3.992 \(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx\)
Optimal. Leaf size=307 \[ -\frac {\tan (c+d x) \sec (c+d x) \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right )}{2 a^2 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\left (a^2 (A+2 C)-4 a b B+6 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac {\tan (c+d x) \left (a^3 B-a^2 b (2 A-C)-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}-\frac {2 b \left (2 a^4 C-3 a^3 b B+4 a^2 A b^2-a^2 b^2 C+2 a b^3 B-3 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}} \]
[Out]
-2*b*(4*A*a^2*b^2-3*A*b^4-3*B*a^3*b+2*B*a*b^3+2*C*a^4-C*a^2*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(
1/2))/a^4/(a-b)^(3/2)/(a+b)^(3/2)/d+1/2*(6*A*b^2-4*a*b*B+a^2*(A+2*C))*arctanh(sin(d*x+c))/a^4/d+(3*A*b^3+a^3*B
-2*a*b^2*B-a^2*b*(2*A-C))*tan(d*x+c)/a^3/(a^2-b^2)/d-1/2*(3*A*b^2-2*a*b*B-a^2*(A-2*C))*sec(d*x+c)*tan(d*x+c)/a
^2/(a^2-b^2)/d+(A*b^2-a*(B*b-C*a))*sec(d*x+c)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))
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Rubi [A] time = 1.40, antiderivative size = 307, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used =
{3055, 3001, 3770, 2659, 205} \[ -\frac {2 b \left (4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 C+2 a b^3 B-3 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\tan (c+d x) \left (-a^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}+\frac {\left (a^2 (A+2 C)-4 a b B+6 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {\tan (c+d x) \sec (c+d x) \left (a^2 (-(A-2 C))-2 a b B+3 A b^2\right )}{2 a^2 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]
[Out]
(-2*b*(4*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 2*a*b^3*B + 2*a^4*C - a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/
2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d
*x]])/(2*a^4*d) + ((3*A*b^3 + a^3*B - 2*a*b^2*B - a^2*b*(2*A - C))*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b
^2 - 2*a*b*B - a^2*(A - 2*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[
c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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 3001
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rule 3055
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-3 A b^2+2 a b B+a^2 (A-2 C)-a (A b-a B+b C) \cos (c+d x)+2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (2 \left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right )+a \left (A b^2-2 a b B+a^2 (A+2 C)\right ) \cos (c+d x)-b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (\left (a^2-b^2\right ) \left (6 A b^2-4 a b B+a^2 (A+2 C)\right )-a b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (b \left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{a^4 \left (a^2-b^2\right )}+\frac {\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) \int \sec (c+d x) \, dx}{2 a^4}\\ &=\frac {\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (2 b \left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right ) d}\\ &=-\frac {2 b \left (4 a^2 A b^2-3 A b^4-3 a^3 b B+2 a b^3 B+2 a^4 C-a^2 b^2 C\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 5.77, size = 389, normalized size = 1.27 \[ \frac {-2 \left (a^2 (A+2 C)-4 a b B+6 A b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \left (a^2 (A+2 C)-4 a b B+6 A b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^2 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a^2 A}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {8 b \left (-2 a^4 C+3 a^3 b B+a^2 b^2 (C-4 A)-2 a b^3 B+3 A b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}+\frac {4 a b^2 \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac {4 a (a B-2 A b) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {4 a (a B-2 A b) \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}}{4 a^4 d} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]
