2)^4*tan(1/2*d*x+1/2*c)^7*A*b^3+1/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*A*b^2-1/d/a^2/(1+tan
(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*B*b-2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*A*b^3+6/d/a^
3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*b^2*B-10/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*
c)^3*A*b-1/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*A*b^2-6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(
1/2*d*x+1/2*c)^3*A*b^3+1/d/a^3*arctan(tan(1/2*d*x+1/2*c))*A*b^2-1/d/a^2*arctan(tan(1/2*d*x+1/2*c))*B*b+6/d/a^3
/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*b^2*B-1/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3
*B*b+1/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*B*b-2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*
x+1/2*c)^7*A*b-5/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*A

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 8.64, size = 5903, normalized size = 24.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(5*A*a^3 - 8*A*b^3 + 8*B*a^3 + 4*A*a*b^2 - 8*A*a^2*b + 8*B*a*b^2 - 4*B*a^2*b))/(4*a^4) -
(tan(c/2 + (d*x)/2)^7*(5*A*a^3 + 8*A*b^3 - 8*B*a^3 + 4*A*a*b^2 + 8*A*a^2*b - 8*B*a*b^2 - 4*B*a^2*b))/(4*a^4) -
(tan(c/2 + (d*x)/2)^3*(9*A*a^3 + 72*A*b^3 - 40*B*a^3 - 12*A*a*b^2 + 40*A*a^2*b - 72*B*a*b^2 + 12*B*a^2*b))/(1
2*a^4) + (tan(c/2 + (d*x)/2)^5*(9*A*a^3 - 72*A*b^3 + 40*B*a^3 - 12*A*a*b^2 - 40*A*a^2*b + 72*B*a*b^2 + 12*B*a^
2*b))/(12*a^4))/(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)) - (atan(((((((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*
b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (tan(c
/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*
a^3*b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5) + (tan(c/2 + (d*
x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^
2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^
2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b
^3 + 16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 4
64*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A
*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i)*1i)/(8*a^5) - (((((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5
+ 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2
- 12*A*a^15*b - 16*B*a^15*b)/a^12 + (tan(c/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(A*a^4*3i
+ A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3
*8i - B*a^3*b*4i))/(8*a^5) - (tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b -
256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 8
1*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B
^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*
b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^
3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i)*1i)/(8*a^5))/((64
*A^3*b^14 - 96*A^3*a*b^13 + 96*A^3*a^2*b^12 - 104*A^3*a^3*b^11 + 104*A^3*a^4*b^10 - 88*A^3*a^5*b^9 + 48*A^3*a^
6*b^8 - 33*A^3*a^7*b^7 + 18*A^3*a^8*b^6 - 9*A^3*a^9*b^5 - 64*B^3*a^3*b^11 + 96*B^3*a^4*b^10 - 96*B^3*a^5*b^9 +
80*B^3*a^6*b^8 - 32*B^3*a^7*b^7 + 16*B^3*a^8*b^6 - 192*A^2*B*a*b^13 + 192*A*B^2*a^2*b^12 - 288*A*B^2*a^3*b^11
+ 288*A*B^2*a^4*b^10 - 264*A*B^2*a^5*b^9 + 168*A*B^2*a^6*b^8 - 120*A*B^2*a^7*b^7 + 48*A*B^2*a^8*b^6 - 24*A*B^
2*a^9*b^5 + 288*A^2*B*a^2*b^12 - 288*A^2*B*a^3*b^11 + 288*A^2*B*a^4*b^10 - 240*A^2*B*a^5*b^9 + 192*A^2*B*a^6*b
^8 - 96*A^2*B*a^7*b^7 + 57*A^2*B*a^8*b^6 - 18*A^2*B*a^9*b^5 + 9*A^2*B*a^10*b^4)/a^12 + (((((12*A*a^16 + 32*A*a
^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a
^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (tan(c/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 2
56*a^11*b^2)*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i +
A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5) + (tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a
*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^
5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 +
256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*
B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a
^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*
b*4i))/(8*a^5) + (((((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2
- 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 + (tan(c/2 +
(d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*
b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5) - (tan(c/2 + (d*x)/2
)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^
4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^
9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 +
16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A
*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A*b^4
*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5)))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*
a^3*b*4i)*1i)/(4*a^5*d) - (b^4*atan(((b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11
- 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A
^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a
^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b
^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 -
368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) + (b^4*((a + b)*(a - b))^(1/2)*
(A*b - B*a)*((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a
^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (b^4*tan(c/2 + (d*
x)/2)*((a + b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))
)/(a^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2) + (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*
a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 2
56*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B
^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a
^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^
6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) - (b^4*((a + b)*(a - b))^(1
/2)*(A*b - B*a)*((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32
*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 + (b^4*tan(c/2 +
(d*x)/2)*((a + b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^
2))))/(a^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2))/((64*A^3*b^14 - 96*A^3*a*b^13 + 96*A^3*a^2*b^12 - 104*A^3*a^3*b^11
+ 104*A^3*a^4*b^10 - 88*A^3*a^5*b^9 + 48*A^3*a^6*b^8 - 33*A^3*a^7*b^7 + 18*A^3*a^8*b^6 - 9*A^3*a^9*b^5 - 64*B
^3*a^3*b^11 + 96*B^3*a^4*b^10 - 96*B^3*a^5*b^9 + 80*B^3*a^6*b^8 - 32*B^3*a^7*b^7 + 16*B^3*a^8*b^6 - 192*A^2*B*
a*b^13 + 192*A*B^2*a^2*b^12 - 288*A*B^2*a^3*b^11 + 288*A*B^2*a^4*b^10 - 264*A*B^2*a^5*b^9 + 168*A*B^2*a^6*b^8
- 120*A*B^2*a^7*b^7 + 48*A*B^2*a^8*b^6 - 24*A*B^2*a^9*b^5 + 288*A^2*B*a^2*b^12 - 288*A^2*B*a^3*b^11 + 288*A^2*
B*a^4*b^10 - 240*A^2*B*a^5*b^9 + 192*A^2*B*a^6*b^8 - 96*A^2*B*a^7*b^7 + 57*A^2*B*a^8*b^6 - 18*A^2*B*a^9*b^5 +
9*A^2*B*a^10*b^4)/a^12 + (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b
^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 -
216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256
*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*B
*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*
b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) + (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*
((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48
*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (b^4*tan(c/2 + (d*x)/2)*((a +
b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2))))/(a^7 - a^5
*b^2)))/(a^7 - a^5*b^2) - (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*
b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6
- 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 25
6*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*
B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6
*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) - (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)
*((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 4
8*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 + (b^4*tan(c/2 + (d*x)/2)*((a +
b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2))))/(a^7 - a^
5*b^2)))/(a^7 - a^5*b^2)))*((a + b)*(a - b))^(1/2)*(A*b - B*a)*2i)/(d*(a^7 - a^5*b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*sec(c + d*x))*cos(c + d*x)**4/(a + b*sec(c + d*x)), x)

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