3.5.75 \(\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \, dx\) [475]

Optimal. Leaf size=185 \[ \frac {1}{16} a \left (5 a^2+18 b^2\right ) x+\frac {b \left (17 a^2+6 b^2\right ) \sin (c+d x)}{6 d}+\frac {a \left (5 a^2+18 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (5 a^2+18 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{6 d}-\frac {b \left (5 a^2+b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {13 a^2 b \sin ^5(c+d x)}{30 d} \]

[Out]

1/16*a*(5*a^2+18*b^2)*x+1/6*b*(17*a^2+6*b^2)*sin(d*x+c)/d+1/16*a*(5*a^2+18*b^2)*cos(d*x+c)*sin(d*x+c)/d+1/24*a
*(5*a^2+18*b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/6*a^2*cos(d*x+c)^5*(a+b*sec(d*x+c))*sin(d*x+c)/d-1/3*b*(5*a^2+b^2)
*sin(d*x+c)^3/d+13/30*a^2*b*sin(d*x+c)^5/d

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Rubi [A]
time = 0.17, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3926, 4132, 2715, 8, 4129, 3092, 380} \begin {gather*} -\frac {b \left (5 a^2+b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {b \left (17 a^2+6 b^2\right ) \sin (c+d x)}{6 d}+\frac {a \left (5 a^2+18 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a \left (5 a^2+18 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a x \left (5 a^2+18 b^2\right )+\frac {13 a^2 b \sin ^5(c+d x)}{30 d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]

[Out]

(a*(5*a^2 + 18*b^2)*x)/16 + (b*(17*a^2 + 6*b^2)*Sin[c + d*x])/(6*d) + (a*(5*a^2 + 18*b^2)*Cos[c + d*x]*Sin[c +
d*x])/(16*d) + (a*(5*a^2 + 18*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^5*(a + b*Sec[c + d
*x])*Sin[c + d*x])/(6*d) - (b*(5*a^2 + b^2)*Sin[c + d*x]^3)/(3*d) + (13*a^2*b*Sin[c + d*x]^5)/(30*d)

Rule 8

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

Rule 380

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 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 3092

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[-f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rule 3926

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^2*Co
t[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*n)), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x]
)^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2*(n + 1))*Csc[e + f*x] - b*(b^2*
n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2]
&& ((IntegerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))

Rule 4129

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*
x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rubi steps

\begin {align*} \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \, dx &=\frac {a^2 \cos ^5(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) \left (13 a^2 b+a \left (5 a^2+18 b^2\right ) \sec (c+d x)+2 b \left (2 a^2+3 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \cos ^5(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) \left (13 a^2 b+2 b \left (2 a^2+3 b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{6} \left (a \left (5 a^2+18 b^2\right )\right ) \int \cos ^4(c+d x) \, dx\\ &=\frac {a \left (5 a^2+18 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^3(c+d x) \left (2 b \left (2 a^2+3 b^2\right )+13 a^2 b \cos ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (a \left (5 a^2+18 b^2\right )\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {a \left (5 a^2+18 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (5 a^2+18 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{16} \left (a \left (5 a^2+18 b^2\right )\right ) \int 1 \, dx-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (13 a^2 b+2 b \left (2 a^2+3 b^2\right )-13 a^2 b x^2\right ) \, dx,x,-\sin (c+d x)\right )}{6 d}\\ &=\frac {1}{16} a \left (5 a^2+18 b^2\right ) x+\frac {a \left (5 a^2+18 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (5 a^2+18 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \left (17 a^2 b \left (1+\frac {6 b^2}{17 a^2}\right )-6 b \left (5 a^2+b^2\right ) x^2+13 a^2 b x^4\right ) \, dx,x,-\sin (c+d x)\right )}{6 d}\\ &=\frac {1}{16} a \left (5 a^2+18 b^2\right ) x+\frac {b \left (17 a^2+6 b^2\right ) \sin (c+d x)}{6 d}+\frac {a \left (5 a^2+18 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (5 a^2+18 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{6 d}-\frac {b \left (5 a^2+b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {13 a^2 b \sin ^5(c+d x)}{30 d}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 159, normalized size = 0.86 \begin {gather*} \frac {300 a^3 c+1080 a b^2 c+300 a^3 d x+1080 a b^2 d x+360 b \left (5 a^2+2 b^2\right ) \sin (c+d x)+45 \left (5 a^3+16 a b^2\right ) \sin (2 (c+d x))+300 a^2 b \sin (3 (c+d x))+80 b^3 \sin (3 (c+d x))+45 a^3 \sin (4 (c+d x))+90 a b^2 \sin (4 (c+d x))+36 a^2 b \sin (5 (c+d x))+5 a^3 \sin (6 (c+d x))}{960 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]