[Out]
((8*b*(3*A*b^4 + 3*a^3*b*B - 2*a*b^3*B - 2*a^4*C + a^2*b^2*(-4*A + C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt
[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 2*(6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)
/2]] + 2*(6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A)/(Cos[(c + d*x)
/2] - Sin[(c + d*x)/2])^2 + (4*a*(-2*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^2
*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(-2*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]) + (4*a*b^2*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(4*a^4
*d)
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fricas [B] time = 77.59, size = 1515, normalized size = 4.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^2,x, algorithm="fricas")
[Out]
[-1/4*(2*((2*C*a^4*b^2 - 3*B*a^3*b^3 + (4*A - C)*a^2*b^4 + 2*B*a*b^5 - 3*A*b^6)*cos(d*x + c)^3 + (2*C*a^5*b -
3*B*a^4*b^2 + (4*A - C)*a^3*b^3 + 2*B*a^2*b^4 - 3*A*a*b^5)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x
+ c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^
2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - (((A + 2*C)*a^6*b - 4*B*a^5*b^2 + 4*(A - C)*a^4*b^3 + 8*B*a^3*
b^4 - (11*A - 2*C)*a^2*b^5 - 4*B*a*b^6 + 6*A*b^7)*cos(d*x + c)^3 + ((A + 2*C)*a^7 - 4*B*a^6*b + 4*(A - C)*a^5*
b^2 + 8*B*a^4*b^3 - (11*A - 2*C)*a^3*b^4 - 4*B*a^2*b^5 + 6*A*a*b^6)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((
(A + 2*C)*a^6*b - 4*B*a^5*b^2 + 4*(A - C)*a^4*b^3 + 8*B*a^3*b^4 - (11*A - 2*C)*a^2*b^5 - 4*B*a*b^6 + 6*A*b^7)*
cos(d*x + c)^3 + ((A + 2*C)*a^7 - 4*B*a^6*b + 4*(A - C)*a^5*b^2 + 8*B*a^4*b^3 - (11*A - 2*C)*a^3*b^4 - 4*B*a^2
*b^5 + 6*A*a*b^6)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(A*a^7 - 2*A*a^5*b^2 + A*a^3*b^4 + 2*(B*a^6*b - (
2*A - C)*a^5*b^2 - 3*B*a^4*b^3 + (5*A - C)*a^3*b^4 + 2*B*a^2*b^5 - 3*A*a*b^6)*cos(d*x + c)^2 + (2*B*a^7 - 3*A*
a^6*b - 4*B*a^5*b^2 + 6*A*a^4*b^3 + 2*B*a^3*b^4 - 3*A*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^8*b - 2*a^6*b^3
+ a^4*b^5)*d*cos(d*x + c)^3 + (a^9 - 2*a^7*b^2 + a^5*b^4)*d*cos(d*x + c)^2), -1/4*(4*((2*C*a^4*b^2 - 3*B*a^3*
b^3 + (4*A - C)*a^2*b^4 + 2*B*a*b^5 - 3*A*b^6)*cos(d*x + c)^3 + (2*C*a^5*b - 3*B*a^4*b^2 + (4*A - C)*a^3*b^3 +
2*B*a^2*b^4 - 3*A*a*b^5)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*
x + c))) - (((A + 2*C)*a^6*b - 4*B*a^5*b^2 + 4*(A - C)*a^4*b^3 + 8*B*a^3*b^4 - (11*A - 2*C)*a^2*b^5 - 4*B*a*b^
6 + 6*A*b^7)*cos(d*x + c)^3 + ((A + 2*C)*a^7 - 4*B*a^6*b + 4*(A - C)*a^5*b^2 + 8*B*a^4*b^3 - (11*A - 2*C)*a^3*
b^4 - 4*B*a^2*b^5 + 6*A*a*b^6)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + (((A + 2*C)*a^6*b - 4*B*a^5*b^2 + 4*(A
- C)*a^4*b^3 + 8*B*a^3*b^4 - (11*A - 2*C)*a^2*b^5 - 4*B*a*b^6 + 6*A*b^7)*cos(d*x + c)^3 + ((A + 2*C)*a^7 - 4*B
*a^6*b + 4*(A - C)*a^5*b^2 + 8*B*a^4*b^3 - (11*A - 2*C)*a^3*b^4 - 4*B*a^2*b^5 + 6*A*a*b^6)*cos(d*x + c)^2)*log
(-sin(d*x + c) + 1) - 2*(A*a^7 - 2*A*a^5*b^2 + A*a^3*b^4 + 2*(B*a^6*b - (2*A - C)*a^5*b^2 - 3*B*a^4*b^3 + (5*A
