3.2.45 \(\int \frac {\cos ^3(c+d x) (A+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^3} \, dx\) [145]

Optimal. Leaf size=216 \[ -\frac {(23 A+6 C) x}{2 a^3}+\frac {4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d} \]

[Out]

-1/2*(23*A+6*C)*x/a^3+4/5*(34*A+9*C)*sin(d*x+c)/a^3/d-1/2*(23*A+6*C)*cos(d*x+c)*sin(d*x+c)/a^3/d-1/5*(A+C)*cos
(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c))^3-1/15*(13*A+3*C)*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2-1/3*(
23*A+6*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a^3+a^3*sec(d*x+c))-4/15*(34*A+9*C)*sin(d*x+c)^3/a^3/d

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Rubi [A]
time = 0.34, antiderivative size = 216, normalized size of antiderivative = 1.00, 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 = {4170, 4105, 3872, 2713, 2715, 8} \begin {gather*} -\frac {4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d}+\frac {4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {(23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {x (23 A+6 C)}{2 a^3}-\frac {(13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((23*A + 6*C)*x)/a^3 + (4*(34*A + 9*C)*Sin[c + d*x])/(5*a^3*d) - ((23*A + 6*C)*Cos[c + d*x]*Sin[c + d*x])
/(2*a^3*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((13*A + 3*C)*Cos[c + d*x]^2
*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((23*A + 6*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a^3 + a^3*Se
c[c + d*x])) - (4*(34*A + 9*C)*Sin[c + d*x]^3)/(15*a^3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

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

Rule 2715

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

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4105

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

Rule 4170

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> Simp[(-a)*(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*
(2*m + 1))), x] + Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*C*n + A*b
*(2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x
] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]

Rubi steps

\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) (-a (8 A+3 C)+5 a A \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos ^3(c+d x) \left (-9 a^2 (7 A+2 C)+4 a^2 (13 A+3 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \cos ^3(c+d x) \left (-12 a^3 (34 A+9 C)+15 a^3 (23 A+6 C) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(23 A+6 C) \int \cos ^2(c+d x) \, dx}{a^3}+\frac {(4 (34 A+9 C)) \int \cos ^3(c+d x) \, dx}{5 a^3}\\ &=-\frac {(23 A+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(23 A+6 C) \int 1 \, dx}{2 a^3}-\frac {(4 (34 A+9 C)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=-\frac {(23 A+6 C) x}{2 a^3}+\frac {4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(455\) vs. \(2(216)=432\).
time = 1.94, size = 455, normalized size = 2.11 \begin {gather*} -\frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (600 (23 A+6 C) d x \cos \left (\frac {d x}{2}\right )+600 (23 A+6 C) d x \cos \left (c+\frac {d x}{2}\right )+6900 A d x \cos \left (c+\frac {3 d x}{2}\right )+1800 C d x \cos \left (c+\frac {3 d x}{2}\right )+6900 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+1800 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+1380 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+360 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+1380 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+360 C d x \cos \left (3 c+\frac {5 d x}{2}\right )-20410 A \sin \left (\frac {d x}{2}\right )-7020 C \sin \left (\frac {d x}{2}\right )+11110 A \sin \left (c+\frac {d x}{2}\right )+4500 C \sin \left (c+\frac {d x}{2}\right )-15380 A \sin \left (c+\frac {3 d x}{2}\right )-4860 C \sin \left (c+\frac {3 d x}{2}\right )+380 A \sin \left (2 c+\frac {3 d x}{2}\right )+900 C \sin \left (2 c+\frac {3 d x}{2}\right )-4777 A \sin \left (2 c+\frac {5 d x}{2}\right )-1452 C \sin \left (2 c+\frac {5 d x}{2}\right )-1625 A \sin \left (3 c+\frac {5 d x}{2}\right )-300 C \sin \left (3 c+\frac {5 d x}{2}\right )-230 A \sin \left (3 c+\frac {7 d x}{2}\right )-60 C \sin \left (3 c+\frac {7 d x}{2}\right )-230 A \sin \left (4 c+\frac {7 d x}{2}\right )-60 C \sin \left (4 c+\frac {7 d x}{2}\right )+20 A \sin \left (4 c+\frac {9 d x}{2}\right )+20 A \sin \left (5 c+\frac {9 d x}{2}\right )-5 A \sin \left (5 c+\frac {11 d x}{2}\right )-5 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{3840 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3840*(Sec[c/2]*Sec[(c + d*x)/2]^5*(600*(23*A + 6*C)*d*x*Cos[(d*x)/2] + 600*(23*A + 6*C)*d*x*Cos[c + (d*x)/2
] + 6900*A*d*x*Cos[c + (3*d*x)/2] + 1800*C*d*x*Cos[c + (3*d*x)/2] + 6900*A*d*x*Cos[2*c + (3*d*x)/2] + 1800*C*d
*x*Cos[2*c + (3*d*x)/2] + 1380*A*d*x*Cos[2*c + (5*d*x)/2] + 360*C*d*x*Cos[2*c + (5*d*x)/2] + 1380*A*d*x*Cos[3*
c + (5*d*x)/2] + 360*C*d*x*Cos[3*c + (5*d*x)/2] - 20410*A*Sin[(d*x)/2] - 7020*C*Sin[(d*x)/2] + 11110*A*Sin[c +
(d*x)/2] + 4500*C*Sin[c + (d*x)/2] - 15380*A*Sin[c + (3*d*x)/2] - 4860*C*Sin[c + (3*d*x)/2] + 380*A*Sin[2*c +
(3*d*x)/2] + 900*C*Sin[2*c + (3*d*x)/2] - 4777*A*Sin[2*c + (5*d*x)/2] - 1452*C*Sin[2*c + (5*d*x)/2] - 1625*A*
Sin[3*c + (5*d*x)/2] - 300*C*Sin[3*c + (5*d*x)/2] - 230*A*Sin[3*c + (7*d*x)/2] - 60*C*Sin[3*c + (7*d*x)/2] - 2
30*A*Sin[4*c + (7*d*x)/2] - 60*C*Sin[4*c + (7*d*x)/2] + 20*A*Sin[4*c + (9*d*x)/2] + 20*A*Sin[5*c + (9*d*x)/2]
- 5*A*Sin[5*c + (11*d*x)/2] - 5*A*Sin[6*c + (11*d*x)/2]))/(a^3*d)

