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

Optimal. Leaf size=210 \[ -\frac{3 a^3 \sin ^7(c+d x)}{7 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \tan (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}-\frac{293 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{603 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{805 a^3 x}{128} \]

[Out]

(-805*a^3*x)/128 - (a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (603*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (293*a^3*
Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^3*Cos[c + d*x]^7*Sin[c +
d*x])/(8*d) - (a^3*Sin[c + d*x]^3)/(3*d) - (2*a^3*Sin[c + d*x]^5)/(5*d) - (3*a^3*Sin[c + d*x]^7)/(7*d) + (3*a^
3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.389447, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3872, 2872, 2637, 2635, 8, 2633, 3770, 3767, 3768} \[ -\frac{3 a^3 \sin ^7(c+d x)}{7 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \tan (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}-\frac{293 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{603 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{805 a^3 x}{128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-805*a^3*x)/128 - (a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (603*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (293*a^3*
Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^3*Cos[c + d*x]^7*Sin[c +
d*x])/(8*d) - (a^3*Sin[c + d*x]^3)/(3*d) - (2*a^3*Sin[c + d*x]^5)/(5*d) - (3*a^3*Sin[c + d*x]^7)/(7*d) + (3*a^
3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, 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 2633

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 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 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]

Rubi steps

\begin{align*} \int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx &=-\int (-a-a \cos (c+d x))^3 \sin ^5(c+d x) \tan ^3(c+d x) \, dx\\ &=-\frac{\int \left (11 a^{11}+6 a^{11} \cos (c+d x)-14 a^{11} \cos ^2(c+d x)-14 a^{11} \cos ^3(c+d x)+6 a^{11} \cos ^4(c+d x)+11 a^{11} \cos ^5(c+d x)+a^{11} \cos ^6(c+d x)-3 a^{11} \cos ^7(c+d x)-a^{11} \cos ^8(c+d x)+a^{11} \sec (c+d x)-3 a^{11} \sec ^2(c+d x)-a^{11} \sec ^3(c+d x)\right ) \, dx}{a^8}\\ &=-11 a^3 x-a^3 \int \cos ^6(c+d x) \, dx+a^3 \int \cos ^8(c+d x) \, dx-a^3 \int \sec (c+d x) \, dx+a^3 \int \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^7(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx-\left (6 a^3\right ) \int \cos (c+d x) \, dx-\left (6 a^3\right ) \int \cos ^4(c+d x) \, dx-\left (11 a^3\right ) \int \cos ^5(c+d x) \, dx+\left (14 a^3\right ) \int \cos ^2(c+d x) \, dx+\left (14 a^3\right ) \int \cos ^3(c+d x) \, dx\\ &=-11 a^3 x-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{6 a^3 \sin (c+d x)}{d}+\frac{7 a^3 \cos (c+d x) \sin (c+d x)}{d}-\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} a^3 \int \sec (c+d x) \, dx-\frac{1}{6} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{8} \left (7 a^3\right ) \int \cos ^6(c+d x) \, dx-\frac{1}{2} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx+\left (7 a^3\right ) \int 1 \, dx-\frac{\left (3 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}+\frac{\left (11 a^3\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac{\left (14 a^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-4 a^3 x-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{19 a^3 \cos (c+d x) \sin (c+d x)}{4 d}-\frac{41 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^7(c+d x)}{7 d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac{1}{8} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{48} \left (35 a^3\right ) \int \cos ^4(c+d x) \, dx-\frac{1}{4} \left (9 a^3\right ) \int 1 \, dx\\ &=-\frac{25 a^3 x}{4}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{71 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac{293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^7(c+d x)}{7 d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac{1}{16} \left (5 a^3\right ) \int 1 \, dx+\frac{1}{64} \left (35 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{105 a^3 x}{16}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{603 a^3 \cos (c+d x) \sin (c+d x)}{128 d}-\frac{293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^7(c+d x)}{7 d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{128} \left (35 a^3\right ) \int 1 \, dx\\ &=-\frac{805 a^3 x}{128}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{603 a^3 \cos (c+d x) \sin (c+d x)}{128 d}-\frac{293 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^7(c+d x)}{7 d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 2.01652, size = 156, normalized size = 0.74 \[ \frac{a^3 \sec ^2(c+d x) \left (173600 \sin (c+d x)+1052520 \sin (2 (c+d x))-11648 \sin (3 (c+d x))+175280 \sin (4 (c+d x))+22784 \sin (5 (c+d x))-18095 \sin (6 (c+d x))-6288 \sin (7 (c+d x))+770 \sin (8 (c+d x))+720 \sin (9 (c+d x))+105 \sin (10 (c+d x))-1352400 (c+d x) \cos (2 (c+d x))-215040 \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))-1352400 c-1352400 d x\right )}{430080 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sec[c + d*x]^2*(-1352400*c - 1352400*d*x - 215040*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 - 1352400*(c + d*x
)*Cos[2*(c + d*x)] + 173600*Sin[c + d*x] + 1052520*Sin[2*(c + d*x)] - 11648*Sin[3*(c + d*x)] + 175280*Sin[4*(c
+ d*x)] + 22784*Sin[5*(c + d*x)] - 18095*Sin[6*(c + d*x)] - 6288*Sin[7*(c + d*x)] + 770*Sin[8*(c + d*x)] + 72
0*Sin[9*(c + d*x)] + 105*Sin[10*(c + d*x)]))/(430080*d)

