3.70 \(\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=210 \[ \frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{2 d}-\frac{a b^2 \tan (c+d x) \sec ^3(c+d x)}{8 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}-\frac{b^3 \sec ^5(c+d x)}{5 d} \]

[Out]

(3*a^3*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(16*d) + (3*a^2*b*Sec[c + d*x]^5)/(5*d)
- (b^3*Sec[c + d*x]^5)/(5*d) + (b^3*Sec[c + d*x]^7)/(7*d) + (3*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a*b^2
*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (a*b^2*Sec[c + d*x]^3*Tan[c + d
*x])/(8*d) + (a*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(2*d)

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Rubi [A]  time = 0.219708, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3090, 3768, 3770, 2606, 30, 2611, 14} \[ \frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{2 d}-\frac{a b^2 \tan (c+d x) \sec ^3(c+d x)}{8 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}-\frac{b^3 \sec ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]

[Out]

(3*a^3*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(16*d) + (3*a^2*b*Sec[c + d*x]^5)/(5*d)
- (b^3*Sec[c + d*x]^5)/(5*d) + (b^3*Sec[c + d*x]^7)/(7*d) + (3*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a*b^2
*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (a*b^2*Sec[c + d*x]^3*Tan[c + d
*x])/(8*d) + (a*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(2*d)

Rule 3090

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Sym
bol] :> Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] &&
IntegerQ[m] && IGtQ[n, 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]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=\int \left (a^3 \sec ^5(c+d x)+3 a^2 b \sec ^5(c+d x) \tan (c+d x)+3 a b^2 \sec ^5(c+d x) \tan ^2(c+d x)+b^3 \sec ^5(c+d x) \tan ^3(c+d x)\right ) \, dx\\ &=a^3 \int \sec ^5(c+d x) \, dx+\left (3 a^2 b\right ) \int \sec ^5(c+d x) \tan (c+d x) \, dx+\left (3 a b^2\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+b^3 \int \sec ^5(c+d x) \tan ^3(c+d x) \, dx\\ &=\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}+\frac{1}{4} \left (3 a^3\right ) \int \sec ^3(c+d x) \, dx-\frac{1}{2} \left (a b^2\right ) \int \sec ^5(c+d x) \, dx+\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,\sec (c+d x)\right )}{d}+\frac{b^3 \operatorname{Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{a b^2 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}+\frac{1}{8} \left (3 a^3\right ) \int \sec (c+d x) \, dx-\frac{1}{8} \left (3 a b^2\right ) \int \sec ^3(c+d x) \, dx+\frac{b^3 \operatorname{Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^2 b \sec ^5(c+d x)}{5 d}-\frac{b^3 \sec ^5(c+d x)}{5 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}+\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{3 a b^2 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{a b^2 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}-\frac{1}{16} \left (3 a b^2\right ) \int \sec (c+d x) \, dx\\ &=\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 a^2 b \sec ^5(c+d x)}{5 d}-\frac{b^3 \sec ^5(c+d x)}{5 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}+\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{3 a b^2 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{a b^2 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}\\ \end{align*}

Mathematica [B]  time = 2.02354, size = 637, normalized size = 3.03 \[ \frac{\sec ^7(c+d x) \left (3584 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-3675 a \left (2 a^2-b^2\right ) \cos (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 )+10752 a^2 b+4340 a^3 \sin (2 (c+d x))+2800 a^3 \sin (4 (c+d x))+420 a^3 \sin (6 (c+d x))-4410 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-1470 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-210 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4410 a^3 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1470 a^3 \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+210 a^3 \cos (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+6790 a b^2 \sin (2 (c+d x))-1400 a b^2 \sin (4 (c+d x))-210 a b^2 \sin (6 (c+d x))+2205 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+735 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-2205 a b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-735 a b^2 \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-105 a b^2 \cos (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1536 b^3\right )}{35840 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^7*(10752*a^2*b + 1536*b^3 + 3584*(3*a^2*b - b^3)*Cos[2*(c + d*x)] - 4410*a^3*Cos[3*(c + d*x)]*Lo
g[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2205*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]
- 1470*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 735*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c +
d*x)/2] - Sin[(c + d*x)/2]] - 210*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*a*b^2*C
os[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3675*a*(2*a^2 - b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x
)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4410*a^3*Cos[3*(c + d*x)]*Log[Cos[(c +
d*x)/2] + Sin[(c + d*x)/2]] - 2205*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1470*a^3*
Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 735*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] +
Sin[(c + d*x)/2]] + 210*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 105*a*b^2*Cos[7*(c + d
*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4340*a^3*Sin[2*(c + d*x)] + 6790*a*b^2*Sin[2*(c + d*x)] + 2800
*a^3*Sin[4*(c + d*x)] - 1400*a*b^2*Sin[4*(c + d*x)] + 420*a^3*Sin[6*(c + d*x)] - 210*a*b^2*Sin[6*(c + d*x)]))/
(35840*d)

