3.420 \(\int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=349 \[ \frac{a b \left (1664 a^4 b^2+2789 a^2 b^4+40 a^6+512 b^6\right ) \cos (c+d x)}{20 d}+\frac{b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left (992 a^2 b^2+120 a^4+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left (624 a^2 b^2+40 a^4+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{b^2 \left (2248 a^4 b^2+2502 a^2 b^4+80 a^6+175 b^6\right ) \sin (c+d x) \cos (c+d x)}{80 d}-\frac{7}{16} b^2 x \left (240 a^4 b^2+120 a^2 b^4+64 a^6+5 b^6\right )+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d} \]

[Out]

(-7*b^2*(64*a^6 + 240*a^4*b^2 + 120*a^2*b^4 + 5*b^6)*x)/16 + (a*b*(40*a^6 + 1664*a^4*b^2 + 2789*a^2*b^4 + 512*
b^6)*Cos[c + d*x])/(20*d) + (b^2*(80*a^6 + 2248*a^4*b^2 + 2502*a^2*b^4 + 175*b^6)*Cos[c + d*x]*Sin[c + d*x])/(
80*d) + (a*b*(40*a^4 + 624*a^2*b^2 + 337*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(40*d) + (b*(120*a^4 + 992*
a^2*b^2 + 175*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(120*d) + (a*b*(30*a^2 + 113*b^2)*Cos[c + d*x]*(a + b*
Sin[c + d*x])^4)/(30*d) + (b*(6*a^2 + 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(6*d) + (a*b*Cos[c + d*x]*(a
+ b*Sin[c + d*x])^6)/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/d

________________________________________________________________________________________

Rubi [A]  time = 0.563405, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2691, 2753, 2734} \[ \frac{a b \left (1664 a^4 b^2+2789 a^2 b^4+40 a^6+512 b^6\right ) \cos (c+d x)}{20 d}+\frac{b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left (992 a^2 b^2+120 a^4+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left (624 a^2 b^2+40 a^4+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{b^2 \left (2248 a^4 b^2+2502 a^2 b^4+80 a^6+175 b^6\right ) \sin (c+d x) \cos (c+d x)}{80 d}-\frac{7}{16} b^2 x \left (240 a^4 b^2+120 a^2 b^4+64 a^6+5 b^6\right )+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]

[Out]

(-7*b^2*(64*a^6 + 240*a^4*b^2 + 120*a^2*b^4 + 5*b^6)*x)/16 + (a*b*(40*a^6 + 1664*a^4*b^2 + 2789*a^2*b^4 + 512*
b^6)*Cos[c + d*x])/(20*d) + (b^2*(80*a^6 + 2248*a^4*b^2 + 2502*a^2*b^4 + 175*b^6)*Cos[c + d*x]*Sin[c + d*x])/(
80*d) + (a*b*(40*a^4 + 624*a^2*b^2 + 337*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(40*d) + (b*(120*a^4 + 992*
a^2*b^2 + 175*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(120*d) + (a*b*(30*a^2 + 113*b^2)*Cos[c + d*x]*(a + b*
Sin[c + d*x])^4)/(30*d) + (b*(6*a^2 + 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(6*d) + (a*b*Cos[c + d*x]*(a
+ b*Sin[c + d*x])^6)/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/d

Rule 2691

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[((g*C
os[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(b + a*Sin[e + f*x]))/(f*g*(p + 1)), x] + Dist[1/(g^2*(p + 1
)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin
[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[
2*m, 2*p] || IntegerQ[m])

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2734

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

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\int (a+b \sin (c+d x))^6 \left (7 b^2+7 a b \sin (c+d x)\right ) \, dx\\ &=\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac{1}{7} \int (a+b \sin (c+d x))^5 \left (91 a b^2+7 b \left (6 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac{1}{42} \int (a+b \sin (c+d x))^4 \left (7 b^2 \left (108 a^2+35 b^2\right )+7 a b \left (30 a^2+113 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac{1}{210} \int (a+b \sin (c+d x))^3 \left (231 a b^2 \left (20 a^2+19 b^2\right )+7 b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac{1}{840} \int (a+b \sin (c+d x))^2 \left (21 b^2 \left (1000 a^4+1828 a^2 b^2+175 b^4\right )+63 a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac{\int (a+b \sin (c+d x)) \left (63 a b^2 \left (1080 a^4+3076 a^2 b^2+849 b^4\right )+63 b \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \sin (c+d x)\right ) \, dx}{2520}\\ &=-\frac{7}{16} b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) x+\frac{a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{20 d}+\frac{b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}\\ \end{align*}