[Out]

(300*a^3*c + 1080*a*b^2*c + 300*a^3*d*x + 1080*a*b^2*d*x + 360*b*(5*a^2 + 2*b^2)*Sin[c + d*x] + 45*(5*a^3 + 16
*a*b^2)*Sin[2*(c + d*x)] + 300*a^2*b*Sin[3*(c + d*x)] + 80*b^3*Sin[3*(c + d*x)] + 45*a^3*Sin[4*(c + d*x)] + 90
*a*b^2*Sin[4*(c + d*x)] + 36*a^2*b*Sin[5*(c + d*x)] + 5*a^3*Sin[6*(c + d*x)])/(960*d)

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Maple [A]
time = 0.14, size = 145, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 b \,a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 b^{2} a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(145\)
default	\(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 b \,a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 b^{2} a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(145\)
risch	\(\frac {5 a^{3} x}{16}+\frac {9 a \,b^{2} x}{8}+\frac {15 a^{2} b \sin \left (d x +c \right )}{8 d}+\frac {3 \sin \left (d x +c \right ) b^{3}}{4 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 b \,a^{2} \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) b^{2} a}{32 d}+\frac {5 b \,a^{2} \sin \left (3 d x +3 c \right )}{16 d}+\frac {\sin \left (3 d x +3 c \right ) b^{3}}{12 d}+\frac {15 a^{3} \sin \left (2 d x +2 c \right )}{64 d}+\frac {3 \sin \left (2 d x +2 c \right ) b^{2} a}{4 d}\)	\(184\)
norman	\(\frac {\left (\frac {5}{16} a^{3}+\frac {9}{8} b^{2} a \right ) x +\left (-\frac {25}{8} a^{3}-\frac {45}{4} b^{2} a \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{4} a^{3}-\frac {9}{2} b^{2} a \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{4} a^{3}-\frac {9}{2} b^{2} a \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{4} a^{3}+\frac {9}{2} b^{2} a \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{4} a^{3}+\frac {9}{2} b^{2} a \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{4} a^{3}+\frac {9}{2} b^{2} a \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{4} a^{3}+\frac {9}{2} b^{2} a \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{16} a^{3}+\frac {9}{8} b^{2} a \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (11 a^{3}-48 b \,a^{2}+30 b^{2} a -16 b^{3}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {\left (11 a^{3}+48 b \,a^{2}+30 b^{2} a +16 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {\left (71 a^{3}-48 b \,a^{2}+54 b^{2} a -80 b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {\left (71 a^{3}+48 b \,a^{2}+54 b^{2} a +80 b^{3}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {\left (665 a^{3}-1104 b \,a^{2}-630 b^{2} a +80 b^{3}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {\left (665 a^{3}+1104 b \,a^{2}-630 b^{2} a -80 b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {\left (1375 a^{3}-2064 b \,a^{2}-90 b^{2} a -560 b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}-\frac {\left (1375 a^{3}+2064 b \,a^{2}-90 b^{2} a +560 b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}\)	\(557\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+3/5*b*a^2*(8/3+cos(d
*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+3*b^2*a*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+1/3*
b^3*(2+cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]
time = 0.26, size = 145, normalized size = 0.78 \begin {gather*} -\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{3}}{960 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/960*(5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^3 - 192*(3*sin(d
*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^2*b - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x +
2*c))*a*b^2 + 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*b^3)/d

