3.584 \(\int \frac{(A+C \cos ^2(c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx\)
Optimal. Leaf size=378 \[ -\frac{b \left (5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 C+12 A b^6\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{b \left (-a^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right ) \tan (c+d x)}{2 a^4 d \left (a^2-b^2\right )^2}+\frac{\left (a^2 (A+2 C)+12 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left (-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2}+\frac{\left (7 a^2 A b^2+3 a^4 C-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2} \]
[Out]
-((b*(12*A*b^6 - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/S
qrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((12*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*
d) - (b*(12*A*b^4 + a^4*(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4
+ a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Se
c[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*Sec[c
+ d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))
________________________________________________________________________________________
Rubi [A] time = 1.87584, antiderivative size = 378, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used =
{3056, 3055, 3001, 3770, 2659, 205} \[ -\frac{b \left (5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 C+12 A b^6\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{b \left (-a^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right ) \tan (c+d x)}{2 a^4 d \left (a^2-b^2\right )^2}+\frac{\left (a^2 (A+2 C)+12 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\left (-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2}+\frac{\left (7 a^2 A b^2+3 a^4 C-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]
[Out]
-((b*(12*A*b^6 - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/S
qrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((12*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*
d) - (b*(12*A*b^4 + a^4*(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4
+ a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Se
c[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*Sec[c
+ d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))
Rule 3056
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 + 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] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), I
nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*
(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)
*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ
[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 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 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 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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 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]
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (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 (2 A b^2-a^2 (A-C)\right )-2 a b (A+C) \cos (c+d x)+3 \left (A b^2+a^2 C\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec (c+d x) \tan (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 (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right )+a b \left (A b^2-a^2 (4 A+3 C)\right ) \cos (c+d x)+2 \left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec (c+d x) \tan (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 b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right )-2 a \left (2 A b^4-a^2 b^2 (4 A+C)-a^4 (A+2 C)\right ) \cos (c+d x)+2 b \left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec (c+d x) \tan (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 \left (12 A b^2+a^2 (A+2 C)\right )+2 a b \left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac{\left (b \left (12 A b^6-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^2}+\frac{\left (12 A b^2+a^2 (A+2 C)\right ) \int \sec (c+d x) \, dx}{2 a^5}\\ &=\frac{\left (12 A b^2+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac{\left (b \left (12 A b^6-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 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^5 \left (a^2-b^2\right )^2 d}\\ &=-\frac{b \left (20 a^4 A b^2-29 a^2 A b^4+12 A b^6+6 a^6 C-5 a^4 b^2 C+2 a^2 b^4 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}+\frac{\left (12 A b^2+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [B] time = 6.3841, size = 856, normalized size = 2.26 \[ \frac{\left (-A a^2-2 C a^2-12 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 ) \left (A \sec ^2(c+d x)+C\right ) \cos ^2(c+d x)}{a^5 d (2 A+C+C \cos (2 c+2 d x))}+\frac{\left (A a^2+2 C a^2+12 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 ) \left (A \sec ^2(c+d x)+C\right ) \cos ^2(c+d x)}{a^5 d (2 A+C+C \cos (2 c+2 d x))}+\frac{2 b \left (6 C a^6+20 A b^2 a^4-5 b^2 C a^4-29 A b^4 a^2+2 b^4 C a^2+12 A b^6\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right ) \left (A \sec ^2(c+d x)+C\right ) \cos ^2(c+d x)}{a^5 \left (a^2-b^2\right )^2 \sqrt{b^2-a^2} d (2 A+C+C \cos (2 c+2 d x))}-\frac{6 A b \left (A \sec ^2(c+d x)+C\right ) \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^2(c+d x)}{a^4 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{\left (A \sec ^2(c+d x)+C\right ) \left (A \sin (c+d x) b^4+a^2 C \sin (c+d x) b^2\right ) \cos ^2(c+d x)}{a^3 (a-b) (a+b) d (a+b \cos (c+d x))^2 (2 A+C+C \cos (2 c+2 d x))}+\frac{\left (A \sec ^2(c+d x)+C\right ) \left (-6 A \sin (c+d x) b^6+9 a^2 A \sin (c+d x) b^4-2 a^2 C \sin (c+d x) b^4+5 a^4 C \sin (c+d x) b^2\right ) \cos ^2(c+d x)}{a^4 (a-b)^2 (a+b)^2 d (a+b \cos (c+d x)) (2 A+C+C \cos (2 c+2 d x))}-\frac{6 A b \left (A \sec ^2(c+d x)+C\right ) \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^2(c+d x)}{a^4 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{A \left (A \sec ^2(c+d x)+C\right ) \cos ^2(c+d x)}{2 a^3 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{A \left (A \sec ^2(c+d x)+C\right ) \cos ^2(c+d x)}{2 a^3 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]