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 235, normalized size = 1.1 \begin{align*}{\frac{23\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{7}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{161\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{48\,d}}+{\frac{805\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{192\,d}}+{\frac{805\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{128\,d}}-{\frac{805\,{a}^{3}x}{128}}-{\frac{805\,{a}^{3}c}{128\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{14\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{10\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{6\,d}}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{2\,d}}-{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

23/8/d*a^3*sin(d*x+c)^7*cos(d*x+c)+161/48*a^3*cos(d*x+c)*sin(d*x+c)^5/d+805/192*a^3*cos(d*x+c)*sin(d*x+c)^3/d+
805/128*a^3*cos(d*x+c)*sin(d*x+c)/d-805/128*a^3*x-805/128/d*a^3*c+1/14*a^3*sin(d*x+c)^7/d+1/10*a^3*sin(d*x+c)^
5/d+1/6*a^3*sin(d*x+c)^3/d+1/2*a^3*sin(d*x+c)/d-1/2/d*a^3*ln(sec(d*x+c)+tan(d*x+c))+3/d*a^3*sin(d*x+c)^9/cos(d
*x+c)+1/2/d*a^3*sin(d*x+c)^9/cos(d*x+c)^2

________________________________________________________________________________________

Maxima [A]  time = 1.80166, size = 393, normalized size = 1.87 \begin{align*} -\frac{1536 \,{\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{3} - 1792 \,{\left (12 \, \sin \left (d x + c\right )^{5} + 40 \, \sin \left (d x + c\right )^{3} - \frac{30 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 180 \, \sin \left (d x + c\right )\right )} a^{3} - 35 \,{\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 6720 \,{\left (105 \, d x + 105 \, c - \frac{87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} a^{3}}{107520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/107520*(1536*(30*sin(d*x + c)^7 + 42*sin(d*x + c)^5 + 70*sin(d*x + c)^3 - 105*log(sin(d*x + c) + 1) + 105*l
og(sin(d*x + c) - 1) + 210*sin(d*x + c))*a^3 - 1792*(12*sin(d*x + c)^5 + 40*sin(d*x + c)^3 - 30*sin(d*x + c)/(
sin(d*x + c)^2 - 1) - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 180*sin(d*x + c))*a^3 - 35*(128*
sin(2*d*x + 2*c)^3 + 840*d*x + 840*c + 3*sin(8*d*x + 8*c) + 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a^3 +
6720*(105*d*x + 105*c - (87*tan(d*x + c)^5 + 136*tan(d*x + c)^3 + 57*tan(d*x + c))/(tan(d*x + c)^6 + 3*tan(d*
x + c)^4 + 3*tan(d*x + c)^2 + 1) - 48*tan(d*x + c))*a^3)/d

________________________________________________________________________________________

Fricas [A]  time = 2.04888, size = 567, normalized size = 2.7 \begin{align*} -\frac{84525 \, a^{3} d x \cos \left (d x + c\right )^{2} + 3360 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3360 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (1680 \, a^{3} \cos \left (d x + c\right )^{9} + 5760 \, a^{3} \cos \left (d x + c\right )^{8} - 280 \, a^{3} \cos \left (d x + c\right )^{7} - 22656 \, a^{3} \cos \left (d x + c\right )^{6} - 20510 \, a^{3} \cos \left (d x + c\right )^{5} + 32512 \, a^{3} \cos \left (d x + c\right )^{4} + 63315 \, a^{3} \cos \left (d x + c\right )^{3} - 15616 \, a^{3} \cos \left (d x + c\right )^{2} + 40320 \, a^{3} \cos \left (d x + c\right ) + 6720 \, a^{3}\right )} \sin \left (d x + c\right )}{13440 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13440*(84525*a^3*d*x*cos(d*x + c)^2 + 3360*a^3*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 3360*a^3*cos(d*x + c)
^2*log(-sin(d*x + c) + 1) - (1680*a^3*cos(d*x + c)^9 + 5760*a^3*cos(d*x + c)^8 - 280*a^3*cos(d*x + c)^7 - 2265
6*a^3*cos(d*x + c)^6 - 20510*a^3*cos(d*x + c)^5 + 32512*a^3*cos(d*x + c)^4 + 63315*a^3*cos(d*x + c)^3 - 15616*
a^3*cos(d*x + c)^2 + 40320*a^3*cos(d*x + c) + 6720*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.2664, size = 329, normalized size = 1.57 \begin{align*} -\frac{84525 \,{\left (d x + c\right )} a^{3} + 6720 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 6720 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{13440 \,{\left (5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 7 \, 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 )}^{2}} + \frac{2 \,{\left (44205 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 303065 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 841981 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1123793 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 487983 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 490749 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 267225 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 44205 \, 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 )}^{8}}}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13440*(84525*(d*x + c)*a^3 + 6720*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 6720*a^3*log(abs(tan(1/2*d*x + 1
/2*c) - 1)) + 13440*(5*a^3*tan(1/2*d*x + 1/2*c)^3 - 7*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2
+ 2*(44205*a^3*tan(1/2*d*x + 1/2*c)^15 + 303065*a^3*tan(1/2*d*x + 1/2*c)^13 + 841981*a^3*tan(1/2*d*x + 1/2*c)
^11 + 1123793*a^3*tan(1/2*d*x + 1/2*c)^9 + 487983*a^3*tan(1/2*d*x + 1/2*c)^7 - 490749*a^3*tan(1/2*d*x + 1/2*c)
^5 - 267225*a^3*tan(1/2*d*x + 1/2*c)^3 - 44205*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d