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

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

[Out]

-7/16*b^2*(64*a^6+240*a^4*b^2+120*a^2*b^4+5*b^6)*x+1/20*a*b*(40*a^6+1664*a^4*b^2+2789*a^2*b^4+512*b^6)*cos(d*x
+c)/d+1/80*b^2*(80*a^6+2248*a^4*b^2+2502*a^2*b^4+175*b^6)*cos(d*x+c)*sin(d*x+c)/d+1/40*a*b*(40*a^4+624*a^2*b^2
+337*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^2/d+1/120*b*(120*a^4+992*a^2*b^2+175*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^3/
d+1/30*a*b*(30*a^2+113*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^4/d+1/6*b*(6*a^2+7*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^5/
d+a*b*cos(d*x+c)*(a+b*sin(d*x+c))^6/d+sec(d*x+c)*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^7/d

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Rubi [A]  time = 0.56, antiderivative size = 349, normalized size of antiderivative = 1.00, 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 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]

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*}

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Mathematica [A]  time = 1.09, size = 313, normalized size = 0.90 \[ \frac {\sec (c+d x) \left (1920 a^8 \sin (c+d x)+15360 a^7 b+53760 a^6 b^2 \sin (c+d x)+161280 a^5 b^3+151200 a^4 b^4 \sin (c+d x)+16800 a^4 b^4 \sin (3 (c+d x))-4480 a^3 b^5 \cos (4 (c+d x))+201600 a^3 b^5+67200 a^2 b^6 \sin (c+d x)+12600 a^2 b^6 \sin (3 (c+d x))-840 a^2 b^6 \sin (5 (c+d x))+1120 \left (48 a^5 b^3+80 a^3 b^5+15 a b^7\right ) \cos (2 (c+d x))-840 b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) (c+d x) \cos (c+d x)-1344 a b^7 \cos (4 (c+d x))+96 a b^7 \cos (6 (c+d x))+33600 a b^7+2625 b^8 \sin (c+d x)+630 b^8 \sin (3 (c+d x))-70 b^8 \sin (5 (c+d x))+5 b^8 \sin (7 (c+d x))\right )}{1920 d} \]

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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fricas [A]  time = 0.52, size = 266, normalized size = 0.76 \[ \frac {384 \, a b^{7} \cos \left (d x + c\right )^{6} + 1920 \, a^{7} b + 13440 \, a^{5} b^{3} + 13440 \, a^{3} b^{5} + 1920 \, a b^{7} - 640 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 105 \, {\left (64 \, a^{6} b^{2} + 240 \, a^{4} b^{4} + 120 \, a^{2} b^{6} + 5 \, b^{8}\right )} d x \cos \left (d x + c\right ) + 1920 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 48 \, a^{8} + 1344 \, a^{6} b^{2} + 3360 \, a^{4} b^{4} + 1344 \, a^{2} b^{6} + 48 \, b^{8} - 2 \, {\left (168 \, a^{2} b^{6} + 19 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (560 \, a^{4} b^{4} + 504 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )} \]

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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giac [B]  time = 0.78, size = 799, normalized size = 2.29 \[ \text {result too large to display} \]

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

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maple [A]  time = 0.42, size = 406, normalized size = 1.16 \[ \frac {a^{8} \tan \left (d x +c \right )+\frac {8 a^{7} b}{\cos \left (d x +c \right )}+28 a^{6} b^{2} \left (\tan \left (d x +c \right )-d x -c \right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d} \]

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 = 0.43, size = 348, normalized size = 1.00 \[ -\frac {6720 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{6} b^{2} + 8400 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 4480 \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{3} b^{5} + 840 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} - 384 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac {5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a b^{7} + 5 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} b^{8} - 13440 \, a^{5} b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - 240 \, a^{8} \tan \left (d x + c\right ) - \frac {1920 \, a^{7} b}{\cos \left (d x + c\right )}}{240 \, d} \]

