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

Optimal. Leaf size=423 \[ -\frac{11 a b \left (10536 a^4 b^2+9588 a^2 b^4+1792 a^6+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (1500 a^2 b^2+784 a^4+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{13 a b \left (348 a^2 b^2+112 a^4+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left (28088 a^4 b^2+15956 a^2 b^4+6272 a^6+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}+\frac{\left (896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right ) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{1}{256} x \left (896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right )-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d} \]

[Out]

((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*x)/256 - (11*a*b*(1792*a^6 + 10536*a^4*b^2 + 958
8*a^2*b^4 + 1289*b^6)*Cos[c + d*x]^3)/(40320*d) + ((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8
)*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (b*(6272*a^6 + 28088*a^4*b^2 + 15956*a^2*b^4 + 735*b^6)*Cos[c + d*x]^3*
(a + b*Sin[c + d*x]))/(13440*d) - (13*a*b*(112*a^4 + 348*a^2*b^2 + 101*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x]
)^2)/(3360*d) - (b*(784*a^4 + 1500*a^2*b^2 + 147*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(2016*d) - (a*b*(
112*a^2 + 109*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(336*d) - (b*(64*a^2 + 21*b^2)*Cos[c + d*x]^3*(a + b
*Sin[c + d*x])^5)/(240*d) - (17*a*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^6)/(90*d) - (b*Cos[c + d*x]^3*(a + b*S
in[c + d*x])^7)/(10*d)

________________________________________________________________________________________

Rubi [A]  time = 1.21706, antiderivative size = 423, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2692, 2862, 2669, 2635, 8} \[ -\frac{11 a b \left (10536 a^4 b^2+9588 a^2 b^4+1792 a^6+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (1500 a^2 b^2+784 a^4+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{13 a b \left (348 a^2 b^2+112 a^4+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left (28088 a^4 b^2+15956 a^2 b^4+6272 a^6+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}+\frac{\left (896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right ) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{1}{256} x \left (896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right )-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*x)/256 - (11*a*b*(1792*a^6 + 10536*a^4*b^2 + 958
8*a^2*b^4 + 1289*b^6)*Cos[c + d*x]^3)/(40320*d) + ((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8
)*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (b*(6272*a^6 + 28088*a^4*b^2 + 15956*a^2*b^4 + 735*b^6)*Cos[c + d*x]^3*
(a + b*Sin[c + d*x]))/(13440*d) - (13*a*b*(112*a^4 + 348*a^2*b^2 + 101*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x]
)^2)/(3360*d) - (b*(784*a^4 + 1500*a^2*b^2 + 147*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(2016*d) - (a*b*(
112*a^2 + 109*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(336*d) - (b*(64*a^2 + 21*b^2)*Cos[c + d*x]^3*(a + b
*Sin[c + d*x])^5)/(240*d) - (17*a*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^6)/(90*d) - (b*Cos[c + d*x]^3*(a + b*S
in[c + d*x])^7)/(10*d)

Rule 2692

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

Rule 2862

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

Rule 2669

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

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]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx &=-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{1}{10} \int \cos ^2(c+d x) (a+b \sin (c+d x))^6 \left (10 a^2+7 b^2+17 a b \sin (c+d x)\right ) \, dx\\ &=-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{1}{90} \int \cos ^2(c+d x) (a+b \sin (c+d x))^5 \left (15 a \left (6 a^2+11 b^2\right )+3 b \left (64 a^2+21 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{1}{720} \int \cos ^2(c+d x) (a+b \sin (c+d x))^4 \left (15 \left (48 a^4+152 a^2 b^2+21 b^4\right )+15 a b \left (112 a^2+109 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{\int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \left (15 a \left (336 a^4+1512 a^2 b^2+583 b^4\right )+15 b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \sin (c+d x)\right ) \, dx}{5040}\\ &=-\frac{b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{\int \cos ^2(c+d x) (a+b \sin (c+d x))^2 \left (45 \left (672 a^6+3808 a^4 b^2+2666 a^2 b^4+147 b^6\right )+585 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \sin (c+d x)\right ) \, dx}{30240}\\ &=-\frac{13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{\int \cos ^2(c+d x) (a+b \sin (c+d x)) \left (45 a \left (3360 a^6+21952 a^4 b^2+22378 a^2 b^4+3361 b^6\right )+45 b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \sin (c+d x)\right ) \, dx}{151200}\\ &=-\frac{b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac{13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{\int \cos ^2(c+d x) \left (4725 \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right )+495 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \sin (c+d x)\right ) \, dx}{604800}\\ &=-\frac{11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}-\frac{b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac{13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{1}{128} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac{\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac{b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac{13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}+\frac{1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int 1 \, dx\\ &=\frac{1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) x-\frac{11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac{\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac{b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac{13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\\ \end{align*}