Mathematica [A]  time = 1.12468, size = 313, normalized size = 0.9 \[ \frac{\sec (c+d x) \left (53760 a^6 b^2 \sin (c+d x)+151200 a^4 b^4 \sin (c+d x)+16800 a^4 b^4 \sin (3 (c+d x))+67200 a^2 b^6 \sin (c+d x)+12600 a^2 b^6 \sin (3 (c+d x))-840 a^2 b^6 \sin (5 (c+d x))-4480 a^3 b^5 \cos (4 (c+d x))-840 b^2 \left (240 a^4 b^2+120 a^2 b^4+64 a^6+5 b^6\right ) (c+d x) \cos (c+d x)+1120 \left (80 a^3 b^5+48 a^5 b^3+15 a b^7\right ) \cos (2 (c+d x))+161280 a^5 b^3+201600 a^3 b^5+15360 a^7 b+1920 a^8 \sin (c+d x)-1344 a b^7 \cos (4 (c+d x))+96 a b^7 \cos (6 (c+d x))+33600 a b^7+2625 b^8 \sin (c+d x)+630 b^8 \sin (3 (c+d x))-70 b^8 \sin (5 (c+d x))+5 b^8 \sin (7 (c+d x))\right )}{1920 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]

[Out]

(Sec[c + d*x]*(15360*a^7*b + 161280*a^5*b^3 + 201600*a^3*b^5 + 33600*a*b^7 - 840*b^2*(64*a^6 + 240*a^4*b^2 + 1
20*a^2*b^4 + 5*b^6)*(c + d*x)*Cos[c + d*x] + 1120*(48*a^5*b^3 + 80*a^3*b^5 + 15*a*b^7)*Cos[2*(c + d*x)] - 4480
*a^3*b^5*Cos[4*(c + d*x)] - 1344*a*b^7*Cos[4*(c + d*x)] + 96*a*b^7*Cos[6*(c + d*x)] + 1920*a^8*Sin[c + d*x] +
53760*a^6*b^2*Sin[c + d*x] + 151200*a^4*b^4*Sin[c + d*x] + 67200*a^2*b^6*Sin[c + d*x] + 2625*b^8*Sin[c + d*x]
+ 16800*a^4*b^4*Sin[3*(c + d*x)] + 12600*a^2*b^6*Sin[3*(c + d*x)] + 630*b^8*Sin[3*(c + d*x)] - 840*a^2*b^6*Sin
[5*(c + d*x)] - 70*b^8*Sin[5*(c + d*x)] + 5*b^8*Sin[7*(c + d*x)]))/(1920*d)

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Maple [A]  time = 0.066, size = 406, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{8}\tan \left ( dx+c \right ) +8\,{\frac{{a}^{7}b}{\cos \left ( dx+c \right ) }}+28\,{a}^{6}{b}^{2} \left ( \tan \left ( dx+c \right ) -dx-c \right ) +56\,{a}^{5}{b}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}+ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +70\,{a}^{4}{b}^{4} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{\cos \left ( dx+c \right ) }}+ \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{3}+3/2\,\sin \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) -3/2\,dx-3/2\,c \right ) +56\,{a}^{3}{b}^{5} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{\cos \left ( dx+c \right ) }}+ \left ( 8/3+ \left ( \sin \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +28\,{a}^{2}{b}^{6} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{\cos \left ( dx+c \right ) }}+ \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\sin \left ( dx+c \right ) }{8}} \right ) \cos \left ( dx+c \right ) -{\frac{15\,dx}{8}}-{\frac{15\,c}{8}} \right ) +8\,a{b}^{7} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{\cos \left ( dx+c \right ) }}+ \left ({\frac{16}{5}}+ \left ( \sin \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}+8/5\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +{b}^{8} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{\cos \left ( dx+c \right ) }}+ \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( dx+c \right ) }{16}} \right ) \cos \left ( dx+c \right ) -{\frac{35\,dx}{16}}-{\frac{35\,c}{16}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x)

