3.812 \(\int \frac {(B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=214 \[ \frac {B \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {b \left (-3 a^3 C+5 a^2 b B-2 b^3 B\right ) \sin (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {\left (-2 a^5 C+6 a^4 b B-a^3 b^2 C-5 a^2 b^3 B+2 b^5 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.80, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3029, 3000, 3055, 3001, 3770, 2659, 205} \[ -\frac {\left (-5 a^2 b^3 B-a^3 b^2 C+6 a^4 b B-2 a^5 C+2 b^5 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {b \left (5 a^2 b B-3 a^3 C-2 b^3 B\right ) \sin (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {b (b B-a C) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {B \tanh ^{-1}(\sin (c+d x))}{a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 2*a^5*C - a^3*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]
])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)) + (B*ArcTanh[Sin[c + d*x]])/(a^3*d) + (b*(b*B - a*C)*Sin[c + d*x])/(2*
a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^
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 3000

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((A*b^2 - a*b*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*
Sin[e + f*x])^(1 + n))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[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[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m
+ n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^
2, 0] && RationalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n,
-1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 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 3029

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])
^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 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 (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=\int \frac {(B+C \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx\\ &=\frac {b (b B-a C) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\int \frac {\left (2 \left (a^2-b^2\right ) B-2 a (b B-a C) \cos (c+d x)+b (b B-a C) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {b (b B-a C) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (2 \left (a^2-b^2\right )^2 B-a \left (4 a^2 b B-b^3 B-2 a^3 C-a b^2 C\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {b (b B-a C) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {B \int \sec (c+d x) \, dx}{a^3}-\frac {\left (6 a^4 b B-5 a^2 b^3 B+2 b^5 B-2 a^5 C-a^3 b^2 C\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac {b (b B-a C) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (6 a^4 b B-5 a^2 b^3 B+2 b^5 B-2 a^5 C-a^3 b^2 C\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^3 \left (a^2-b^2\right )^2 d}\\ &=-\frac {\left (6 a^4 b B-5 a^2 b^3 B+2 b^5 B-2 a^5 C-a^3 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^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {B \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac {b (b B-a C) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 1.30, size = 269, normalized size = 1.26 \[ \frac {\cos (c+d x) (B \sec (c+d x)+C) \left (\frac {a^2 b (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {a b \left (-3 a^3 C+5 a^2 b B-2 b^3 B\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac {2 \left (2 a^5 C-6 a^4 b B+a^3 b^2 C+5 a^2 b^3 B-2 b^5 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}-2 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2 a^3 d (B+C \cos (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[c + d*x]*(C + B*Sec[c + d*x])*((-2*(-6*a^4*b*B + 5*a^2*b^3*B - 2*b^5*B + 2*a^5*C + a^3*b^2*C)*ArcTanh[((a
- b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 2*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] +
2*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*b*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c
+ d*x])^2) + (a*b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x]))))/
(2*a^3*d*(B + C*Cos[c + d*x]))

