3.27 \(\int (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \sec ^7(c+d x) \, dx\)

Optimal. Leaf size=225 \[ \frac{a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{120 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]

[Out]

(a^3*(23*A + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 30*C
)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((31*A + 30*C)
*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*
Tan[c + d*x])/(10*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)

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Rubi [A]  time = 0.617659, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3044, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{120 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]

[Out]

(a^3*(23*A + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 30*C
)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((31*A + 30*C)
*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*
Tan[c + d*x])/(10*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)

Rule 3044

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

Rule 2975

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

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(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))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

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 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]

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 8

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

Rubi steps

\begin{align*} \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx &=\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x))^3 (3 a A+2 a (A+3 C) \cos (c+d x)) \sec ^6(c+d x) \, dx}{6 a}\\ &=\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x))^2 \left (a^2 (31 A+30 C)+2 a^2 (8 A+15 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{30 a}\\ &=\frac{(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x)) \left (3 a^3 (73 A+90 C)+18 a^3 (7 A+10 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a}\\ &=\frac{(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int \left (3 a^4 (73 A+90 C)+\left (18 a^4 (7 A+10 C)+3 a^4 (73 A+90 C)\right ) \cos (c+d x)+18 a^4 (7 A+10 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a}\\ &=\frac{a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{\int \left (45 a^4 (23 A+30 C)+24 a^4 (34 A+45 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{360 a}\\ &=\frac{a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{8} \left (a^3 (23 A+30 C)\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{15} \left (a^3 (34 A+45 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a^3 (23 A+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{16} \left (a^3 (23 A+30 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (a^3 (34 A+45 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac{a^3 (23 A+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 (73 A+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac{(31 A+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac{A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 1.94335, size = 358, normalized size = 1.59 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (480 (23 A+30 C) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-160 (34 A+45 C) \sin (c)+2250 A \sin (2 c+d x)+7680 A \sin (c+2 d x)-480 A \sin (3 c+2 d x)+1955 A \sin (2 c+3 d x)+1955 A \sin (4 c+3 d x)+3264 A \sin (3 c+4 d x)+345 A \sin (4 c+5 d x)+345 A \sin (6 c+5 d x)+544 A \sin (5 c+6 d x)+30 (75 A+38 C) \sin (d x)+1140 C \sin (2 c+d x)+8160 C \sin (c+2 d x)-2640 C \sin (3 c+2 d x)+1590 C \sin (2 c+3 d x)+1590 C \sin (4 c+3 d x)+4080 C \sin (3 c+4 d x)-240 C \sin (5 c+4 d x)+450 C \sin (4 c+5 d x)+450 C \sin (6 c+5 d x)+720 C \sin (5 c+6 d x))\right )}{61440 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]

[Out]

-(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^6*(480*(23*A + 30*C)*Cos[c + d*x]^6*(Log[Cos[(c + d
*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-160*(34*A + 45*C)*Sin[c] + 3
0*(75*A + 38*C)*Sin[d*x] + 2250*A*Sin[2*c + d*x] + 1140*C*Sin[2*c + d*x] + 7680*A*Sin[c + 2*d*x] + 8160*C*Sin[
c + 2*d*x] - 480*A*Sin[3*c + 2*d*x] - 2640*C*Sin[3*c + 2*d*x] + 1955*A*Sin[2*c + 3*d*x] + 1590*C*Sin[2*c + 3*d
*x] + 1955*A*Sin[4*c + 3*d*x] + 1590*C*Sin[4*c + 3*d*x] + 3264*A*Sin[3*c + 4*d*x] + 4080*C*Sin[3*c + 4*d*x] -
240*C*Sin[5*c + 4*d*x] + 345*A*Sin[4*c + 5*d*x] + 450*C*Sin[4*c + 5*d*x] + 345*A*Sin[6*c + 5*d*x] + 450*C*Sin[
6*c + 5*d*x] + 544*A*Sin[5*c + 6*d*x] + 720*C*Sin[5*c + 6*d*x])))/(61440*d)

