3.372 \(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=287 \[ \frac {4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac {4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac {(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {(44 A-21 B+8 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac {(44 A-21 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac {(178 A-87 B+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(16 A-9 B+2 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4} \]

[Out]

-1/2*(44*A-21*B+8*C)*arctanh(sin(d*x+c))/a^4/d+4/35*(454*A-216*B+83*C)*tan(d*x+c)/a^4/d-1/2*(44*A-21*B+8*C)*se
c(d*x+c)*tan(d*x+c)/a^4/d-1/105*(178*A-87*B+31*C)*sec(d*x+c)^2*tan(d*x+c)/a^4/d/(1+cos(d*x+c))^2-1/3*(44*A-21*
B+8*C)*sec(d*x+c)^2*tan(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*(A-B+C)*sec(d*x+c)^2*tan(d*x+c)/d/(a+a*cos(d*x+c))^4-1
/35*(16*A-9*B+2*C)*sec(d*x+c)^2*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^3+4/105*(454*A-216*B+83*C)*tan(d*x+c)^3/a^4/d

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Rubi [A]  time = 0.80, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3041, 2978, 2748, 3767, 3768, 3770} \[ \frac {4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac {4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac {(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {(44 A-21 B+8 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac {(44 A-21 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac {(178 A-87 B+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(16 A-9 B+2 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((44*A - 21*B + 8*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + (4*(454*A - 216*B + 83*C)*Tan[c + d*x])/(35*a^4*d) -
((44*A - 21*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((178*A - 87*B + 31*C)*Sec[c + d*x]^2*Tan[c + d*x]
)/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((44*A - 21*B + 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^4*d*(1 + Cos[c + d
*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((16*A - 9*B + 2*C)*Sec[c + d
*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (4*(454*A - 216*B + 83*C)*Tan[c + d*x]^3)/(105*a^4*d)

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2978

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

Rule 3041

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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 ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(a (10 A-3 B+3 C)-a (6 A-6 B-C) \cos (c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (7 a^2 (14 A-6 B+3 C)-5 a^2 (16 A-9 B+2 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (3 a^3 (276 A-129 B+52 C)-4 a^3 (178 A-87 B+31 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \left (12 a^4 (454 A-216 B+83 C)-105 a^4 (44 A-21 B+8 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{105 a^8}\\ &=-\frac {(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(44 A-21 B+8 C) \int \sec ^3(c+d x) \, dx}{a^4}+\frac {(4 (454 A-216 B+83 C)) \int \sec ^4(c+d x) \, dx}{35 a^4}\\ &=-\frac {(44 A-21 B+8 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(44 A-21 B+8 C) \int \sec (c+d x) \, dx}{2 a^4}-\frac {(4 (454 A-216 B+83 C)) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 a^4 d}\\ &=-\frac {(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac {4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac {(44 A-21 B+8 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}\\ \end {align*}

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Mathematica [A]  time = 1.71, size = 304, normalized size = 1.06 \[ \frac {26880 (44 A-21 B+8 C) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+2 \sin \left (\frac {1}{2} (c+d x)\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) (14 (28252 A-13353 B+5224 C) \cos (c+d x)+56 (5218 A-2472 B+961 C) \cos (2 (c+d x))+173316 A \cos (3 (c+d x))+79264 A \cos (4 (c+d x))+24436 A \cos (5 (c+d x))+3632 A \cos (6 (c+d x))+217696 A-82239 B \cos (3 (c+d x))-37656 B \cos (4 (c+d x))-11619 B \cos (5 (c+d x))-1728 B \cos (6 (c+d x))-102504 B+31832 C \cos (3 (c+d x))+14528 C \cos (4 (c+d x))+4472 C \cos (5 (c+d x))+664 C \cos (6 (c+d x))+39952 C)}{3360 a^4 d (\cos (c+d x)+1)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26880*(44*A - 21*B + 8*C)*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2]]) + 2*Cos[(c + d*x)/2]*(217696*A - 102504*B + 39952*C + 14*(28252*A - 13353*B + 5224*C)*Co
s[c + d*x] + 56*(5218*A - 2472*B + 961*C)*Cos[2*(c + d*x)] + 173316*A*Cos[3*(c + d*x)] - 82239*B*Cos[3*(c + d*
x)] + 31832*C*Cos[3*(c + d*x)] + 79264*A*Cos[4*(c + d*x)] - 37656*B*Cos[4*(c + d*x)] + 14528*C*Cos[4*(c + d*x)
] + 24436*A*Cos[5*(c + d*x)] - 11619*B*Cos[5*(c + d*x)] + 4472*C*Cos[5*(c + d*x)] + 3632*A*Cos[6*(c + d*x)] -
1728*B*Cos[6*(c + d*x)] + 664*C*Cos[6*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2])/(3360*a^4*d*(1 + Cos[c + d*
x])^4)