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Maple [A]  time = 0.133, size = 328, normalized size = 1.6 \begin{align*}{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{3\,{a}^{2}b}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a{b}^{2}\sin \left ( dx+c \right ) }{16\,d}}-{\frac{3\,a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{7\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{3\,{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d\cos \left ( dx+c \right ) }}-{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{35\,d}}-{\frac{2\,{b}^{3}\cos \left ( dx+c \right ) }{35\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^8*(a*cos(d*x+c)+b*sin(d*x+c))^3,x)

[Out]

1/4*a^3*sec(d*x+c)^3*tan(d*x+c)/d+3/8*a^3*sec(d*x+c)*tan(d*x+c)/d+3/8/d*a^3*ln(sec(d*x+c)+tan(d*x+c))+3/5/d*a^
2*b/cos(d*x+c)^5+1/2/d*a*b^2*sin(d*x+c)^3/cos(d*x+c)^6+3/8/d*a*b^2*sin(d*x+c)^3/cos(d*x+c)^4+3/16/d*a*b^2*sin(
d*x+c)^3/cos(d*x+c)^2+3/16*a*b^2*sin(d*x+c)/d-3/16/d*a*b^2*ln(sec(d*x+c)+tan(d*x+c))+1/7/d*b^3*sin(d*x+c)^4/co
s(d*x+c)^7+3/35/d*b^3*sin(d*x+c)^4/cos(d*x+c)^5+1/35/d*b^3*sin(d*x+c)^4/cos(d*x+c)^3-1/35/d*b^3*sin(d*x+c)^4/c
os(d*x+c)-1/35/d*cos(d*x+c)*sin(d*x+c)^2*b^3-2/35*b^3*cos(d*x+c)/d

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Maxima [A]  time = 1.22831, size = 281, normalized size = 1.34 \begin{align*} \frac{35 \, a b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \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} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 70 \, 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 )} + \frac{672 \, a^{2} b}{\cos \left (d x + c\right )^{5}} - \frac{32 \,{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} b^{3}}{\cos \left (d x + c\right )^{7}}}{1120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/1120*(35*a*b^2*(2*(3*sin(d*x + c)^5 - 8*sin(d*x + c)^3 - 3*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4
+ 3*sin(d*x + c)^2 - 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 70*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)) +
672*a^2*b/cos(d*x + c)^5 - 32*(7*cos(d*x + c)^2 - 5)*b^3/cos(d*x + c)^7)/d

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Fricas [A]  time = 0.554504, size = 410, normalized size = 1.95 \begin{align*} \frac{105 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 160 \, b^{3} + 224 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 70 \,{\left (3 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 8 \, a b^{2} \cos \left (d x + c\right ) + 2 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1120 \, d \cos \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/1120*(105*(2*a^3 - a*b^2)*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(2*a^3 - a*b^2)*cos(d*x + c)^7*log(-sin
(d*x + c) + 1) + 160*b^3 + 224*(3*a^2*b - b^3)*cos(d*x + c)^2 + 70*(3*(2*a^3 - a*b^2)*cos(d*x + c)^5 + 8*a*b^2
*cos(d*x + c) + 2*(2*a^3 - a*b^2)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^7)

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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(sec(d*x+c)**8*(a*cos(d*x+c)+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.20132, size = 628, normalized size = 2.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/560*(105*(2*a^3 - a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(2*a^3 - a*b^2)*log(abs(tan(1/2*d*x + 1/2*
c) - 1)) + 2*(350*a^3*tan(1/2*d*x + 1/2*c)^13 + 105*a*b^2*tan(1/2*d*x + 1/2*c)^13 - 1680*a^2*b*tan(1/2*d*x + 1
/2*c)^12 - 840*a^3*tan(1/2*d*x + 1/2*c)^11 + 1540*a*b^2*tan(1/2*d*x + 1/2*c)^11 + 3360*a^2*b*tan(1/2*d*x + 1/2
*c)^10 - 1120*b^3*tan(1/2*d*x + 1/2*c)^10 + 630*a^3*tan(1/2*d*x + 1/2*c)^9 + 1085*a*b^2*tan(1/2*d*x + 1/2*c)^9
- 5040*a^2*b*tan(1/2*d*x + 1/2*c)^8 - 1120*b^3*tan(1/2*d*x + 1/2*c)^8 + 6720*a^2*b*tan(1/2*d*x + 1/2*c)^6 - 2
240*b^3*tan(1/2*d*x + 1/2*c)^6 - 630*a^3*tan(1/2*d*x + 1/2*c)^5 - 1085*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 3696*a^2
*b*tan(1/2*d*x + 1/2*c)^4 - 448*b^3*tan(1/2*d*x + 1/2*c)^4 + 840*a^3*tan(1/2*d*x + 1/2*c)^3 - 1540*a*b^2*tan(1
/2*d*x + 1/2*c)^3 + 672*a^2*b*tan(1/2*d*x + 1/2*c)^2 - 224*b^3*tan(1/2*d*x + 1/2*c)^2 - 350*a^3*tan(1/2*d*x +
1/2*c) - 105*a*b^2*tan(1/2*d*x + 1/2*c) - 336*a^2*b + 32*b^3)/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d