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

Optimal. Leaf size=404 \[ -\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {4 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{105 d}+\frac {b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{105 d}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x \]

[Out]

b^8*x+4/105*a*b*(24*a^6-88*a^4*b^2+125*a^2*b^4-96*b^6)*cos(d*x+c)/d+1/105*b^2*(48*a^6-152*a^4*b^2+174*a^2*b^4-
105*b^6)*cos(d*x+c)*sin(d*x+c)/d+2/105*a*b*(24*a^4-40*a^2*b^2+9*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^2/d+2/105*b*(
24*a^4+8*a^2*b^2-35*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^3/d+1/7*sec(d*x+c)^7*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^7/
d-2/105*sec(d*x+c)^3*(a+b*sin(d*x+c))^5*(b*(6*a^2-7*b^2)-a*(12*a^2-11*b^2)*sin(d*x+c))/d-1/35*sec(d*x+c)^5*(a+
b*sin(d*x+c))^6*(a*b-(6*a^2-7*b^2)*sin(d*x+c))/d-2/105*sec(d*x+c)*(a+b*sin(d*x+c))^4*(3*a*b*(12*a^2-11*b^2)-(2
4*a^4+8*a^2*b^2-35*b^4)*sin(d*x+c))/d

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Rubi [A]  time = 0.82, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2691, 2861, 2753, 2734} \[ \frac {4 a b \left (-88 a^4 b^2+125 a^2 b^4+24 a^6-96 b^6\right ) \cos (c+d x)}{105 d}+\frac {b^2 \left (-152 a^4 b^2+174 a^2 b^4+48 a^6-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{105 d}+\frac {2 b \left (8 a^2 b^2+24 a^4-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {2 a b \left (-40 a^2 b^2+24 a^4+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (8 a^2 b^2+24 a^4-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b^8*x + (4*a*b*(24*a^6 - 88*a^4*b^2 + 125*a^2*b^4 - 96*b^6)*Cos[c + d*x])/(105*d) + (b^2*(48*a^6 - 152*a^4*b^2
+ 174*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(105*d) + (2*a*b*(24*a^4 - 40*a^2*b^2 + 9*b^4)*Cos[c + d*
x]*(a + b*Sin[c + d*x])^2)/(105*d) + (2*b*(24*a^4 + 8*a^2*b^2 - 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(
105*d) + (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(7*d) - (2*Sec[c + d*x]^3*(a + b*Sin[c +
d*x])^5*(b*(6*a^2 - 7*b^2) - a*(12*a^2 - 11*b^2)*Sin[c + d*x]))/(105*d) - (Sec[c + d*x]^5*(a + b*Sin[c + d*x]
)^6*(a*b - (6*a^2 - 7*b^2)*Sin[c + d*x]))/(35*d) - (2*Sec[c + d*x]*(a + b*Sin[c + d*x])^4*(3*a*b*(12*a^2 - 11*
b^2) - (24*a^4 + 8*a^2*b^2 - 35*b^4)*Sin[c + d*x]))/(105*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 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]

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 2861

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

Rubi steps

\begin {align*} \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {1}{7} \int \sec ^6(c+d x) (a+b \sin (c+d x))^6 \left (-6 a^2+7 b^2+a b \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {1}{35} \int \sec ^4(c+d x) (a+b \sin (c+d x))^5 \left (2 a \left (12 a^2-11 b^2\right )-2 b \left (6 a^2-7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {1}{105} \int \sec ^2(c+d x) (a+b \sin (c+d x))^4 \left (-2 \left (24 a^4+8 a^2 b^2-35 b^4\right )+6 a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {1}{105} \int (a+b \sin (c+d x))^3 \left (24 a b^2 \left (12 a^2-11 b^2\right )-8 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {1}{420} \int (a+b \sin (c+d x))^2 \left (24 b^2 \left (24 a^4-52 a^2 b^2+35 b^4\right )-24 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac {\int (a+b \sin (c+d x)) \left (24 a b^2 \left (24 a^4-76 a^2 b^2+87 b^4\right )-24 b \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \sin (c+d x)\right ) \, dx}{1260}\\ &=b^8 x+\frac {4 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{105 d}+\frac {b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{105 d}+\frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}\\ \end {align*}