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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mupad [B]  time = 7.73, size = 767, normalized size = 2.20 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (240\,a^7\,b+1120\,a^5\,b^3+\frac {1792\,a^3\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (96\,a^7\,b+224\,a^5\,b^3\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^8+56\,a^6\,b^2+210\,a^4\,b^4+105\,a^2\,b^6+\frac {35\,b^8}{8}\right )+\frac {256\,a\,b^7}{5}+16\,a^7\,b+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (96\,a^7\,b+1120\,a^5\,b^3+\frac {4480\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (240\,a^7\,b+2240\,a^5\,b^3+2688\,a^3\,b^5+\frac {2304\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (320\,a^7\,b+2240\,a^5\,b^3+\frac {6272\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (2\,a^8+56\,a^6\,b^2+210\,a^4\,b^4+105\,a^2\,b^6+\frac {35\,b^8}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^8+336\,a^6\,b^2+980\,a^4\,b^4+490\,a^2\,b^6+\frac {245\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (12\,a^8+336\,a^6\,b^2+980\,a^4\,b^4+490\,a^2\,b^6+\frac {245\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (30\,a^8+840\,a^6\,b^2+2030\,a^4\,b^4+791\,a^2\,b^6+\frac {791\,b^8}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (30\,a^8+840\,a^6\,b^2+2030\,a^4\,b^4+791\,a^2\,b^6+\frac {791\,b^8}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (40\,a^8+1120\,a^6\,b^2+2520\,a^4\,b^4+812\,a^2\,b^6+\frac {25\,b^8}{2}\right )+\frac {896\,a^3\,b^5}{3}+224\,a^5\,b^3+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}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,b^2\,\mathrm {atan}\left (\frac {7\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (64\,a^6+240\,a^4\,b^2+120\,a^2\,b^4+5\,b^6\right )}{448\,a^6\,b^2+1680\,a^4\,b^4+840\,a^2\,b^6+35\,b^8}\right )\,\left (64\,a^6+240\,a^4\,b^2+120\,a^2\,b^4+5\,b^6\right )}{8\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)^8*(240*a^7*b + (1792*a^3*b^5)/3 + 1120*a^5*b^3) + tan(c/2 + (d*x)/2)^10*(96*a^7*b + 224*a^
5*b^3) + tan(c/2 + (d*x)/2)*(2*a^8 + (35*b^8)/8 + 105*a^2*b^6 + 210*a^4*b^4 + 56*a^6*b^2) + (256*a*b^7)/5 + 16
*a^7*b + tan(c/2 + (d*x)/2)^2*(256*a*b^7 + 96*a^7*b + (4480*a^3*b^5)/3 + 1120*a^5*b^3) + tan(c/2 + (d*x)/2)^4*
((2304*a*b^7)/5 + 240*a^7*b + 2688*a^3*b^5 + 2240*a^5*b^3) + tan(c/2 + (d*x)/2)^6*(256*a*b^7 + 320*a^7*b + (62
72*a^3*b^5)/3 + 2240*a^5*b^3) + tan(c/2 + (d*x)/2)^13*(2*a^8 + (35*b^8)/8 + 105*a^2*b^6 + 210*a^4*b^4 + 56*a^6
*b^2) + tan(c/2 + (d*x)/2)^3*(12*a^8 + (245*b^8)/12 + 490*a^2*b^6 + 980*a^4*b^4 + 336*a^6*b^2) + tan(c/2 + (d*
x)/2)^11*(12*a^8 + (245*b^8)/12 + 490*a^2*b^6 + 980*a^4*b^4 + 336*a^6*b^2) + tan(c/2 + (d*x)/2)^5*(30*a^8 + (7
91*b^8)/24 + 791*a^2*b^6 + 2030*a^4*b^4 + 840*a^6*b^2) + tan(c/2 + (d*x)/2)^9*(30*a^8 + (791*b^8)/24 + 791*a^2
*b^6 + 2030*a^4*b^4 + 840*a^6*b^2) + tan(c/2 + (d*x)/2)^7*(40*a^8 + (25*b^8)/2 + 812*a^2*b^6 + 2520*a^4*b^4 +
1120*a^6*b^2) + (896*a^3*b^5)/3 + 224*a^5*b^3 + 16*a^7*b*tan(c/2 + (d*x)/2)^12)/(d*(5*tan(c/2 + (d*x)/2)^2 + 9
*tan(c/2 + (d*x)/2)^4 + 5*tan(c/2 + (d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 - 9*tan(c/2 + (d*x)/2)^10 - 5*tan(c/2
+ (d*x)/2)^12 - tan(c/2 + (d*x)/2)^14 + 1)) - (7*b^2*atan((7*b^2*tan(c/2 + (d*x)/2)*(64*a^6 + 5*b^6 + 120*a^2*
b^4 + 240*a^4*b^2))/(35*b^8 + 840*a^2*b^6 + 1680*a^4*b^4 + 448*a^6*b^2))*(64*a^6 + 5*b^6 + 120*a^2*b^4 + 240*a
^4*b^2))/(8*d)

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

[Out]

Timed out

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