Mathematica [A]  time = 1.05263, size = 457, normalized size = 1.08 \[ \frac{-564480 a^6 b^2 \sin (4 (c+d x))-705600 a^4 b^4 \sin (2 (c+d x))-705600 a^4 b^4 \sin (4 (c+d x))+235200 a^4 b^4 \sin (6 (c+d x))-282240 a^2 b^6 \sin (2 (c+d x))-141120 a^2 b^6 \sin (4 (c+d x))+94080 a^2 b^6 \sin (6 (c+d x))-17640 a^2 b^6 \sin (8 (c+d x))+451584 a^5 b^3 \cos (5 (c+d x))+338688 a^3 b^5 \cos (5 (c+d x))-80640 a^3 b^5 \cos (7 (c+d x))-40320 a b \left (112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right ) \cos (c+d x)-26880 \left (28 a^5 b^3+7 a^3 b^5+16 a^7 b\right ) \cos (3 (c+d x))+2257920 a^6 b^2 c+2822400 a^4 b^4 c+705600 a^2 b^6 c+2257920 a^6 b^2 d x+2822400 a^4 b^4 d x+705600 a^2 b^6 d x+161280 a^8 \sin (2 (c+d x))+322560 a^8 c+322560 a^8 d x+32256 a b^7 \cos (5 (c+d x))-14400 a b^7 \cos (7 (c+d x))+2240 a b^7 \cos (9 (c+d x))-8820 b^8 \sin (2 (c+d x))-2520 b^8 \sin (4 (c+d x))+2730 b^8 \sin (6 (c+d x))-945 b^8 \sin (8 (c+d x))+126 b^8 \sin (10 (c+d x))+17640 b^8 c+17640 b^8 d x}{645120 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(322560*a^8*c + 2257920*a^6*b^2*c + 2822400*a^4*b^4*c + 705600*a^2*b^6*c + 17640*b^8*c + 322560*a^8*d*x + 2257
920*a^6*b^2*d*x + 2822400*a^4*b^4*d*x + 705600*a^2*b^6*d*x + 17640*b^8*d*x - 40320*a*b*(32*a^6 + 112*a^4*b^2 +
70*a^2*b^4 + 7*b^6)*Cos[c + d*x] - 26880*(16*a^7*b + 28*a^5*b^3 + 7*a^3*b^5)*Cos[3*(c + d*x)] + 451584*a^5*b^
3*Cos[5*(c + d*x)] + 338688*a^3*b^5*Cos[5*(c + d*x)] + 32256*a*b^7*Cos[5*(c + d*x)] - 80640*a^3*b^5*Cos[7*(c +
d*x)] - 14400*a*b^7*Cos[7*(c + d*x)] + 2240*a*b^7*Cos[9*(c + d*x)] + 161280*a^8*Sin[2*(c + d*x)] - 705600*a^4
*b^4*Sin[2*(c + d*x)] - 282240*a^2*b^6*Sin[2*(c + d*x)] - 8820*b^8*Sin[2*(c + d*x)] - 564480*a^6*b^2*Sin[4*(c
+ d*x)] - 705600*a^4*b^4*Sin[4*(c + d*x)] - 141120*a^2*b^6*Sin[4*(c + d*x)] - 2520*b^8*Sin[4*(c + d*x)] + 2352
00*a^4*b^4*Sin[6*(c + d*x)] + 94080*a^2*b^6*Sin[6*(c + d*x)] + 2730*b^8*Sin[6*(c + d*x)] - 17640*a^2*b^6*Sin[8
*(c + d*x)] - 945*b^8*Sin[8*(c + d*x)] + 126*b^8*Sin[10*(c + d*x)])/(645120*d)

