3.10.7 \(\int \frac {\cos ^4(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{a+b \sec (c+d x)} \, dx\) [907]

Optimal. Leaf size=276 \[ \frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d} \]

[Out]

1/8*(8*A*b^4-4*a^3*b*B-8*a*b^3*B+4*a^2*b^2*(A+2*C)+a^4*(3*A+4*C))*x/a^5-1/3*(3*A*b^3-2*a^3*B-3*a*b^2*B+a^2*b*(
2*A+3*C))*sin(d*x+c)/a^4/d+1/8*(4*A*b^2-4*a*b*B+a^2*(3*A+4*C))*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^3*(A*b^2-a*(B*b-C*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 = 0.78, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {4189, 4004, 3916, 2738, 214} \begin {gather*} -\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \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 {\sin (c+d x) \cos (c+d x) \left (a^2 (3 A+4 C)-4 a b B+4 A b^2\right )}{8 a^3 d}-\frac {\sin (c+d x) \left (-2 a^3 B+a^2 b (2 A+3 C)-3 a b^2 B+3 A b^3\right )}{3 a^4 d}+\frac {x \left (a^4 (3 A+4 C)-4 a^3 b B+4 a^2 b^2 (A+2 C)-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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x)/(8*a^5) - (2*b^3*(A*b^2 - a*(b*B
- a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - ((3*A*b^3 - 2*
a^3*B - 3*a*b^2*B + a^2*b*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*d) + ((4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*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 214

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

Rule 2738

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 3916

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

Rule 4004

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 4189

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) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{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)-a (3 A+4 C) \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 (4 b (A b-a B)+a^2 (3 A+4 C)\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 (4 A b^2-4 a b B+a^2 (3 A+4 C)\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 (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right )+a \left (4 A b^2-4 a b B-3 a^2 (3 A+4 C)\right ) \sec (c+d x)-3 b \left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{24 a^3}\\ &=-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\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 (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )+3 a b \left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{24 a^4}\\ &=\frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\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 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5}\\ &=\frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\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^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^5}\\ &=\frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\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^2 \left (A b^2-a (b B-a C)\right )\right ) \text {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 (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \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 (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\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 = 0.87, size = 235, normalized size = 0.85 \begin {gather*} \frac {12 \left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) (c+d x)+\frac {192 b^3 \left (A b^2+a (-b B+a C)\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+24 a \left (-4 A b^3+3 a^3 B+4 a b^2 B-a^2 b (3 A+4 C)\right ) \sin (c+d x)+24 a^2 \left (A b^2-a b B+a^2 (A+C)\right ) \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*(8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*(c + d*x) + (192*b^3*(A*b^2 + a*
(-(b*B) + a*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 24*a*(-4*A*b^3 + 3*a^3
*B + 4*a*b^2*B - a^2*b*(3*A + 4*C))*Sin[c + d*x] + 24*a^2*(A*b^2 - a*b*B + a^2*(A + C))*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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Maple [A]
time = 0.48, size = 456, normalized size = 1.65 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)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/a^5*(((-5/8*A*a^4-A*a^3*b-1/2*a^2*A*b^2-a*A*b^3+a^4*B+1/2*a^3*b*B+a^2*b^2*B-1/2*a^4*C-a^3*b*C)*tan(1/2*
d*x+1/2*c)^7+(3*a^2*b^2*B+1/2*a^3*b*B-3*a*A*b^3-3*a^3*b*C-5/3*A*a^3*b+3/8*A*a^4+5/3*a^4*B-1/2*a^4*C-1/2*a^2*A*
b^2)*tan(1/2*d*x+1/2*c)^5+(-3/8*A*a^4+1/2*a^2*A*b^2-1/2*a^3*b*B+1/2*a^4*C+3*a^2*b^2*B-3*a*A*b^3-3*a^3*b*C-5/3*
A*a^3*b+5/3*a^4*B)*tan(1/2*d*x+1/2*c)^3+(5/8*A*a^4+1/2*a^2*A*b^2-1/2*a^3*b*B+1/2*a^4*C-A*a^3*b-a*A*b^3+a^4*B+a
^2*b^2*B-a^3*b*C)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^4+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+4*C*a^4+8*C*a^2*b^2)*arctan(tan(1/2*d*x+1/2*c)))-2*b^3*(A*b^2-B*a*b+C*a^2)/a^5/((a+b)*(a-b))^(1/2)*arc
tanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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)+C*sec(d*x+c)^2)/(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 de