- C)*a^3*b^4 + 2*B*a^2*b^5 - 3*A*a*b^6)*cos(d*x + c)^2 + (2*B*a^7 - 3*A*a^6*b - 4*B*a^5*b^2 + 6*A*a^4*b^3 + 2
*B*a^3*b^4 - 3*A*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^8*b - 2*a^6*b^3 + a^4*b^5)*d*cos(d*x + c)^3 + (a^9 -
2*a^7*b^2 + a^5*b^4)*d*cos(d*x + c)^2)]
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giac [A] time = 1.50, size = 423, normalized size = 1.38 \[ \frac {\frac {4 \, {\left (2 \, C a^{4} b - 3 \, B a^{3} b^{2} + 4 \, A a^{2} b^{3} - C a^{2} b^{3} + 2 \, B a b^{4} - 3 \, A b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, {\left (C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}} + \frac {{\left (A a^{2} + 2 \, C a^{2} - 4 \, B a b + 6 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {{\left (A a^{2} + 2 \, C a^{2} - 4 \, B a b + 6 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^2,x, algorithm="giac")
[Out]
1/2*(4*(2*C*a^4*b - 3*B*a^3*b^2 + 4*A*a^2*b^3 - C*a^2*b^3 + 2*B*a*b^4 - 3*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)))/((a^6 - a^4
*b^2)*sqrt(a^2 - b^2)) + 4*(C*a^2*b^2*tan(1/2*d*x + 1/2*c) - B*a*b^3*tan(1/2*d*x + 1/2*c) + A*b^4*tan(1/2*d*x
+ 1/2*c))/((a^5 - a^3*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)) + (A*a^2 + 2*C*a^2 -
4*B*a*b + 6*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - (A*a^2 + 2*C*a^2 - 4*B*a*b + 6*A*b^2)*log(abs(tan
(1/2*d*x + 1/2*c) - 1))/a^4 + 2*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan(1/2*d*x + 1/2*c)^3 + 4*A*b*tan(1/2*d*x
+ 1/2*c)^3 + A*a*tan(1/2*d*x + 1/2*c) + 2*B*a*tan(1/2*d*x + 1/2*c) - 4*A*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*
x + 1/2*c)^2 - 1)^2*a^3))/d
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maple [B] time = 0.30, size = 914, normalized size = 2.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^2,x)
[Out]
-8/d/a^2/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^3+2/d*b^4/a^
3/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)*A-2/d*b^3/a^2/(a^2-b^2)*tan
(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)*B+2/d*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(
a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)*C-1/d/a^2/(tan(1/2*d*x+1/2*c)-1)*B+1/2/d/a^2*A/(tan(1/2*d*x
+1/2*c)-1)^2-1/d/a^2/(tan(1/2*d*x+1/2*c)+1)*B-1/2/d/a^2*A/(tan(1/2*d*x+1/2*c)+1)^2+1/2/d/a^2*A/(tan(1/2*d*x+1/
2*c)-1)-1/2/d/a^2*A*ln(tan(1/2*d*x+1/2*c)-1)+1/2/d/a^2*A*ln(tan(1/2*d*x+1/2*c)+1)+2/d*b^3/a^2/(a-b)/(a+b)/((a-
b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C+6/d*b^5/a^4/(a-b)/(a+b)/((a-b)*(a+b))^(
1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+6/d*b^2/a/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan(t
an(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-4/d*b^4/a^3/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+
1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+1/2/d/a^2*A/(tan(1/2*d*x+1/2*c)+1)-1/d/a^2*ln(tan(1/2*d*x+1/2*c)-1)*C+1/d/
a^2*ln(tan(1/2*d*x+1/2*c)+1)*C-4/d/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b
))^(1/2))*C*b-2/d/a^3*ln(tan(1/2*d*x+1/2*c)+1)*B*b+2/d/a^3/(tan(1/2*d*x+1/2*c)-1)*A*b-3/d/a^4*ln(tan(1/2*d*x+1
/2*c)-1)*A*b^2+2/d/a^3*ln(tan(1/2*d*x+1/2*c)-1)*B*b+2/d/a^3/(tan(1/2*d*x+1/2*c)+1)*A*b+3/d/a^4*ln(tan(1/2*d*x+
1/2*c)+1)*A*b^2
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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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^2,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*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?