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Mathematica [A]  time = 1.43, size = 479, normalized size = 1.19 \[ \frac {\sec ^7(c+d x) \left (1680 a^8 \sin (c+d x)+1008 a^8 \sin (3 (c+d x))+336 a^8 \sin (5 (c+d x))+48 a^8 \sin (7 (c+d x))+7680 a^7 b+23520 a^6 b^2 \sin (c+d x)-4704 a^6 b^2 \sin (3 (c+d x))-1568 a^6 b^2 \sin (5 (c+d x))-224 a^6 b^2 \sin (7 (c+d x))-37632 a^5 b^3 \cos (2 (c+d x))+16128 a^5 b^3+44100 a^4 b^4 \sin (c+d x)-20580 a^4 b^4 \sin (3 (c+d x))+2940 a^4 b^4 \sin (5 (c+d x))+420 a^4 b^4 \sin (7 (c+d x))-12544 a^3 b^5 \cos (2 (c+d x))+15680 a^3 b^5 \cos (4 (c+d x))+25536 a^3 b^5+14700 a^2 b^6 \sin (c+d x)-8820 a^2 b^6 \sin (3 (c+d x))+2940 a^2 b^6 \sin (5 (c+d x))-420 a^2 b^6 \sin (7 (c+d x))-14448 a b^7 \cos (2 (c+d x))-3360 a b^7 \cos (4 (c+d x))-1680 a b^7 \cos (6 (c+d x))-5088 a b^7-1176 b^8 \sin (3 (c+d x))-392 b^8 \sin (5 (c+d x))-176 b^8 \sin (7 (c+d x))+3675 b^8 (c+d x) \cos (c+d x)+2205 b^8 (c+d x) \cos (3 (c+d x))+735 b^8 (c+d x) \cos (5 (c+d x))+105 b^8 (c+d x) \cos (7 (c+d x))\right )}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^7*(7680*a^7*b + 16128*a^5*b^3 + 25536*a^3*b^5 - 5088*a*b^7 + 3675*b^8*(c + d*x)*Cos[c + d*x] - 3
7632*a^5*b^3*Cos[2*(c + d*x)] - 12544*a^3*b^5*Cos[2*(c + d*x)] - 14448*a*b^7*Cos[2*(c + d*x)] + 2205*b^8*(c +
d*x)*Cos[3*(c + d*x)] + 15680*a^3*b^5*Cos[4*(c + d*x)] - 3360*a*b^7*Cos[4*(c + d*x)] + 735*b^8*(c + d*x)*Cos[5
*(c + d*x)] - 1680*a*b^7*Cos[6*(c + d*x)] + 105*b^8*(c + d*x)*Cos[7*(c + d*x)] + 1680*a^8*Sin[c + d*x] + 23520
*a^6*b^2*Sin[c + d*x] + 44100*a^4*b^4*Sin[c + d*x] + 14700*a^2*b^6*Sin[c + d*x] + 1008*a^8*Sin[3*(c + d*x)] -
4704*a^6*b^2*Sin[3*(c + d*x)] - 20580*a^4*b^4*Sin[3*(c + d*x)] - 8820*a^2*b^6*Sin[3*(c + d*x)] - 1176*b^8*Sin[
3*(c + d*x)] + 336*a^8*Sin[5*(c + d*x)] - 1568*a^6*b^2*Sin[5*(c + d*x)] + 2940*a^4*b^4*Sin[5*(c + d*x)] + 2940
*a^2*b^6*Sin[5*(c + d*x)] - 392*b^8*Sin[5*(c + d*x)] + 48*a^8*Sin[7*(c + d*x)] - 224*a^6*b^2*Sin[7*(c + d*x)]
+ 420*a^4*b^4*Sin[7*(c + d*x)] - 420*a^2*b^6*Sin[7*(c + d*x)] - 176*b^8*Sin[7*(c + d*x)]))/(6720*d)