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fricas [B]  time = 22.22, size = 1400, normalized size = 6.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*((2*C*a^7 - 6*B*a^6*b + C*a^5*b^2 + 5*B*a^4*b^3 - 2*B*a^2*b^5 + (2*C*a^5*b^2 - 6*B*a^4*b^3 + C*a^3*b^4 +
5*B*a^2*b^5 - 2*B*b^7)*cos(d*x + c)^2 + 2*(2*C*a^6*b - 6*B*a^5*b^2 + C*a^4*b^3 + 5*B*a^3*b^4 - 2*B*a*b^6)*cos(
d*x + c))*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)) + 2*(B*a^8 - 3*B*a^
6*b^2 + 3*B*a^4*b^4 - B*a^2*b^6 + (B*a^6*b^2 - 3*B*a^4*b^4 + 3*B*a^2*b^6 - B*b^8)*cos(d*x + c)^2 + 2*(B*a^7*b
- 3*B*a^5*b^3 + 3*B*a^3*b^5 - B*a*b^7)*cos(d*x + c))*log(sin(d*x + c) + 1) - 2*(B*a^8 - 3*B*a^6*b^2 + 3*B*a^4*
b^4 - B*a^2*b^6 + (B*a^6*b^2 - 3*B*a^4*b^4 + 3*B*a^2*b^6 - B*b^8)*cos(d*x + c)^2 + 2*(B*a^7*b - 3*B*a^5*b^3 +
3*B*a^3*b^5 - B*a*b^7)*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(4*C*a^7*b - 6*B*a^6*b^2 - 5*C*a^5*b^3 + 9*B*a
^4*b^4 + C*a^3*b^5 - 3*B*a^2*b^6 + (3*C*a^6*b^2 - 5*B*a^5*b^3 - 3*C*a^4*b^4 + 7*B*a^3*b^5 - 2*B*a*b^7)*cos(d*x
+ c))*sin(d*x + c))/((a^9*b^2 - 3*a^7*b^4 + 3*a^5*b^6 - a^3*b^8)*d*cos(d*x + c)^2 + 2*(a^10*b - 3*a^8*b^3 + 3
*a^6*b^5 - a^4*b^7)*d*cos(d*x + c) + (a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d), 1/2*((2*C*a^7 - 6*B*a^6*b +
C*a^5*b^2 + 5*B*a^4*b^3 - 2*B*a^2*b^5 + (2*C*a^5*b^2 - 6*B*a^4*b^3 + C*a^3*b^4 + 5*B*a^2*b^5 - 2*B*b^7)*cos(d*
x + c)^2 + 2*(2*C*a^6*b - 6*B*a^5*b^2 + C*a^4*b^3 + 5*B*a^3*b^4 - 2*B*a*b^6)*cos(d*x + c))*sqrt(a^2 - b^2)*arc
tan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) + (B*a^8 - 3*B*a^6*b^2 + 3*B*a^4*b^4 - B*a^2*b^6 + (
B*a^6*b^2 - 3*B*a^4*b^4 + 3*B*a^2*b^6 - B*b^8)*cos(d*x + c)^2 + 2*(B*a^7*b - 3*B*a^5*b^3 + 3*B*a^3*b^5 - B*a*b
^7)*cos(d*x + c))*log(sin(d*x + c) + 1) - (B*a^8 - 3*B*a^6*b^2 + 3*B*a^4*b^4 - B*a^2*b^6 + (B*a^6*b^2 - 3*B*a^
4*b^4 + 3*B*a^2*b^6 - B*b^8)*cos(d*x + c)^2 + 2*(B*a^7*b - 3*B*a^5*b^3 + 3*B*a^3*b^5 - B*a*b^7)*cos(d*x + c))*
log(-sin(d*x + c) + 1) - (4*C*a^7*b - 6*B*a^6*b^2 - 5*C*a^5*b^3 + 9*B*a^4*b^4 + C*a^3*b^5 - 3*B*a^2*b^6 + (3*C
*a^6*b^2 - 5*B*a^5*b^3 - 3*C*a^4*b^4 + 7*B*a^3*b^5 - 2*B*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2 - 3*a^7*
b^4 + 3*a^5*b^6 - a^3*b^8)*d*cos(d*x + c)^2 + 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*cos(d*x + c) + (a
^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d)]

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giac [B]  time = 0.51, size = 481, normalized size = 2.25 \[ \frac {\frac {{\left (2 \, C a^{5} - 6 \, B a^{4} b + C a^{3} b^{2} + 5 \, B a^{2} b^{3} - 2 \, B 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^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {B \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} - \frac {4 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\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 )}^{2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*C*a^5 - 6*B*a^4*b + C*a^3*b^2 + 5*B*a^2*b^3 - 2*B*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^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(a
^2 - b^2)) + B*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 - B*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^3 - (4*C*a^4*b*
tan(1/2*d*x + 1/2*c)^3 - 6*B*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^2*b^3
*tan(1/2*d*x + 1/2*c)^3 - C*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 3*B*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 2*B*b^5*tan(1/
2*d*x + 1/2*c)^3 + 4*C*a^4*b*tan(1/2*d*x + 1/2*c) - 6*B*a^3*b^2*tan(1/2*d*x + 1/2*c) + 3*C*a^3*b^2*tan(1/2*d*x
+ 1/2*c) - 5*B*a^2*b^3*tan(1/2*d*x + 1/2*c) - C*a^2*b^3*tan(1/2*d*x + 1/2*c) + 3*B*a*b^4*tan(1/2*d*x + 1/2*c)
+ 2*B*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - 2*a^4*b^2 + a^2*b^4)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*
c)^2 + a + b)^2))/d

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maple [B]  time = 0.26, size = 1045, normalized size = 4.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^3,x)

[Out]

6/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*b^2*B+1/d
/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^3/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-2/d/
a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^4/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-4/d
/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*a*b-1/d/(a
*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*b^2*C+6/d/(a*ta
n(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-1/d/a/(a*tan(1/2*d*x+1
/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^3/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-2/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-t
an(1/2*d*x+1/2*c)^2*b+a+b)^2*b^4/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-4/d*a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+
1/2*c)^2*b+a+b)^2*b/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C+1/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)
^2*b^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-6/d*a*b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1
/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+5/d/a/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-
b)/((a-b)*(a+b))^(1/2))*B*b^3-2/d/a^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/
((a-b)*(a+b))^(1/2))*B*b^5+2/d*a^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a
-b)*(a+b))^(1/2))*C+1/d/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^
(1/2))*b^2*C-1/d/a^3*B*ln(tan(1/2*d*x+1/2*c)-1)+1/d/a^3*B*ln(tan(1/2*d*x+1/2*c)+1)

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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((B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^3,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 = 10.94, size = 6911, normalized size = 32.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^2*(a + b*cos(c + d*x))^3),x)

[Out]