[Out]

1/d*(a^8*tan(d*x+c)+8*a^7*b/cos(d*x+c)+28*a^6*b^2*(tan(d*x+c)-d*x-c)+56*a^5*b^3*(sin(d*x+c)^4/cos(d*x+c)+(2+si
n(d*x+c)^2)*cos(d*x+c))+70*a^4*b^4*(sin(d*x+c)^5/cos(d*x+c)+(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)-3/2*d*x-3
/2*c)+56*a^3*b^5*(sin(d*x+c)^6/cos(d*x+c)+(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))+28*a^2*b^6*(sin(d*x+
c)^7/cos(d*x+c)+(sin(d*x+c)^5+5/4*sin(d*x+c)^3+15/8*sin(d*x+c))*cos(d*x+c)-15/8*d*x-15/8*c)+8*a*b^7*(sin(d*x+c
)^8/cos(d*x+c)+(16/5+sin(d*x+c)^6+6/5*sin(d*x+c)^4+8/5*sin(d*x+c)^2)*cos(d*x+c))+b^8*(sin(d*x+c)^9/cos(d*x+c)+
(sin(d*x+c)^7+7/6*sin(d*x+c)^5+35/24*sin(d*x+c)^3+35/16*sin(d*x+c))*cos(d*x+c)-35/16*d*x-35/16*c))

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Maxima [A]  time = 1.48209, size = 470, normalized size = 1.35 \begin{align*} -\frac{6720 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{6} b^{2} + 8400 \,{\left (3 \, d x + 3 \, c - \frac{\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 4480 \,{\left (\cos \left (d x + c\right )^{3} - \frac{3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{3} b^{5} + 840 \,{\left (15 \, d x + 15 \, c - \frac{9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} - 384 \,{\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac{5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a b^{7} + 5 \,{\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 )} b^{8} - 13440 \, a^{5} b^{3}{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - 240 \, a^{8} \tan \left (d x + c\right ) - \frac{1920 \, a^{7} b}{\cos \left (d x + c\right )}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/240*(6720*(d*x + c - tan(d*x + c))*a^6*b^2 + 8400*(3*d*x + 3*c - tan(d*x + c)/(tan(d*x + c)^2 + 1) - 2*tan(
d*x + c))*a^4*b^4 + 4480*(cos(d*x + c)^3 - 3/cos(d*x + c) - 6*cos(d*x + c))*a^3*b^5 + 840*(15*d*x + 15*c - (9*
tan(d*x + c)^3 + 7*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1) - 8*tan(d*x + c))*a^2*b^6 - 384*(cos(
d*x + c)^5 - 5*cos(d*x + c)^3 + 5/cos(d*x + c) + 15*cos(d*x + c))*a*b^7 + 5*(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*ta
n(d*x + c))*b^8 - 13440*a^5*b^3*(1/cos(d*x + c) + cos(d*x + c)) - 240*a^8*tan(d*x + c) - 1920*a^7*b/cos(d*x +
c))/d

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Fricas [A]  time = 2.81196, size = 651, normalized size = 1.87 \begin{align*} \frac{384 \, a b^{7} \cos \left (d x + c\right )^{6} + 1920 \, a^{7} b + 13440 \, a^{5} b^{3} + 13440 \, a^{3} b^{5} + 1920 \, a b^{7} - 640 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 105 \,{\left (64 \, a^{6} b^{2} + 240 \, a^{4} b^{4} + 120 \, a^{2} b^{6} + 5 \, b^{8}\right )} d x \cos \left (d x + c\right ) + 1920 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 5 \,{\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 48 \, a^{8} + 1344 \, a^{6} b^{2} + 3360 \, a^{4} b^{4} + 1344 \, a^{2} b^{6} + 48 \, b^{8} - 2 \,{\left (168 \, a^{2} b^{6} + 19 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (560 \, a^{4} b^{4} + 504 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/240*(384*a*b^7*cos(d*x + c)^6 + 1920*a^7*b + 13440*a^5*b^3 + 13440*a^3*b^5 + 1920*a*b^7 - 640*(7*a^3*b^5 + 3
*a*b^7)*cos(d*x + c)^4 - 105*(64*a^6*b^2 + 240*a^4*b^4 + 120*a^2*b^6 + 5*b^8)*d*x*cos(d*x + c) + 1920*(7*a^5*b
^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 + 5*(8*b^8*cos(d*x + c)^6 + 48*a^8 + 1344*a^6*b^2 + 3360*a^4*b^4 + 1
344*a^2*b^6 + 48*b^8 - 2*(168*a^2*b^6 + 19*b^8)*cos(d*x + c)^4 + 3*(560*a^4*b^4 + 504*a^2*b^6 + 29*b^8)*cos(d*
x + c)^2)*sin(d*x + c))/(d*cos(d*x + c))