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fricas [A]  time = 0.47, size = 422, normalized size = 1.47 \[ -\frac {105 \, {\left ({\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (454 \, A - 216 \, B + 83 \, C\right )} \cos \left (d x + c\right )^{6} + {\left (24436 \, A - 11619 \, B + 4472 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (7184 \, A - 3411 \, B + 1318 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3196 \, A - 1509 \, B + 592 \, C\right )} \cos \left (d x + c\right )^{3} + 70 \, {\left (14 \, A - 6 \, B + 3 \, C\right )} \cos \left (d x + c\right )^{2} - 35 \, {\left (4 \, A - 3 \, B\right )} \cos \left (d x + c\right ) + 70 \, A\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{7} + 4 \, a^{4} d \cos \left (d x + c\right )^{6} + 6 \, a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + a^{4} d \cos \left (d x + c\right )^{3}\right )}} \]

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)^4/(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/420*(105*((44*A - 21*B + 8*C)*cos(d*x + c)^7 + 4*(44*A - 21*B + 8*C)*cos(d*x + c)^6 + 6*(44*A - 21*B + 8*C)
*cos(d*x + c)^5 + 4*(44*A - 21*B + 8*C)*cos(d*x + c)^4 + (44*A - 21*B + 8*C)*cos(d*x + c)^3)*log(sin(d*x + c)
+ 1) - 105*((44*A - 21*B + 8*C)*cos(d*x + c)^7 + 4*(44*A - 21*B + 8*C)*cos(d*x + c)^6 + 6*(44*A - 21*B + 8*C)*
cos(d*x + c)^5 + 4*(44*A - 21*B + 8*C)*cos(d*x + c)^4 + (44*A - 21*B + 8*C)*cos(d*x + c)^3)*log(-sin(d*x + c)
+ 1) - 2*(16*(454*A - 216*B + 83*C)*cos(d*x + c)^6 + (24436*A - 11619*B + 4472*C)*cos(d*x + c)^5 + 4*(7184*A -
3411*B + 1318*C)*cos(d*x + c)^4 + 4*(3196*A - 1509*B + 592*C)*cos(d*x + c)^3 + 70*(14*A - 6*B + 3*C)*cos(d*x
+ c)^2 - 35*(4*A - 3*B)*cos(d*x + c) + 70*A)*sin(d*x + c))/(a^4*d*cos(d*x + c)^7 + 4*a^4*d*cos(d*x + c)^6 + 6*
a^4*d*cos(d*x + c)^5 + 4*a^4*d*cos(d*x + c)^4 + a^4*d*cos(d*x + c)^3)

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giac [A]  time = 0.62, size = 407, normalized size = 1.42 \[ -\frac {\frac {420 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (44 \, A - 21 \, B + 8 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {280 \, {\left (78 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 124 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C \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 )}^{3} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 231 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 189 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21945 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11655 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, 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)^4/(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