((tan(c/2 + (d*x)/2)^3*(2*B*b^4 - 6*B*a^2*b^2 + C*a^2*b^2 - B*a*b^3 + 4*C*a^3*b))/((a^2*b - a^3)*(a + b)^2) -
(tan(c/2 + (d*x)/2)*(2*B*b^4 - 6*B*a^2*b^2 - C*a^2*b^2 + B*a*b^3 + 4*C*a^3*b))/((a + b)*(a^4 - 2*a^3*b + a^2*b
^2)))/(d*(2*a*b + tan(c/2 + (d*x)/2)^2*(2*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b + b^2) + a^2 + b^2)
) - (B*atan(-((B*((B*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*
a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b
^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^
11*b^2) - (8*B*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 +
48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/(a^3*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 +
3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))))/a^3 - (8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^
2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 3
2*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 +
8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))*1i)/a^3 -
(B*((B*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*
B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^1
4*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (8*
B*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5
+ 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/(a^3*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3
*a^8*b^3 - 3*a^9*b^2))))/a^3 + (8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B
^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3
+ 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3
))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))*1i)/a^3)/((16*(4*B^3*b
^9 + 4*B*C^2*a^9 - 4*B^2*C*a^9 - 2*B^3*a*b^8 + 12*B^3*a^8*b - 18*B^3*a^2*b^7 + 13*B^3*a^3*b^6 + 36*B^3*a^4*b^5
- 26*B^3*a^5*b^4 - 34*B^3*a^6*b^3 + 24*B^3*a^7*b^2 - 20*B^2*C*a^8*b + B*C^2*a^5*b^4 + 4*B*C^2*a^7*b^2 - 2*B^2
*C*a^2*b^7 - 2*B^2*C*a^3*b^6 + 2*B^2*C*a^4*b^5 + 2*B^2*C*a^6*b^3 + 6*B^2*C*a^7*b^2))/(a^12*b + a^13 - a^6*b^7
- a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (B*((B*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 +
2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*
a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^
7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (8*B*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*
a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/(a^3*
(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))))/a^3 - (8*tan(c/2 + (d*x
)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*
B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2
- 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^
5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))/a^3 + (B*((B*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 1
8*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^
9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^
5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (8*B*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8
*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/(a^3*(a^10*b + a^11 -
a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))))/a^3 + (8*tan(c/2 + (d*x)/2)*(4*B^2*a^10
+ 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48
*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b -
4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 -
3*a^8*b^3 - 3*a^9*b^2)))/a^3))*2i)/(a^3*d) - (atan(((((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a
^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a
^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*
a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)
- ((-(a + b)^5*(a - b)^5)^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*
b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 -
6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10
*b^3 - 3*a^11*b^2) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a
^3*b^2 - 6*B*a^4*b)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 +
32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2
)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^5 - 2*B*b^5 + 5
*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))
*(-(a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*1i)/(2*(a^13 - a^3*b^1
0 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*
C^2*a^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*
B^2*a^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2
*B*C*a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*
b^2) + ((-(a + b)^5*(a - b)^5)^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B
*a^9*b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*
b^3 - 6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3
*a^10*b^3 - 3*a^11*b^2) + (4*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3
+ C*a^3*b^2 - 6*B*a^4*b)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*
b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^1
1*b^2)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^5 - 2*B*b^
5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b
^2)))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*1i)/(2*(a^13 - a^
3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))/((16*(4*B^3*b^9 + 4*B*C^2*a^9 - 4*B^2*C*a^9 - 2*B
^3*a*b^8 + 12*B^3*a^8*b - 18*B^3*a^2*b^7 + 13*B^3*a^3*b^6 + 36*B^3*a^4*b^5 - 26*B^3*a^5*b^4 - 34*B^3*a^6*b^3 +
24*B^3*a^7*b^2 - 20*B^2*C*a^8*b + B*C^2*a^5*b^4 + 4*B*C^2*a^7*b^2 - 2*B^2*C*a^2*b^7 - 2*B^2*C*a^3*b^6 + 2*B^2
*C*a^4*b^5 + 2*B^2*C*a^6*b^3 + 6*B^2*C*a^7*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 -
3*a^10*b^3 - 3*a^11*b^2) - (((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B^2
*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3 +
24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3))
/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) - ((-(a + b)^5*(a - b)^5)
^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B
*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14
*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (4*t
an(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*(8*a^
15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^
3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)*(a^10*b + a^11 - a^4*b^
7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 -
6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(-(a + b)^5*(a - b)^5)^(
1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 +
10*a^9*b^4 - 5*a^11*b^2)) + (((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B
^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3
+ 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3
))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) + ((-(a + b)^5*(a - b)^
5)^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6
*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^
14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (4
*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*(8*
a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*
b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)*(a^10*b + a^11 - a^4*
b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2
- 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*(-(a + b)^5*(a - b)^5)
^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6
+ 10*a^9*b^4 - 5*a^11*b^2))))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B
*a^4*b)*1i)/(d*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (B + C \cos {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**2/(a+b*cos(d*x+c))**3,x)

[Out]

Integral((B + C*cos(c + d*x))*cos(c + d*x)*sec(c + d*x)**2/(a + b*cos(c + d*x))**3, x)

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