3.475 \(\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \, dx\)

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

[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)

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Rubi [A]  time = 0.232106, antiderivative size = 185, normalized size of antiderivative = 1., 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 = {3841, 4047, 2635, 8, 4044, 3013, 373} \[ -\frac{b \left (5 a^2+b^2\right ) \sin ^3(c+d x)}{3 d}+\frac{b \left (17 a^2+6 b^2\right ) \sin (c+d x)}{6 d}+\frac{a \left (5 a^2+18 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \left (5 a^2+18 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left (5 a^2+18 b^2\right )+\frac{13 a^2 b \sin ^5(c+d x)}{30 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))}{6 d} \]

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 3841

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(a^2*C
ot[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 4047

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]

Rule 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 4044

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 3013

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 373

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]

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{\operatorname{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{\operatorname{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*}

Mathematica [A]  time = 0.336222, size = 159, normalized size = 0.86 \[ \frac{360 b \left (5 a^2+2 b^2\right ) \sin (c+d x)+45 \left (5 a^3+16 a b^2\right ) \sin (2 (c+d x))+300 a^2 b \sin (3 (c+d x))+36 a^2 b \sin (5 (c+d x))+45 a^3 \sin (4 (c+d x))+5 a^3 \sin (6 (c+d x))+300 a^3 c+300 a^3 d x+90 a b^2 \sin (4 (c+d x))+1080 a b^2 c+1080 a b^2 d x+80 b^3 \sin (3 (c+d x))}{960 d} \]

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 = 0.058, size = 145, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{3\,{a}^{2}b\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+3\,a{b}^{2} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{{b}^{3} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}

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)

[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*a^2*b*(8/3+cos(d
*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+3*a*b^2*(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*(cos(d*x+c)^2+2)*sin(d*x+c))

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Maxima [A]  time = 1.1925, size = 196, normalized size = 1.06 \begin{align*} -\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{align*}

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 = 1.75999, size = 324, normalized size = 1.75 \begin{align*} \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{align*}

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

[Out]

Timed out

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Giac [B]  time = 1.28646, size = 582, normalized size = 3.15 \begin{align*} \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{align*}

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