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mupad [B] time = 12.36, size = 9931, normalized size = 32.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b*cos(c + d*x))^2),x)
[Out]
(atan(((((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^
3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 +
12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*tan(c/2 +
(d*x)/2)*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8
*a^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^2)))*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b))/a^4 - (8*tan(c/2 + (
d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 12
0*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^
2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 +
8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 -
8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^
5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*
A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*
b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2))*(3*A*b^2 + a^2*(A/
2 + C) - 2*B*a*b)*1i)/a^4 - (((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^
11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13
*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10
*b^2) + (8*tan(c/2 + (d*x)/2)*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*
b^4 - 16*a^11*b^3 - 8*a^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^2)))*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b))
/a^4 + (8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b -
120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^
2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^
3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*
a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^
4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 -
100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*
a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2
))*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b)*1i)/a^4)/((16*(108*A^3*b^11 - 54*A^3*a*b^10 + 8*C^3*a^10*b - 216*A^3*a^
2*b^9 + 81*A^3*a^3*b^8 + 63*A^3*a^4*b^7 - 9*A^3*a^5*b^6 + 41*A^3*a^6*b^5 - 4*A^3*a^7*b^4 + 4*A^3*a^8*b^3 - 32*
B^3*a^3*b^8 + 16*B^3*a^4*b^7 + 80*B^3*a^5*b^6 - 24*B^3*a^6*b^5 - 48*B^3*a^7*b^4 + 4*C^3*a^6*b^5 - 4*C^3*a^7*b^
4 - 12*C^3*a^8*b^3 + 8*C^3*a^9*b^2 - 216*A^2*B*a*b^10 + 8*A*C^2*a^10*b + 2*A^2*C*a^10*b + 144*A*B^2*a^2*b^9 -
72*A*B^2*a^3*b^8 - 336*A*B^2*a^4*b^7 + 108*A*B^2*a^5*b^6 + 168*A*B^2*a^6*b^5 - 6*A*B^2*a^7*b^4 + 24*A*B^2*a^8*
b^3 + 108*A^2*B*a^2*b^9 + 468*A^2*B*a^3*b^8 - 162*A^2*B*a^4*b^7 - 186*A^2*B*a^5*b^6 + 15*A^2*B*a^6*b^5 - 63*A^
2*B*a^7*b^4 + 3*A^2*B*a^8*b^3 - 3*A^2*B*a^9*b^2 + 36*A*C^2*a^4*b^7 - 30*A*C^2*a^5*b^6 - 96*A*C^2*a^6*b^5 + 52*
A*C^2*a^7*b^4 + 52*A*C^2*a^8*b^3 + 108*A^2*C*a^2*b^9 - 72*A^2*C*a^3*b^8 - 252*A^2*C*a^4*b^7 + 111*A^2*C*a^5*b^
6 + 105*A^2*C*a^6*b^5 - 5*A^2*C*a^7*b^4 + 37*A^2*C*a^8*b^3 - 2*A^2*C*a^9*b^2 - 24*B*C^2*a^5*b^6 + 20*B*C^2*a^6
*b^5 + 68*B*C^2*a^7*b^4 - 36*B*C^2*a^8*b^3 - 44*B*C^2*a^9*b^2 + 48*B^2*C*a^4*b^7 - 32*B^2*C*a^5*b^6 - 128*B^2*
C*a^6*b^5 + 52*B^2*C*a^7*b^4 + 80*B^2*C*a^8*b^3 - 144*A*B*C*a^3*b^8 + 96*A*B*C*a^4*b^7 + 360*A*B*C*a^5*b^6 - 1
52*A*B*C*a^6*b^5 - 188*A*B*C*a^7*b^4 + 4*A*B*C*a^8*b^3 - 28*A*B*C*a^9*b^2))/(a^11*b + a^12 - a^9*b^3 - a^10*b^
2) + (((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3
+ 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 1
2*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*tan(c/2 + (d
*x)/2)*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a
^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^2)))*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b))/a^4 - (8*tan(c/2 + (d*
x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*
A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*
b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*
C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8
*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5
+ 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*
C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^
5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2))*(3*A*b^2 + a^2*(A/2
+ C) - 2*B*a*b))/a^4 + (((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^
4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2
- 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2)
+ (8*tan(c/2 + (d*x)/2)*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 -
16*a^11*b^3 - 8*a^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^2)))*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b))/a^4
+ (8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*
A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8
*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 1
6*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10
- 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6
- 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A
*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b
^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2))*(3
*A*b^2 + a^2*(A/2 + C) - 2*B*a*b))/a^4))*(3*A*b^2 + a^2*(A/2 + C) - 2*B*a*b)*2i)/(a^4*d) - ((tan(c/2 + (d*x)/2
)*(A*a^4 + 6*A*b^4 + 2*B*a^4 - 5*A*a^2*b^2 - 2*B*a^2*b^2 + 2*C*a^2*b^2 + 3*A*a*b^3 - 3*A*a^3*b - 4*B*a*b^3 + 2
*B*a^3*b))/((a^3*b - a^4)*(a + b)) + (tan(c/2 + (d*x)/2)^5*(A*a^4 + 6*A*b^4 - 2*B*a^4 - 5*A*a^2*b^2 + 2*B*a^2*
b^2 + 2*C*a^2*b^2 - 3*A*a*b^3 + 3*A*a^3*b - 4*B*a*b^3 + 2*B*a^3*b))/((a^3*b - a^4)*(a + b)) + (2*tan(c/2 + (d*
x)/2)^3*(A*a^4 - 6*A*b^4 + 3*A*a^2*b^2 - 2*C*a^2*b^2 + 4*B*a*b^3 - 2*B*a^3*b))/(a*(a^2*b - a^3)*(a + b)))/(d*(
a + b - tan(c/2 + (d*x)/2)^2*(a + 3*b) - tan(c/2 + (d*x)/2)^4*(a - 3*b) + tan(c/2 + (d*x)/2)^6*(a - b))) - (b*
atan(((b*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*
b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11