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Maple [A]  time = 0.082, size = 497, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^8*(-1/10*sin(d*x+c)^7*cos(d*x+c)^3-7/80*sin(d*x+c)^5*cos(d*x+c)^3-7/96*sin(d*x+c)^3*cos(d*x+c)^3-7/128*
cos(d*x+c)^3*sin(d*x+c)+7/256*cos(d*x+c)*sin(d*x+c)+7/256*d*x+7/256*c)+8*a*b^7*(-1/9*sin(d*x+c)^6*cos(d*x+c)^3
-2/21*sin(d*x+c)^4*cos(d*x+c)^3-8/105*sin(d*x+c)^2*cos(d*x+c)^3-16/315*cos(d*x+c)^3)+28*a^2*b^6*(-1/8*sin(d*x+
c)^5*cos(d*x+c)^3-5/48*sin(d*x+c)^3*cos(d*x+c)^3-5/64*cos(d*x+c)^3*sin(d*x+c)+5/128*cos(d*x+c)*sin(d*x+c)+5/12
8*d*x+5/128*c)+56*a^3*b^5*(-1/7*sin(d*x+c)^4*cos(d*x+c)^3-4/35*sin(d*x+c)^2*cos(d*x+c)^3-8/105*cos(d*x+c)^3)+7
0*a^4*b^4*(-1/6*sin(d*x+c)^3*cos(d*x+c)^3-1/8*cos(d*x+c)^3*sin(d*x+c)+1/16*cos(d*x+c)*sin(d*x+c)+1/16*d*x+1/16
*c)+56*a^5*b^3*(-1/5*sin(d*x+c)^2*cos(d*x+c)^3-2/15*cos(d*x+c)^3)+28*a^6*b^2*(-1/4*cos(d*x+c)^3*sin(d*x+c)+1/8
*cos(d*x+c)*sin(d*x+c)+1/8*d*x+1/8*c)-8/3*a^7*b*cos(d*x+c)^3+a^8*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 1.01069, size = 454, normalized size = 1.07 \begin{align*} -\frac{1720320 \, a^{7} b \cos \left (d x + c\right )^{3} - 161280 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8} - 564480 \,{\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{6} b^{2} - 2408448 \,{\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{5} b^{3} + 235200 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} b^{4} + 344064 \,{\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{3} b^{5} + 5880 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 120 \, d x - 120 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{6} - 16384 \,{\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a b^{7} - 21 \,{\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c - 45 \, \sin \left (8 \, d x + 8 \, c\right ) - 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{8}}{645120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/645120*(1720320*a^7*b*cos(d*x + c)^3 - 161280*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^8 - 564480*(4*d*x + 4*c -
sin(4*d*x + 4*c))*a^6*b^2 - 2408448*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^5*b^3 + 235200*(4*sin(2*d*x + 2*c)
^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^4*b^4 + 344064*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x
+ c)^3)*a^3*b^5 + 5880*(64*sin(2*d*x + 2*c)^3 - 120*d*x - 120*c + 3*sin(8*d*x + 8*c) + 24*sin(4*d*x + 4*c))*a^
2*b^6 - 16384*(35*cos(d*x + c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 105*cos(d*x + c)^3)*a*b^7 - 21*(9
6*sin(2*d*x + 2*c)^5 - 640*sin(2*d*x + 2*c)^3 + 840*d*x + 840*c - 45*sin(8*d*x + 8*c) - 120*sin(4*d*x + 4*c))*
b^8)/d