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Maple [A]
time = 0.66, size = 182, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {10 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-\frac {17 A}{4}-\frac {C}{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {19 A}{3}-C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {11 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-4 \left (23 A +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(182\)
default	\(\frac {\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {10 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-\frac {17 A}{4}-\frac {C}{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {19 A}{3}-C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {11 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-4 \left (23 A +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(182\)
risch	\(-\frac {23 A x}{2 a^{3}}-\frac {3 x C}{a^{3}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {3 i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {27 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{3} d}+\frac {27 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{3} d}-\frac {3 i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {2 i \left (225 A \,{\mathrm e}^{4 i \left (d x +c \right )}+90 C \,{\mathrm e}^{4 i \left (d x +c \right )}+810 A \,{\mathrm e}^{3 i \left (d x +c \right )}+300 C \,{\mathrm e}^{3 i \left (d x +c \right )}+1160 A \,{\mathrm e}^{2 i \left (d x +c \right )}+420 C \,{\mathrm e}^{2 i \left (d x +c \right )}+760 \,{\mathrm e}^{i \left (d x +c \right )} A +270 C \,{\mathrm e}^{i \left (d x +c \right )}+197 A +72 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\)	\(293\)
norman	\(\frac {\frac {\left (23 A +6 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (23 A +6 C \right ) x}{2 a}+\frac {\left (A +C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}-\frac {\left (11 A +6 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}-\frac {2 \left (19 A +5 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (23 A +6 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (23 A +6 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {\left (93 A +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (127 A +39 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}+\frac {\left (199 A +59 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {\left (207 A +52 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{2}}\)	\(296\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/4/d/a^3*(1/5*A*tan(1/2*d*x+1/2*c)^5+1/5*C*tan(1/2*d*x+1/2*c)^5-10/3*A*tan(1/2*d*x+1/2*c)^3-2*C*tan(1/2*d*x+1
/2*c)^3+49*A*tan(1/2*d*x+1/2*c)+17*C*tan(1/2*d*x+1/2*c)-16*((-17/4*A-1/2*C)*tan(1/2*d*x+1/2*c)^5+(-19/3*A-C)*t
an(1/2*d*x+1/2*c)^3+(-11/4*A-1/2*C)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^3-4*(23*A+6*C)*arctan(tan(1/2
*d*x+1/2*c)))