*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7
*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A
*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B
*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7
- 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B
*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*
b^2) + (b*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 -
14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 1
2*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^
3 - a^10*b^2) + (8*b*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^
10*b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/((a^
8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C
*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*A
*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b)*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)
+ (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b -
120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^
2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^
3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*
a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^
4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 -
100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*
a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2
) - (b*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14
*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B
*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 -
a^10*b^2) - (8*b*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*
b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/((a^8*b
+ a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^
2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*A*b^
4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b)*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))/(
(16*(108*A^3*b^11 - 54*A^3*a*b^10 + 8*C^3*a^10*b - 216*A^3*a^2*b^9 + 81*A^3*a^3*b^8 + 63*A^3*a^4*b^7 - 9*A^3*a
^5*b^6 + 41*A^3*a^6*b^5 - 4*A^3*a^7*b^4 + 4*A^3*a^8*b^3 - 32*B^3*a^3*b^8 + 16*B^3*a^4*b^7 + 80*B^3*a^5*b^6 - 2
4*B^3*a^6*b^5 - 48*B^3*a^7*b^4 + 4*C^3*a^6*b^5 - 4*C^3*a^7*b^4 - 12*C^3*a^8*b^3 + 8*C^3*a^9*b^2 - 216*A^2*B*a*
b^10 + 8*A*C^2*a^10*b + 2*A^2*C*a^10*b + 144*A*B^2*a^2*b^9 - 72*A*B^2*a^3*b^8 - 336*A*B^2*a^4*b^7 + 108*A*B^2*
a^5*b^6 + 168*A*B^2*a^6*b^5 - 6*A*B^2*a^7*b^4 + 24*A*B^2*a^8*b^3 + 108*A^2*B*a^2*b^9 + 468*A^2*B*a^3*b^8 - 162
*A^2*B*a^4*b^7 - 186*A^2*B*a^5*b^6 + 15*A^2*B*a^6*b^5 - 63*A^2*B*a^7*b^4 + 3*A^2*B*a^8*b^3 - 3*A^2*B*a^9*b^2 +
36*A*C^2*a^4*b^7 - 30*A*C^2*a^5*b^6 - 96*A*C^2*a^6*b^5 + 52*A*C^2*a^7*b^4 + 52*A*C^2*a^8*b^3 + 108*A^2*C*a^2*
b^9 - 72*A^2*C*a^3*b^8 - 252*A^2*C*a^4*b^7 + 111*A^2*C*a^5*b^6 + 105*A^2*C*a^6*b^5 - 5*A^2*C*a^7*b^4 + 37*A^2*
C*a^8*b^3 - 2*A^2*C*a^9*b^2 - 24*B*C^2*a^5*b^6 + 20*B*C^2*a^6*b^5 + 68*B*C^2*a^7*b^4 - 36*B*C^2*a^8*b^3 - 44*B
*C^2*a^9*b^2 + 48*B^2*C*a^4*b^7 - 32*B^2*C*a^5*b^6 - 128*B^2*C*a^6*b^5 + 52*B^2*C*a^7*b^4 + 80*B^2*C*a^8*b^3 -
144*A*B*C*a^3*b^8 + 96*A*B*C*a^4*b^7 + 360*A*B*C*a^5*b^6 - 152*A*B*C*a^6*b^5 - 188*A*B*C*a^7*b^4 + 4*A*B*C*a^
8*b^3 - 28*A*B*C*a^9*b^2))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) + (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2
*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4
*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 6
4*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*
b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b
- 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*
B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b
^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*
B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (b*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*A*a^
15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*
B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*
a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) + (8*b*tan(c/2 + (d*x)/2)*(-(a + b)^
3*(a - b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3*A*b^4 - 2*C*
a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/((a^8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6 + 3
*a^6*b^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6
+ 3*a^6*b^4 - 3*a^8*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b
^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2) - (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10
+ 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 -
26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*
a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 -
20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*
B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*
b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 3
2*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^
7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (b*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*A*a^15 + 4
*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*
b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b
^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*b*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a -
b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3*A*b^4 - 2*C*a^4 -
4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/((a^8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6 + 3*a^6*b
^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a
^6*b^4 - 3*a^8*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3
*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^
2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b)*2i)/(d*(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+b*cos(d*x+c))**2,x)
[Out]
Integral((A + B*cos(c + d*x) + C*cos(c + d*x)**2)*sec(c + d*x)**3/(a + b*cos(c + d*x))**2, x)
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