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Fricas [A]
time = 3.72, size = 765, normalized size = 2.77 \begin {gather*} \left [\frac {3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{6} - 4 \, B a^{5} b + {\left (A + 4 \, C\right )} a^{4} b^{2} - 4 \, B a^{3} b^{3} + 4 \, {\left (A - 2 \, C\right )} a^{2} b^{4} + 8 \, B a b^{5} - 8 \, A b^{6}\right )} d x + 12 \, {\left (C a^{2} b^{3} - 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} - 8 \, {\left (2 \, A + 3 \, C\right )} a^{5} b + 8 \, B a^{4} b^{2} - 8 \, {\left (A - 3 \, C\right )} 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 ({\left (3 \, A + 4 \, C\right )} a^{6} - 4 \, B a^{5} b + {\left (A - 4 \, C\right )} 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 ({\left (3 \, A + 4 \, C\right )} a^{6} - 4 \, B a^{5} b + {\left (A + 4 \, C\right )} a^{4} b^{2} - 4 \, B a^{3} b^{3} + 4 \, {\left (A - 2 \, C\right )} a^{2} b^{4} + 8 \, B a b^{5} - 8 \, A b^{6}\right )} d x - 24 \, {\left (C a^{2} b^{3} - 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} - 8 \, {\left (2 \, A + 3 \, C\right )} a^{5} b + 8 \, B a^{4} b^{2} - 8 \, {\left (A - 3 \, C\right )} 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 ({\left (3 \, A + 4 \, C\right )} a^{6} - 4 \, B a^{5} b + {\left (A - 4 \, C\right )} 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 ] \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)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

[1/24*(3*((3*A + 4*C)*a^6 - 4*B*a^5*b + (A + 4*C)*a^4*b^2 - 4*B*a^3*b^3 + 4*(A - 2*C)*a^2*b^4 + 8*B*a*b^5 - 8*
A*b^6)*d*x + 12*(C*a^2*b^3 - 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 - 8*(2*A + 3*C)*a^5*b + 8*B*a^4*b^2 - 8*(A - 3*C)*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 + 4*C)*a^6 - 4*B*a^5*b + (A - 4*C)*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 + 4*C)*a^6 - 4*B*a^5*b + (A + 4*C)*a^4*b^2 - 4*B*a^3*b^3 + 4*(A - 2*C)*a^2*b^4 + 8
*B*a*b^5 - 8*A*b^6)*d*x - 24*(C*a^2*b^3 - B*a*b^4 + A*b^5)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*
x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (16*B*a^6 - 8*(2*A + 3*C)*a^5*b + 8*B*a^4*b^2 - 8*(A - 3*C)*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 + 4*C)*a^6 - 4*B*a^5*b + (A - 4*C)*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \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)+C*sec(d*x+c)**2)/(a+b*sec(d*x+c)),x)

[Out]

Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos(c + d*x)**4/(a + b*sec(c + d*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (257) = 514\).
time = 0.49, size = 801, normalized size = 2.90 \begin {gather*} \frac {\frac {3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} - 4 \, B a^{3} b + 4 \, A a^{2} b^{2} + 8 \, C a^{2} b^{2} - 8 \, B a b^{3} + 8 \, A b^{4}\right )} {\left (d x + c\right )}}{a^{5}} - \frac {48 \, {\left (C a^{2} b^{3} - 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} + 12 \, C 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} + 24 \, C 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} + 12 \, C 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} + 72 \, C 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} - 12 \, C 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} + 72 \, C 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 ) - 12 \, C 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 ) + 24 \, C 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} \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)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

1/24*(3*(3*A*a^4 + 4*C*a^4 - 4*B*a^3*b + 4*A*a^2*b^2 + 8*C*a^2*b^2 - 8*B*a*b^3 + 8*A*b^4)*(d*x + c)/a^5 - 48*(
C*a^2*b^3 - B*a*b^4 + A*b^5)*(pi*floor(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 + 12*C*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 + 24*C*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*tan(1/
2*d*x + 1/2*c)^5 + 12*C*a^3*tan(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 + 72*C*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 - 12*C*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 +
72*C*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) - 12*C*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) + 24*C*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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Mupad [B]
time = 15.02, size = 2500, normalized size = 9.06 \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) + C/cos(c + d*x)^2))/(a + b/cos(c + d*x)),x)

[Out]