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fricas [A]  time = 0.53, size = 306, normalized size = 0.76 \[ \frac {105 \, b^{8} d x \cos \left (d x + c\right )^{7} - 840 \, a b^{7} \cos \left (d x + c\right )^{6} + 120 \, a^{7} b + 840 \, a^{5} b^{3} + 840 \, a^{3} b^{5} + 120 \, a b^{7} + 280 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 168 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, a^{8} + 420 \, a^{6} b^{2} + 1050 \, a^{4} b^{4} + 420 \, a^{2} b^{6} + 15 \, b^{8} + 4 \, {\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} - 105 \, a^{2} b^{6} - 44 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} + 630 \, a^{2} b^{6} + 61 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (3 \, a^{8} - 14 \, a^{6} b^{2} - 280 \, a^{4} b^{4} - 210 \, a^{2} b^{6} - 11 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*b^8*d*x*cos(d*x + c)^7 - 840*a*b^7*cos(d*x + c)^6 + 120*a^7*b + 840*a^5*b^3 + 840*a^3*b^5 + 120*a*b
^7 + 280*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 - 168*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 + (15*a^
8 + 420*a^6*b^2 + 1050*a^4*b^4 + 420*a^2*b^6 + 15*b^8 + 4*(12*a^8 - 56*a^6*b^2 + 105*a^4*b^4 - 105*a^2*b^6 - 4
4*b^8)*cos(d*x + c)^6 + 2*(12*a^8 - 56*a^6*b^2 + 105*a^4*b^4 + 630*a^2*b^6 + 61*b^8)*cos(d*x + c)^4 + 6*(3*a^8
- 14*a^6*b^2 - 280*a^4*b^4 - 210*a^2*b^6 - 11*b^8)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^7)

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giac [A]  time = 0.70, size = 726, normalized size = 1.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*(d*x + c)*b^8 - 2*(105*a^8*tan(1/2*d*x + 1/2*c)^13 - 105*b^8*tan(1/2*d*x + 1/2*c)^13 + 840*a^7*b*ta
n(1/2*d*x + 1/2*c)^12 - 210*a^8*tan(1/2*d*x + 1/2*c)^11 + 3920*a^6*b^2*tan(1/2*d*x + 1/2*c)^11 + 770*b^8*tan(1
/2*d*x + 1/2*c)^11 + 11760*a^5*b^3*tan(1/2*d*x + 1/2*c)^10 + 903*a^8*tan(1/2*d*x + 1/2*c)^9 + 3136*a^6*b^2*tan
(1/2*d*x + 1/2*c)^9 + 23520*a^4*b^4*tan(1/2*d*x + 1/2*c)^9 - 2471*b^8*tan(1/2*d*x + 1/2*c)^9 + 4200*a^7*b*tan(
1/2*d*x + 1/2*c)^8 + 11760*a^5*b^3*tan(1/2*d*x + 1/2*c)^8 + 31360*a^3*b^5*tan(1/2*d*x + 1/2*c)^8 - 636*a^8*tan
(1/2*d*x + 1/2*c)^7 + 12768*a^6*b^2*tan(1/2*d*x + 1/2*c)^7 + 20160*a^4*b^4*tan(1/2*d*x + 1/2*c)^7 + 26880*a^2*
b^6*tan(1/2*d*x + 1/2*c)^7 + 4572*b^8*tan(1/2*d*x + 1/2*c)^7 + 23520*a^5*b^3*tan(1/2*d*x + 1/2*c)^6 + 15680*a^
3*b^5*tan(1/2*d*x + 1/2*c)^6 + 13440*a*b^7*tan(1/2*d*x + 1/2*c)^6 + 903*a^8*tan(1/2*d*x + 1/2*c)^5 + 3136*a^6*
b^2*tan(1/2*d*x + 1/2*c)^5 + 23520*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 2471*b^8*tan(1/2*d*x + 1/2*c)^5 + 2520*a^7
*b*tan(1/2*d*x + 1/2*c)^4 + 4704*a^5*b^3*tan(1/2*d*x + 1/2*c)^4 + 9408*a^3*b^5*tan(1/2*d*x + 1/2*c)^4 - 8064*a
*b^7*tan(1/2*d*x + 1/2*c)^4 - 210*a^8*tan(1/2*d*x + 1/2*c)^3 + 3920*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 770*b^8*t
an(1/2*d*x + 1/2*c)^3 + 2352*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 - 3136*a^3*b^5*tan(1/2*d*x + 1/2*c)^2 + 2688*a*b^7
*tan(1/2*d*x + 1/2*c)^2 + 105*a^8*tan(1/2*d*x + 1/2*c) - 105*b^8*tan(1/2*d*x + 1/2*c) + 120*a^7*b - 336*a^5*b^
3 + 448*a^3*b^5 - 384*a*b^7)/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d