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Maxima [A]
time = 0.51, size = 365, normalized size = 1.69 \begin {gather*} \frac {A {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1380 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 3 \, C {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \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 {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(A*(20*(33*sin(d*x + c)/(cos(d*x + c) + 1) + 76*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 51*sin(d*x + c)^5/(
cos(d*x + c) + 1)^5)/(a^3 + 3*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1
)^4 + a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (735*sin(d*x + c)/(cos(d*x + c) + 1) - 50*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 1380*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a
^3) + 3*C*(40*sin(d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (85*sin(d*x
+ c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 -
120*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3))/d

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Fricas [A]
time = 2.41, size = 201, normalized size = 0.93 \begin {gather*} -\frac {15 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (23 \, A + 6 \, C\right )} d x - {\left (10 \, A \cos \left (d x + c\right )^{5} - 15 \, A \cos \left (d x + c\right )^{4} + 5 \, {\left (19 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (869 \, A + 234 \, C\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (143 \, A + 38 \, C\right )} \cos \left (d x + c\right ) + 544 \, A + 144 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*(23*A + 6*C)*d*x*cos(d*x + c)^3 + 45*(23*A + 6*C)*d*x*cos(d*x + c)^2 + 45*(23*A + 6*C)*d*x*cos(d*x +
c) + 15*(23*A + 6*C)*d*x - (10*A*cos(d*x + c)^5 - 15*A*cos(d*x + c)^4 + 5*(19*A + 6*C)*cos(d*x + c)^3 + (869*
A + 234*C)*cos(d*x + c)^2 + 9*(143*A + 38*C)*cos(d*x + c) + 544*A + 144*C)*sin(d*x + c))/(a^3*d*cos(d*x + c)^3
+ 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(A*cos(c + d*x)**3/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x) + Integral(C*cos(c
+ d*x)**3*sec(c + d*x)**2/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x))/a**3

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Giac [A]
time = 0.51, size = 228, normalized size = 1.06 \begin {gather*} -\frac {\frac {30 \, {\left (d x + c\right )} {\left (23 \, A + 6 \, C\right )}}{a^{3}} - \frac {20 \, {\left (51 \, A \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} + 76 \, A \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} + 33 \, A \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^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 50 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*(d*x + c)*(23*A + 6*C)/a^3 - 20*(51*A*tan(1/2*d*x + 1/2*c)^5 + 6*C*tan(1/2*d*x + 1/2*c)^5 + 76*A*tan
(1/2*d*x + 1/2*c)^3 + 12*C*tan(1/2*d*x + 1/2*c)^3 + 33*A*tan(1/2*d*x + 1/2*c) + 6*C*tan(1/2*d*x + 1/2*c))/((ta
n(1/2*d*x + 1/2*c)^2 + 1)^3*a^3) - (3*A*a^12*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^12*tan(1/2*d*x + 1/2*c)^5 - 50*A*a
^12*tan(1/2*d*x + 1/2*c)^3 - 30*C*a^12*tan(1/2*d*x + 1/2*c)^3 + 735*A*a^12*tan(1/2*d*x + 1/2*c) + 255*C*a^12*t
an(1/2*d*x + 1/2*c))/a^15)/d

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Mupad [B]
time = 2.68, size = 231, normalized size = 1.07 \begin {gather*} \frac {\left (17\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {76\,A}{3}+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}-\frac {x\,\left (23\,A+6\,C\right )}{2\,a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{3\,a^3}+\frac {6\,A+2\,C}{12\,a^3}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^3}+\frac {6\,A+2\,C}{a^3}+\frac {15\,A-C}{4\,a^3}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A+C\right )}{20\,a^3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)^5*(17*A + 2*C) + tan(c/2 + (d*x)/2)^3*((76*A)/3 + 4*C) + tan(c/2 + (d*x)/2)*(11*A + 2*C))/
(d*(3*a^3*tan(c/2 + (d*x)/2)^2 + 3*a^3*tan(c/2 + (d*x)/2)^4 + a^3*tan(c/2 + (d*x)/2)^6 + a^3)) - (x*(23*A + 6*
C))/(2*a^3) - (tan(c/2 + (d*x)/2)^3*((A + C)/(3*a^3) + (6*A + 2*C)/(12*a^3)))/d + (tan(c/2 + (d*x)/2)*((5*(A +
C))/(2*a^3) + (6*A + 2*C)/a^3 + (15*A - C)/(4*a^3)))/d + (tan(c/2 + (d*x)/2)^5*(A + C))/(20*a^3*d)

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