- ((tan(c/2 + (d*x)/2)^7*(5*A*a^3 + 8*A*b^3 - 8*B*a^3 + 4*C*a^3 + 4*A*a*b^2 + 8*A*a^2*b - 8*B*a*b^2 - 4*B*a^2*
b + 8*C*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*C*a^3 - 12*A*a*b^2 + 40*A*
a^2*b - 72*B*a*b^2 + 12*B*a^2*b + 72*C*a^2*b))/(12*a^4) + (tan(c/2 + (d*x)/2)^5*(72*A*b^3 - 9*A*a^3 - 40*B*a^3
+ 12*C*a^3 + 12*A*a*b^2 + 40*A*a^2*b - 72*B*a*b^2 - 12*B*a^2*b + 72*C*a^2*b))/(12*a^4) - (tan(c/2 + (d*x)/2)*
(5*A*a^3 - 8*A*b^3 + 8*B*a^3 + 4*C*a^3 + 4*A*a*b^2 - 8*A*a^2*b + 8*B*a*b^2 - 4*B*a^2*b - 8*C*a^2*b))/(4*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)) - (a
tan(((((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 48*C^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 - 128*C^2*a^4*b^7 + 256*C^2*a^5*b^6 - 2
56*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^11 + 256*A*B*a*b^10 - 24*A*B*a
^10*b - 72*A*C*a^10*b - 32*B*C*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 - 256*A*C*a^2*b^9 + 512*A*C*a^3*b^8 - 5
12*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^8*b^3 + 152*A*C*a^9*b^2 + 256
*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*b^4 - 224*B*C*a^8*b^3 + 96*B*
C*a^9*b^2))/(2*a^8) + (((12*A*a^16 + 16*C*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 + 32*C*a^12*b^4 - 48*C*a^13*b^3
+ 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*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^2*((A*b^2*1i)/2 + C*b^2*1i) + A*b^4*1i + a^4*((A*3i)/8 + (C*1i)/2) - B*a*b^3*1i - (B*a^
3*b*1i)/2))/(2*a^13))*(a^2*((A*b^2*1i)/2 + C*b^2*1i) + A*b^4*1i + a^4*((A*3i)/8 + (C*1i)/2) - B*a*b^3*1i - (B*
a^3*b*1i)/2))/a^5)*(a^2*((A*b^2*1i)/2 + C*b^2*1i) + A*b^4*1i + a^4*((A*3i)/8 + (C*1i)/2) - B*a*b^3*1i - (B*a^3
*b*1i)/2)*1i)/a^5 + (((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b^10 - 27*A^2*a
^10*b - 48*C^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 - 128*C^2*a^4*b^7 + 256*
C^2*a^5*b^6 - 256*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^11 + 256*A*B*a*
b^10 - 24*A*B*a^10*b - 72*A*C*a^10*b - 32*B*C*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 - 256*A*C*a^2*b^9 + 512*
A*C*a^3*b^8 - 512*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^8*b^3 + 152*A*
C*a^9*b^2 + 256*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*b^4 - 224*B*C*
a^8*b^3 + 96*B*C*a^9*b^2))/(2*a^8) - (((12*A*a^16 + 16*C*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 + 32*C*a^12*b^4
- 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*a^15*b)/a^12 + (tan(c/2 + (d*x)/2)*(128*a^1
2*b + 128*a^10*b^3 - 256*a^11*b^2)*(a^2*((A*b^2*1i)/2 + C*b^2*1i) + A*b^4*1i + a^4*((A*3i)/8 + (C*1i)/2) - B*a
*b^3*1i - (B*a^3*b*1i)/2))/(2*a^13))*(a^2*((A*b^2*1i)/2 + C*b^2*1i) + A*b^4*1i + a^4*((A*3i)/8 + (C*1i)/2) - B
*a*b^3*1i - (B*a^3*b*1i)/2))/a^5)*(a^2*((A*b^2*1i)/2 + C*b^2*1i) + A*b^4*1i + a^4*((A*3i)/8 + (C*1i)/2) - B*a*
b^3*1i - (B*a^3*b*1i)/2)*1i)/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^1
1 + 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 + 64*C^3*a^6*b^8 - 96*
C^3*a^7*b^7 + 96*C^3*a^8*b^6 - 80*C^3*a^9*b^5 + 32*C^3*a^10*b^4 - 16*C^3*a^11*b^3 - 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
+ 192*A*C^2*a^4*b^10 - 288*A*C^2*a^5*b^9 + 288*A*C^2*a^6*b^8 - 264*A*C^2*a^7*b^7 + 168*A*C^2*a^8*b^6 - 120*A*
C^2*a^9*b^5 + 48*A*C^2*a^10*b^4 - 24*A*C^2*a^11*b^3 + 192*A^2*C*a^2*b^12 - 288*A^2*C*a^3*b^11 + 288*A^2*C*a^4*
b^10 - 288*A^2*C*a^5*b^9 + 240*A^2*C*a^6*b^8 - ...

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