[Out]
(2*b*(20*a^4*A*b^2 - 29*a^2*A*b^4 + 12*A*b^6 + 6*a^6*C - 5*a^4*b^2*C + 2*a^2*b^4*C)*ArcTanh[((a - b)*Tan[(c +
d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(a^5*(a^2 - b^2)^2*Sqrt[-a^2 + b^2]*d*(2*A +
C + C*Cos[2*c + 2*d*x])) + ((-(a^2*A) - 12*A*b^2 - 2*a^2*C)*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*
x)/2]]*(C + A*Sec[c + d*x]^2))/(a^5*d*(2*A + C + C*Cos[2*c + 2*d*x])) + ((a^2*A + 12*A*b^2 + 2*a^2*C)*Cos[c +
d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^5*d*(2*A + C + C*Cos[2*c + 2*d*x]))
+ (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(2*a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(
c + d*x)/2])^2) - (6*A*b*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^4*d*(2*A + C + C*Cos[2*c +
2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(2*a^3*d*(2*A + C
+ C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (6*A*b*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*
Sin[(c + d*x)/2])/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (Cos[c + d*x]
^2*(C + A*Sec[c + d*x]^2)*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a - b)*(a + b)*d*(a + b*Cos[c +
d*x])^2*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(9*a^2*A*b^4*Sin[c + d*x] -
6*A*b^6*Sin[c + d*x] + 5*a^4*b^2*C*Sin[c + d*x] - 2*a^2*b^4*C*Sin[c + d*x]))/(a^4*(a - b)^2*(a + b)^2*d*(a + b
*Cos[c + d*x])*(2*A + C + C*Cos[2*c + 2*d*x]))
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Maple [B] time = 0.099, size = 1497, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3,x)
[Out]
1/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^5/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A
+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-b)^2*tan(1/2*d*x+1/2*c)*b^2*C-20/d/a*b^3/(
a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+29/d/a^3*b^5/(a^
4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-1/d/a^3*ln(tan(1/2
*d*x+1/2*c)-1)*C+1/2/d/a^3*A/(tan(1/2*d*x+1/2*c)-1)^2+1/d/a^3*ln(tan(1/2*d*x+1/2*c)+1)*C-1/2/d/a^3*A/(tan(1/2*
d*x+1/2*c)+1)^2+1/2/d/a^3*A/(tan(1/2*d*x+1/2*c)-1)+1/2/d/a^3*A/(tan(1/2*d*x+1/2*c)+1)-1/2/d/a^3*A*ln(tan(1/2*d
*x+1/2*c)-1)+1/2/d/a^3*A*ln(tan(1/2*d*x+1/2*c)+1)+3/d*A/a^4/(tan(1/2*d*x+1/2*c)+1)*b-6/d/a^5*ln(tan(1/2*d*x+1/
2*c)-1)*A*b^2+3/d*A/a^4/(tan(1/2*d*x+1/2*c)-1)*b+6/d/a^5*ln(tan(1/2*d*x+1/2*c)+1)*A*b^2+5/d*b^3/a/(a^4-2*a^2*b
^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-12/d*b^7/a^5/(a^4-2*a^2*b^2
+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-2/d*b^5/a^3/(a^4-2*a^2*b^2+b^
4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+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*C-6/d*b*a/(a^4-2*a^2*b^2+b^4)/((a+b)*
(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-2/d*b^4/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/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-1/d*b^5/a^3/(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-b)^2*tan(1/2*d*x+1/2*c)*A-6/d*b^6/a^4/(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-b)^2*tan(1/2*d*x+1/2*c)*A-1/d*b^3/a/(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-b)^2*tan(1/2*d*x+1/2*c)*C-2/d*b^4/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/(a+b
)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-6/d*b^6/a^4/(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*A+1/d*b^3/a/(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+10/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/(a-b)/(a^2+2*a*b
+b^2)*tan(1/2*d*x+1/2*c)^3*A*b^4+10/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/(a+b)/(a-b)^2*
tan(1/2*d*x+1/2*c)*A*b^4
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3,x, algorithm="fricas")
[Out]
Timed out
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+b*cos(d*x+c))**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.83124, size = 1608, normalized size = 4.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3,x, algorithm="giac")
[Out]
1/2*(2*(6*C*a^6*b + 20*A*a^4*b^3 - 5*C*a^4*b^3 - 29*A*a^2*b^5 + 2*C*a^2*b^5 + 12*A*b^7)*(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^
9 - 2*a^7*b^2 + a^5*b^4)*sqrt(a^2 - b^2)) + 2*(A*a^7*tan(1/2*d*x + 1/2*c)^7 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^7
- 13*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c)
^7 - 5*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 3*C*a^3*b^4*tan(1/2*d*x + 1/2*
c)^7 - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 2*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 18*A*a*b^6*tan(1/2*d*x + 1/2
*c)^7 + 12*A*b^7*tan(1/2*d*x + 1/2*c)^7 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^5 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 +
5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5
+ 15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)
^5 + 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^6*tan(1/2*d*x + 1/2*c
)^5 - 36*A*b^7*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^3 - 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 5*
A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 -
15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^3
- 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^
3 + 36*A*b^7*tan(1/2*d*x + 1/2*c)^3 + A*a^7*tan(1/2*d*x + 1/2*c) - 4*A*a^6*b*tan(1/2*d*x + 1/2*c) - 13*A*a^5*b
^2*tan(1/2*d*x + 1/2*c) + 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c) + 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c) + 5*C*a^4*b^3*ta
n(1/2*d*x + 1/2*c) + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c) - 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c) + 17*A*a^2*b^5*tan(1
/2*d*x + 1/2*c) - 2*C*a^2*b^5*tan(1/2*d*x + 1/2*c) - 18*A*a*b^6*tan(1/2*d*x + 1/2*c) - 12*A*b^7*tan(1/2*d*x +
1/2*c))/((a^8 - 2*a^6*b^2 + a^4*b^4)*(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 + 2*b*tan(1/2*d*x +
1/2*c)^2 - a - b)^2) + (A*a^2 + 2*C*a^2 + 12*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 - (A*a^2 + 2*C*a^2
+ 12*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^5)/d