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maple [A]  time = 0.41, size = 567, normalized size = 1.40 \[ \frac {-a^{8} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{7 \cos \left (d x +c \right )^{7}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{35}\right )+\frac {4 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}-\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )}-\frac {\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}\right )+b^{8} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-a^8*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/35*sec(d*x+c)^2)*tan(d*x+c)+8/7*a^7*b/cos(d*x+c)^7+28*a
^6*b^2*(1/7*sin(d*x+c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3/cos(d*x+c)^3)+56*a^5*b
^3*(1/7*sin(d*x+c)^4/cos(d*x+c)^7+3/35*sin(d*x+c)^4/cos(d*x+c)^5+1/35*sin(d*x+c)^4/cos(d*x+c)^3-1/35*sin(d*x+c
)^4/cos(d*x+c)-1/35*(2+sin(d*x+c)^2)*cos(d*x+c))+70*a^4*b^4*(1/7*sin(d*x+c)^5/cos(d*x+c)^7+2/35*sin(d*x+c)^5/c
os(d*x+c)^5)+56*a^3*b^5*(1/7*sin(d*x+c)^6/cos(d*x+c)^7+1/35*sin(d*x+c)^6/cos(d*x+c)^5-1/105*sin(d*x+c)^6/cos(d
*x+c)^3+1/35*sin(d*x+c)^6/cos(d*x+c)+1/35*(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))+4*a^2*b^6*sin(d*x+c)
^7/cos(d*x+c)^7+8*a*b^7*(1/7*sin(d*x+c)^8/cos(d*x+c)^7-1/35*sin(d*x+c)^8/cos(d*x+c)^5+1/35*sin(d*x+c)^8/cos(d*
x+c)^3-1/7*sin(d*x+c)^8/cos(d*x+c)-1/7*(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*(
1/7*tan(d*x+c)^7-1/5*tan(d*x+c)^5+1/3*tan(d*x+c)^3-tan(d*x+c)+d*x+c))

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maxima [A]  time = 0.43, size = 310, normalized size = 0.77 \[ \frac {420 \, a^{2} b^{6} \tan \left (d x + c\right )^{7} + 3 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 210 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 7 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} + {\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} b^{8} - \frac {168 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{7}} + \frac {56 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 15\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{7}} - \frac {24 \, {\left (35 \, \cos \left (d x + c\right )^{6} - 35 \, \cos \left (d x + c\right )^{4} + 21 \, \cos \left (d x + c\right )^{2} - 5\right )} a b^{7}}{\cos \left (d x + c\right )^{7}} + \frac {120 \, a^{7} b}{\cos \left (d x + c\right )^{7}}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(420*a^2*b^6*tan(d*x + c)^7 + 3*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x +
c))*a^8 + 28*(15*tan(d*x + c)^7 + 42*tan(d*x + c)^5 + 35*tan(d*x + c)^3)*a^6*b^2 + 210*(5*tan(d*x + c)^7 + 7*
tan(d*x + c)^5)*a^4*b^4 + (15*tan(d*x + c)^7 - 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 105*d*x + 105*c - 105*t
an(d*x + c))*b^8 - 168*(7*cos(d*x + c)^2 - 5)*a^5*b^3/cos(d*x + c)^7 + 56*(35*cos(d*x + c)^4 - 42*cos(d*x + c)
^2 + 15)*a^3*b^5/cos(d*x + c)^7 - 24*(35*cos(d*x + c)^6 - 35*cos(d*x + c)^4 + 21*cos(d*x + c)^2 - 5)*a*b^7/cos
(d*x + c)^7 + 120*a^7*b/cos(d*x + c)^7)/d