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Fricas [A]  time = 3.26176, size = 780, normalized size = 1.84 \begin{align*} \frac{71680 \, a b^{7} \cos \left (d x + c\right )^{9} - 92160 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{7} + 129024 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{5} - 215040 \,{\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{3} + 315 \,{\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} d x + 21 \,{\left (384 \, b^{8} \cos \left (d x + c\right )^{9} - 48 \,{\left (280 \, a^{2} b^{6} + 31 \, b^{8}\right )} \cos \left (d x + c\right )^{7} + 8 \,{\left (5600 \, a^{4} b^{4} + 4760 \, a^{2} b^{6} + 263 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - 10 \,{\left (2688 \, a^{6} b^{2} + 7840 \, a^{4} b^{4} + 3304 \, a^{2} b^{6} + 121 \, b^{8}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80640*(71680*a*b^7*cos(d*x + c)^9 - 92160*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^7 + 129024*(7*a^5*b^3 + 14*a^3*
b^5 + 3*a*b^7)*cos(d*x + c)^5 - 215040*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^3 + 315*(128*a^8 +
896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*d*x + 21*(384*b^8*cos(d*x + c)^9 - 48*(280*a^2*b^6 + 31*b^8
)*cos(d*x + c)^7 + 8*(5600*a^4*b^4 + 4760*a^2*b^6 + 263*b^8)*cos(d*x + c)^5 - 10*(2688*a^6*b^2 + 7840*a^4*b^4
+ 3304*a^2*b^6 + 121*b^8)*cos(d*x + c)^3 + 15*(128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*cos
(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 44.4758, size = 1115, normalized size = 2.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**8*x*sin(c + d*x)**2/2 + a**8*x*cos(c + d*x)**2/2 + a**8*sin(c + d*x)*cos(c + d*x)/(2*d) - 8*a**7
*b*cos(c + d*x)**3/(3*d) + 7*a**6*b**2*x*sin(c + d*x)**4/2 + 7*a**6*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2 + 7
*a**6*b**2*x*cos(c + d*x)**4/2 + 7*a**6*b**2*sin(c + d*x)**3*cos(c + d*x)/(2*d) - 7*a**6*b**2*sin(c + d*x)*cos
(c + d*x)**3/(2*d) - 56*a**5*b**3*sin(c + d*x)**2*cos(c + d*x)**3/(3*d) - 112*a**5*b**3*cos(c + d*x)**5/(15*d)
+ 35*a**4*b**4*x*sin(c + d*x)**6/8 + 105*a**4*b**4*x*sin(c + d*x)**4*cos(c + d*x)**2/8 + 105*a**4*b**4*x*sin(
c + d*x)**2*cos(c + d*x)**4/8 + 35*a**4*b**4*x*cos(c + d*x)**6/8 + 35*a**4*b**4*sin(c + d*x)**5*cos(c + d*x)/(
8*d) - 35*a**4*b**4*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) - 35*a**4*b**4*sin(c + d*x)*cos(c + d*x)**5/(8*d) -
56*a**3*b**5*sin(c + d*x)**4*cos(c + d*x)**3/(3*d) - 224*a**3*b**5*sin(c + d*x)**2*cos(c + d*x)**5/(15*d) - 64
*a**3*b**5*cos(c + d*x)**7/(15*d) + 35*a**2*b**6*x*sin(c + d*x)**8/32 + 35*a**2*b**6*x*sin(c + d*x)**6*cos(c +
d*x)**2/8 + 105*a**2*b**6*x*sin(c + d*x)**4*cos(c + d*x)**4/16 + 35*a**2*b**6*x*sin(c + d*x)**2*cos(c + d*x)*
*6/8 + 35*a**2*b**6*x*cos(c + d*x)**8/32 + 35*a**2*b**6*sin(c + d*x)**7*cos(c + d*x)/(32*d) - 511*a**2*b**6*si
n(c + d*x)**5*cos(c + d*x)**3/(96*d) - 385*a**2*b**6*sin(c + d*x)**3*cos(c + d*x)**5/(96*d) - 35*a**2*b**6*sin
(c + d*x)*cos(c + d*x)**7/(32*d) - 8*a*b**7*sin(c + d*x)**6*cos(c + d*x)**3/(3*d) - 16*a*b**7*sin(c + d*x)**4*
cos(c + d*x)**5/(5*d) - 64*a*b**7*sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - 128*a*b**7*cos(c + d*x)**9/(315*d)
+ 7*b**8*x*sin(c + d*x)**10/256 + 35*b**8*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 35*b**8*x*sin(c + d*x)**6*co
s(c + d*x)**4/128 + 35*b**8*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 35*b**8*x*sin(c + d*x)**2*cos(c + d*x)**8/
256 + 7*b**8*x*cos(c + d*x)**10/256 + 7*b**8*sin(c + d*x)**9*cos(c + d*x)/(256*d) - 79*b**8*sin(c + d*x)**7*co
s(c + d*x)**3/(384*d) - 7*b**8*sin(c + d*x)**5*cos(c + d*x)**5/(30*d) - 49*b**8*sin(c + d*x)**3*cos(c + d*x)**
7/(384*d) - 7*b**8*sin(c + d*x)*cos(c + d*x)**9/(256*d), Ne(d, 0)), (x*(a + b*sin(c))**8*cos(c)**2, True))