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Maple [A]  time = 0.114, size = 257, normalized size = 1.1 \begin{align*}{\frac{34\,A{a}^{3}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{17\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+3\,{\frac{{a}^{3}C\tan \left ( dx+c \right ) }{d}}+{\frac{23\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{23\,A{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{23\,A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{15\,{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{15\,{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{3\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x)

[Out]

34/15/d*A*a^3*tan(d*x+c)+17/15/d*A*a^3*tan(d*x+c)*sec(d*x+c)^2+3/d*a^3*C*tan(d*x+c)+23/24/d*A*a^3*tan(d*x+c)*s
ec(d*x+c)^3+23/16/d*A*a^3*sec(d*x+c)*tan(d*x+c)+23/16/d*A*a^3*ln(sec(d*x+c)+tan(d*x+c))+15/8/d*a^3*C*sec(d*x+c
)*tan(d*x+c)+15/8/d*a^3*C*ln(sec(d*x+c)+tan(d*x+c))+3/5/d*A*a^3*tan(d*x+c)*sec(d*x+c)^4+1/d*a^3*C*tan(d*x+c)*s
ec(d*x+c)^2+1/6/d*A*a^3*tan(d*x+c)*sec(d*x+c)^5+1/4/d*a^3*C*tan(d*x+c)*sec(d*x+c)^3

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Maxima [A]  time = 1.00758, size = 516, normalized size = 2.29 \begin{align*} \frac{96 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{3} + 160 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 480 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 5 \, A a^{3}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, A a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, C a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, C a^{3}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, C a^{3} \tan \left (d x + c\right )}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="maxima")

[Out]

1/480*(96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^3 + 160*(tan(d*x + c)^3 + 3*tan(d*x + c
))*A*a^3 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 - 5*A*a^3*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 3
3*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log
(sin(d*x + c) - 1)) - 90*A*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1)
- 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 30*C*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*
x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 360*C*a^3*(2*sin(d*x +
c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*C*a^3*tan(d*x + c))/d

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Fricas [A]  time = 1.50272, size = 474, normalized size = 2.11 \begin{align*} \frac{15 \,{\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (34 \, A + 45 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \,{\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \,{\left (17 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \,{\left (23 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 144 \, A a^{3} \cos \left (d x + c\right ) + 40 \, A a^{3}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="fricas")

[Out]

1/480*(15*(23*A + 30*C)*a^3*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 15*(23*A + 30*C)*a^3*cos(d*x + c)^6*log(-si
n(d*x + c) + 1) + 2*(16*(34*A + 45*C)*a^3*cos(d*x + c)^5 + 15*(23*A + 30*C)*a^3*cos(d*x + c)^4 + 16*(17*A + 15
*C)*a^3*cos(d*x + c)^3 + 10*(23*A + 6*C)*a^3*cos(d*x + c)^2 + 144*A*a^3*cos(d*x + c) + 40*A*a^3)*sin(d*x + c))
/(d*cos(d*x + c)^6)

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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+a*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)*sec(d*x+c)**7,x)

[Out]

Timed out

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Giac [A]  time = 1.26656, size = 378, normalized size = 1.68 \begin{align*} \frac{15 \,{\left (23 \, A a^{3} + 30 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (23 \, A a^{3} + 30 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (345 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 450 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1955 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 2550 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5940 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 5814 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 7500 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5130 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1575 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1470 \, C a^{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((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="giac")

[Out]

1/240*(15*(23*A*a^3 + 30*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(23*A*a^3 + 30*C*a^3)*log(abs(tan(1/2*
d*x + 1/2*c) - 1)) - 2*(345*A*a^3*tan(1/2*d*x + 1/2*c)^11 + 450*C*a^3*tan(1/2*d*x + 1/2*c)^11 - 1955*A*a^3*tan
(1/2*d*x + 1/2*c)^9 - 2550*C*a^3*tan(1/2*d*x + 1/2*c)^9 + 4554*A*a^3*tan(1/2*d*x + 1/2*c)^7 + 5940*C*a^3*tan(1
/2*d*x + 1/2*c)^7 - 5814*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 7500*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 3165*A*a^3*tan(1/2
*d*x + 1/2*c)^3 + 5130*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 1575*A*a^3*tan(1/2*d*x + 1/2*c) - 1470*C*a^3*tan(1/2*d*x
+ 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d