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mupad [B]  time = 8.85, size = 546, normalized size = 1.35 \[ b^8\,x-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-4\,a^8+\frac {224\,a^6\,b^2}{3}+\frac {44\,b^8}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-4\,a^8+\frac {224\,a^6\,b^2}{3}+\frac {44\,b^8}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (2\,a^8-2\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {224\,a^5\,b^3}{5}-\frac {896\,a^3\,b^5}{15}+\frac {256\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (448\,a^5\,b^3+\frac {896\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (80\,a^7\,b+224\,a^5\,b^3+\frac {1792\,a^3\,b^5}{3}\right )-\frac {256\,a\,b^7}{35}+\frac {16\,a^7\,b}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (48\,a^7\,b+\frac {448\,a^5\,b^3}{5}+\frac {896\,a^3\,b^5}{5}-\frac {768\,a\,b^7}{5}\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^8-2\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-\frac {424\,a^8}{35}+\frac {1216\,a^6\,b^2}{5}+384\,a^4\,b^4+512\,a^2\,b^6+\frac {3048\,b^8}{35}\right )+\frac {128\,a^3\,b^5}{15}-\frac {32\,a^5\,b^3}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {86\,a^8}{5}+\frac {896\,a^6\,b^2}{15}+448\,a^4\,b^4-\frac {706\,b^8}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {86\,a^8}{5}+\frac {896\,a^6\,b^2}{15}+448\,a^4\,b^4-\frac {706\,b^8}{15}\right )+224\,a^5\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^8*x - (tan(c/2 + (d*x)/2)^3*((44*b^8)/3 - 4*a^8 + (224*a^6*b^2)/3) + tan(c/2 + (d*x)/2)^11*((44*b^8)/3 - 4*a
^8 + (224*a^6*b^2)/3) + tan(c/2 + (d*x)/2)^13*(2*a^8 - 2*b^8) + tan(c/2 + (d*x)/2)^2*((256*a*b^7)/5 - (896*a^3
*b^5)/15 + (224*a^5*b^3)/5) + tan(c/2 + (d*x)/2)^6*(256*a*b^7 + (896*a^3*b^5)/3 + 448*a^5*b^3) + tan(c/2 + (d*
x)/2)^8*(80*a^7*b + (1792*a^3*b^5)/3 + 224*a^5*b^3) - (256*a*b^7)/35 + (16*a^7*b)/7 + tan(c/2 + (d*x)/2)^4*(48
*a^7*b - (768*a*b^7)/5 + (896*a^3*b^5)/5 + (448*a^5*b^3)/5) + tan(c/2 + (d*x)/2)*(2*a^8 - 2*b^8) + tan(c/2 + (
d*x)/2)^7*((3048*b^8)/35 - (424*a^8)/35 + 512*a^2*b^6 + 384*a^4*b^4 + (1216*a^6*b^2)/5) + (128*a^3*b^5)/15 - (
32*a^5*b^3)/5 + tan(c/2 + (d*x)/2)^5*((86*a^8)/5 - (706*b^8)/15 + 448*a^4*b^4 + (896*a^6*b^2)/15) + tan(c/2 +
(d*x)/2)^9*((86*a^8)/5 - (706*b^8)/15 + 448*a^4*b^4 + (896*a^6*b^2)/15) + 224*a^5*b^3*tan(c/2 + (d*x)/2)^10 +
16*a^7*b*tan(c/2 + (d*x)/2)^12)/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6
- 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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