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Fricas [A]
time = 2.64, size = 132, normalized size = 0.71 \begin {gather*} \frac {15 \, {\left (5 \, a^{3} + 18 \, a b^{2}\right )} d x + {\left (40 \, a^{3} \cos \left (d x + c\right )^{5} + 144 \, a^{2} b \cos \left (d x + c\right )^{4} + 10 \, {\left (5 \, a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 384 \, a^{2} b + 160 \, b^{3} + 16 \, {\left (12 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/240*(15*(5*a^3 + 18*a*b^2)*d*x + (40*a^3*cos(d*x + c)^5 + 144*a^2*b*cos(d*x + c)^4 + 10*(5*a^3 + 18*a*b^2)*c
os(d*x + c)^3 + 384*a^2*b + 160*b^3 + 16*(12*a^2*b + 5*b^3)*cos(d*x + c)^2 + 15*(5*a^3 + 18*a*b^2)*cos(d*x + c
))*sin(d*x + c))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(a+b*sec(d*x+c))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (171) = 342\).
time = 0.46, size = 431, normalized size = 2.33 \begin {gather*} \frac {15 \, {\left (5 \, a^{3} + 18 \, a b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (165 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 450 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 630 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 880 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3744 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 180 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1440 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3744 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 180 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1440 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 630 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 880 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 165 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 450 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, b^{3} \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 )}^{6}}}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/240*(15*(5*a^3 + 18*a*b^2)*(d*x + c) - 2*(165*a^3*tan(1/2*d*x + 1/2*c)^11 - 720*a^2*b*tan(1/2*d*x + 1/2*c)^1
1 + 450*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 240*b^3*tan(1/2*d*x + 1/2*c)^11 - 25*a^3*tan(1/2*d*x + 1/2*c)^9 - 1680
*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 630*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 880*b^3*tan(1/2*d*x + 1/2*c)^9 + 450*a^3*ta
n(1/2*d*x + 1/2*c)^7 - 3744*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 180*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 1440*b^3*tan(1/2
*d*x + 1/2*c)^7 - 450*a^3*tan(1/2*d*x + 1/2*c)^5 - 3744*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 180*a*b^2*tan(1/2*d*x +
1/2*c)^5 - 1440*b^3*tan(1/2*d*x + 1/2*c)^5 + 25*a^3*tan(1/2*d*x + 1/2*c)^3 - 1680*a^2*b*tan(1/2*d*x + 1/2*c)^
3 - 630*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 880*b^3*tan(1/2*d*x + 1/2*c)^3 - 165*a^3*tan(1/2*d*x + 1/2*c) - 720*a^2
*b*tan(1/2*d*x + 1/2*c) - 450*a*b^2*tan(1/2*d*x + 1/2*c) - 240*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)
^2 + 1)^6)/d

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Mupad [B]
time = 3.30, size = 350, normalized size = 1.89 \begin {gather*} \frac {\left (-\frac {11\,a^3}{8}+6\,a^2\,b-\frac {15\,a\,b^2}{4}+2\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,a^3}{24}+14\,a^2\,b-\frac {21\,a\,b^2}{4}+\frac {22\,b^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {15\,a^3}{4}+\frac {156\,a^2\,b}{5}-\frac {3\,a\,b^2}{2}+12\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,a^3}{4}+\frac {156\,a^2\,b}{5}+\frac {3\,a\,b^2}{2}+12\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {5\,a^3}{24}+14\,a^2\,b+\frac {21\,a\,b^2}{4}+\frac {22\,b^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,a^3}{8}+6\,a^2\,b+\frac {15\,a\,b^2}{4}+2\,b^3\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,a^2+18\,b^2\right )}{8\,\left (\frac {5\,a^3}{8}+\frac {9\,a\,b^2}{4}\right )}\right )\,\left (5\,a^2+18\,b^2\right )}{8\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*(a + b/cos(c + d*x))^3,x)

[Out]

(tan(c/2 + (d*x)/2)^3*((21*a*b^2)/4 + 14*a^2*b - (5*a^3)/24 + (22*b^3)/3) - tan(c/2 + (d*x)/2)^11*((15*a*b^2)/
4 - 6*a^2*b + (11*a^3)/8 - 2*b^3) + tan(c/2 + (d*x)/2)^9*(14*a^2*b - (21*a*b^2)/4 + (5*a^3)/24 + (22*b^3)/3) +
tan(c/2 + (d*x)/2)^5*((3*a*b^2)/2 + (156*a^2*b)/5 + (15*a^3)/4 + 12*b^3) - tan(c/2 + (d*x)/2)^7*((3*a*b^2)/2
- (156*a^2*b)/5 + (15*a^3)/4 - 12*b^3) + tan(c/2 + (d*x)/2)*((15*a*b^2)/4 + 6*a^2*b + (11*a^3)/8 + 2*b^3))/(d*
(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(
c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (a*atan((a*tan(c/2 + (d*x)/2)*(5*a^2 + 18*b^2))/(8*((9*a*b^2
)/4 + (5*a^3)/8)))*(5*a^2 + 18*b^2))/(8*d)

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