-1/840*(420*(44*A - 21*B + 8*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 420*(44*A - 21*B + 8*C)*log(abs(tan(1
/2*d*x + 1/2*c) - 1))/a^4 + 280*(78*A*tan(1/2*d*x + 1/2*c)^5 - 27*B*tan(1/2*d*x + 1/2*c)^5 + 6*C*tan(1/2*d*x +
1/2*c)^5 - 124*A*tan(1/2*d*x + 1/2*c)^3 + 48*B*tan(1/2*d*x + 1/2*c)^3 - 12*C*tan(1/2*d*x + 1/2*c)^3 + 54*A*ta
n(1/2*d*x + 1/2*c) - 21*B*tan(1/2*d*x + 1/2*c) + 6*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^3*a^4
) - (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 - 15*B*a^24*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 +
231*A*a^24*tan(1/2*d*x + 1/2*c)^5 - 189*B*a^24*tan(1/2*d*x + 1/2*c)^5 + 147*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 20
65*A*a^24*tan(1/2*d*x + 1/2*c)^3 - 1365*B*a^24*tan(1/2*d*x + 1/2*c)^3 + 805*C*a^24*tan(1/2*d*x + 1/2*c)^3 + 21
945*A*a^24*tan(1/2*d*x + 1/2*c) - 11655*B*a^24*tan(1/2*d*x + 1/2*c) + 5145*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/
d

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maple [B]  time = 0.28, size = 626, normalized size = 2.18 \[ \frac {9 B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {11 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {59 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {13 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {23 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {209 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {9 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {7 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {A}{3 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {C}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {A}{3 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{d \,a^{4}}-\frac {5 A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {9 B}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{d \,a^{4}}+\frac {5 A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{2 d \,a^{4}}-\frac {13 A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{2 d \,a^{4}}-\frac {13 A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {22 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}}+\frac {22 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{4}} \]

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)^4/(a+a*cos(d*x+c))^4,x)

[Out]

9/2/d/a^4/(tan(1/2*d*x+1/2*c)-1)*B+7/40/d/a^4*C*tan(1/2*d*x+1/2*c)^5+23/24/d/a^4*C*tan(1/2*d*x+1/2*c)^3+49/8/d
/a^4*C*tan(1/2*d*x+1/2*c)+59/24/d/a^4*tan(1/2*d*x+1/2*c)^3*A-13/8/d/a^4*B*tan(1/2*d*x+1/2*c)^3+209/8/d/a^4*A*t
an(1/2*d*x+1/2*c)-111/8/d/a^4*B*tan(1/2*d*x+1/2*c)+1/56/d/a^4*tan(1/2*d*x+1/2*c)^7*A-1/56/d/a^4*B*tan(1/2*d*x+
1/2*c)^7+11/40/d/a^4*A*tan(1/2*d*x+1/2*c)^5-9/40/d/a^4*B*tan(1/2*d*x+1/2*c)^5+1/56/d/a^4*C*tan(1/2*d*x+1/2*c)^
7-1/3/d/a^4*A/(tan(1/2*d*x+1/2*c)-1)^3-1/2/d/a^4/(tan(1/2*d*x+1/2*c)+1)^2*B+1/2/d/a^4/(tan(1/2*d*x+1/2*c)-1)^2
*B-1/d/a^4/(tan(1/2*d*x+1/2*c)-1)*C-1/d/a^4/(tan(1/2*d*x+1/2*c)+1)*C-1/3/d/a^4*A/(tan(1/2*d*x+1/2*c)+1)^3+4/d/
a^4*ln(tan(1/2*d*x+1/2*c)-1)*C-5/2/d/a^4*A/(tan(1/2*d*x+1/2*c)-1)^2+9/2/d/a^4/(tan(1/2*d*x+1/2*c)+1)*B-4/d/a^4
*ln(tan(1/2*d*x+1/2*c)+1)*C+5/2/d/a^4*A/(tan(1/2*d*x+1/2*c)+1)^2-21/2/d/a^4*ln(tan(1/2*d*x+1/2*c)-1)*B-13/d/a^
4*A/(tan(1/2*d*x+1/2*c)-1)+21/2/d/a^4*ln(tan(1/2*d*x+1/2*c)+1)*B-13/d/a^4*A/(tan(1/2*d*x+1/2*c)+1)-22/d/a^4*A*
ln(tan(1/2*d*x+1/2*c)+1)+22/d/a^4*A*ln(tan(1/2*d*x+1/2*c)-1)