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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)**2*(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [B]  time = 1.17102, size = 1079, normalized size = 3.09 \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)^2*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/240*(105*(64*a^6*b^2 + 240*a^4*b^4 + 120*a^2*b^6 + 5*b^8)*(d*x + c) + 480*(a^8*tan(1/2*d*x + 1/2*c) + 28*a^
6*b^2*tan(1/2*d*x + 1/2*c) + 70*a^4*b^4*tan(1/2*d*x + 1/2*c) + 28*a^2*b^6*tan(1/2*d*x + 1/2*c) + b^8*tan(1/2*d
*x + 1/2*c) + 8*a^7*b + 56*a^5*b^3 + 56*a^3*b^5 + 8*a*b^7)/(tan(1/2*d*x + 1/2*c)^2 - 1) + 2*(8400*a^4*b^4*tan(
1/2*d*x + 1/2*c)^11 + 5880*a^2*b^6*tan(1/2*d*x + 1/2*c)^11 + 285*b^8*tan(1/2*d*x + 1/2*c)^11 - 13440*a^5*b^3*t
an(1/2*d*x + 1/2*c)^10 - 13440*a^3*b^5*tan(1/2*d*x + 1/2*c)^10 - 1920*a*b^7*tan(1/2*d*x + 1/2*c)^10 + 25200*a^
4*b^4*tan(1/2*d*x + 1/2*c)^9 + 24360*a^2*b^6*tan(1/2*d*x + 1/2*c)^9 + 1295*b^8*tan(1/2*d*x + 1/2*c)^9 - 67200*
a^5*b^3*tan(1/2*d*x + 1/2*c)^8 - 94080*a^3*b^5*tan(1/2*d*x + 1/2*c)^8 - 13440*a*b^7*tan(1/2*d*x + 1/2*c)^8 + 1
6800*a^4*b^4*tan(1/2*d*x + 1/2*c)^7 + 18480*a^2*b^6*tan(1/2*d*x + 1/2*c)^7 + 1650*b^8*tan(1/2*d*x + 1/2*c)^7 -
134400*a^5*b^3*tan(1/2*d*x + 1/2*c)^6 - 224000*a^3*b^5*tan(1/2*d*x + 1/2*c)^6 - 42240*a*b^7*tan(1/2*d*x + 1/2
*c)^6 - 16800*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 18480*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 1650*b^8*tan(1/2*d*x + 1
/2*c)^5 - 134400*a^5*b^3*tan(1/2*d*x + 1/2*c)^4 - 241920*a^3*b^5*tan(1/2*d*x + 1/2*c)^4 - 49920*a*b^7*tan(1/2*
d*x + 1/2*c)^4 - 25200*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 24360*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 1295*b^8*tan(1/
2*d*x + 1/2*c)^3 - 67200*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 - 120960*a^3*b^5*tan(1/2*d*x + 1/2*c)^2 - 23424*a*b^7*
tan(1/2*d*x + 1/2*c)^2 - 8400*a^4*b^4*tan(1/2*d*x + 1/2*c) - 5880*a^2*b^6*tan(1/2*d*x + 1/2*c) - 285*b^8*tan(1
/2*d*x + 1/2*c) - 13440*a^5*b^3 - 22400*a^3*b^5 - 4224*a*b^7)/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d