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Giac [A]  time = 1.17001, size = 491, normalized size = 1.16 \begin{align*} \frac{a b^{7} \cos \left (9 \, d x + 9 \, c\right )}{288 \, d} + \frac{b^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{1}{256} \,{\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} x - \frac{{\left (28 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac{{\left (28 \, a^{5} b^{3} + 21 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac{{\left (16 \, a^{7} b + 28 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{24 \, d} - \frac{{\left (32 \, a^{7} b + 112 \, a^{5} b^{3} + 70 \, a^{3} b^{5} + 7 \, a b^{7}\right )} \cos \left (d x + c\right )}{16 \, d} - \frac{{\left (56 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac{{\left (1120 \, a^{4} b^{4} + 448 \, a^{2} b^{6} + 13 \, b^{8}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac{{\left (224 \, a^{6} b^{2} + 280 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac{{\left (128 \, a^{8} - 560 \, a^{4} b^{4} - 224 \, a^{2} b^{6} - 7 \, b^{8}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*a*b^7*cos(9*d*x + 9*c)/d + 1/5120*b^8*sin(10*d*x + 10*c)/d + 1/256*(128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4
+ 280*a^2*b^6 + 7*b^8)*x - 1/224*(28*a^3*b^5 + 5*a*b^7)*cos(7*d*x + 7*c)/d + 1/40*(28*a^5*b^3 + 21*a^3*b^5 +
2*a*b^7)*cos(5*d*x + 5*c)/d - 1/24*(16*a^7*b + 28*a^5*b^3 + 7*a^3*b^5)*cos(3*d*x + 3*c)/d - 1/16*(32*a^7*b + 1
12*a^5*b^3 + 70*a^3*b^5 + 7*a*b^7)*cos(d*x + c)/d - 1/2048*(56*a^2*b^6 + 3*b^8)*sin(8*d*x + 8*c)/d + 1/3072*(1
120*a^4*b^4 + 448*a^2*b^6 + 13*b^8)*sin(6*d*x + 6*c)/d - 1/256*(224*a^6*b^2 + 280*a^4*b^4 + 56*a^2*b^6 + b^8)*
sin(4*d*x + 4*c)/d + 1/512*(128*a^8 - 560*a^4*b^4 - 224*a^2*b^6 - 7*b^8)*sin(2*d*x + 2*c)/d