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maxima [B]  time = 0.37, size = 690, normalized size = 2.40 \[ \frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} - \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {18480 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {18480 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - 3 \, B {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, 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)^4/(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

1/840*(A*(560*(27*sin(d*x + c)/(cos(d*x + c) + 1) - 62*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 39*sin(d*x + c)^5
/(cos(d*x + c) + 1)^5)/(a^4 - 3*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^4*sin(d*x + c)^4/(cos(d*x + c) +
1)^4 - a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (21945*sin(d*x + c)/(cos(d*x + c) + 1) + 2065*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 + 231*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4
- 18480*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 18480*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4) -
3*B*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) - 9*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^4 - 2*a^4*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 + a^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) + 4
55*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(cos(d*x +
c) + 1)^7)/a^4 - 2940*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 2940*log(sin(d*x + c)/(cos(d*x + c) + 1)
- 1)/a^4) + C*(1680*sin(d*x + c)/((a^4 - a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (5145*
sin(d*x + c)/(cos(d*x + c) + 1) + 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 3360*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 3360
*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4))/d

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mupad [B]  time = 1.24, size = 345, normalized size = 1.20 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {7\,A-5\,B+3\,C}{40\,a^4}+\frac {A-B+C}{10\,a^4}\right )}{d}-\frac {\left (26\,A-9\,B+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (16\,B-\frac {124\,A}{3}-4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A-7\,B+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {21\,A-9\,B+C}{24\,a^4}+\frac {7\,A-5\,B+3\,C}{6\,a^4}+\frac {5\,\left (A-B+C\right )}{12\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,A-9\,B+C}{2\,a^4}+\frac {5\,\left (7\,A-5\,B+3\,C\right )}{4\,a^4}-\frac {5\,B-35\,A+5\,C}{8\,a^4}+\frac {5\,\left (A-B+C\right )}{2\,a^4}\right )}{d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (44\,A-21\,B+8\,C\right )}{a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d} \]

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)^4*(a + a*cos(c + d*x))^4),x)

[Out]

(tan(c/2 + (d*x)/2)^5*((7*A - 5*B + 3*C)/(40*a^4) + (A - B + C)/(10*a^4)))/d - (tan(c/2 + (d*x)/2)*(18*A - 7*B
+ 2*C) + tan(c/2 + (d*x)/2)^5*(26*A - 9*B + 2*C) - tan(c/2 + (d*x)/2)^3*((124*A)/3 - 16*B + 4*C))/(d*(3*a^4*t
an(c/2 + (d*x)/2)^2 - 3*a^4*tan(c/2 + (d*x)/2)^4 + a^4*tan(c/2 + (d*x)/2)^6 - a^4)) + (tan(c/2 + (d*x)/2)^3*((
21*A - 9*B + C)/(24*a^4) + (7*A - 5*B + 3*C)/(6*a^4) + (5*(A - B + C))/(12*a^4)))/d + (tan(c/2 + (d*x)/2)*((21
*A - 9*B + C)/(2*a^4) + (5*(7*A - 5*B + 3*C))/(4*a^4) - (5*B - 35*A + 5*C)/(8*a^4) + (5*(A - B + C))/(2*a^4)))
/d - (atanh(tan(c/2 + (d*x)/2))*(44*A - 21*B + 8*C))/(a^4*d) + (tan(c/2 + (d*x)/2)^7*(A - B + C))/(56*a^4*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**4/(a+a*cos(d*x+c))**4,x)

[Out]

Timed out

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