3.726 \(\int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=124 \[ -\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{15 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {x}{8 a^2}-\frac {\cos ^9(c+d x)}{5 d (a \sin (c+d x)+a)^2} \]

[Out]

-1/8*x/a^2-2/35*cos(d*x+c)^7/a^2/d-1/8*cos(d*x+c)*sin(d*x+c)/a^2/d-1/12*cos(d*x+c)^3*sin(d*x+c)/a^2/d-1/15*cos
(d*x+c)^5*sin(d*x+c)/a^2/d-1/5*cos(d*x+c)^9/d/(a+a*sin(d*x+c))^2

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Rubi [A]  time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2859, 2682, 2635, 8} \[ -\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{15 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {x}{8 a^2}-\frac {\cos ^9(c+d x)}{5 d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(8*a^2) - (2*Cos[c + d*x]^7)/(35*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]^3*Sin[c + d
*x])/(12*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(15*a^2*d) - Cos[c + d*x]^9/(5*d*(a + a*Sin[c + d*x])^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 2682

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

Rule 2859

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

Rubi steps

\begin {align*} \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \int \frac {\cos ^8(c+d x)}{a+a \sin (c+d x)} \, dx}{5 a}\\ &=-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \int \cos ^6(c+d x) \, dx}{5 a^2}\\ &=-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\int \cos ^4(c+d x) \, dx}{3 a^2}\\ &=-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\int \cos ^2(c+d x) \, dx}{4 a^2}\\ &=-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\int 1 \, dx}{8 a^2}\\ &=-\frac {x}{8 a^2}-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}\\ \end {align*}

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Mathematica [B]  time = 4.86, size = 418, normalized size = 3.37 \[ -\frac {1680 d x \sin \left (\frac {c}{2}\right )-1155 \sin \left (\frac {c}{2}+d x\right )+1155 \sin \left (\frac {3 c}{2}+d x\right )+210 \sin \left (\frac {3 c}{2}+2 d x\right )+210 \sin \left (\frac {5 c}{2}+2 d x\right )-525 \sin \left (\frac {5 c}{2}+3 d x\right )+525 \sin \left (\frac {7 c}{2}+3 d x\right )-210 \sin \left (\frac {7 c}{2}+4 d x\right )-210 \sin \left (\frac {9 c}{2}+4 d x\right )-63 \sin \left (\frac {9 c}{2}+5 d x\right )+63 \sin \left (\frac {11 c}{2}+5 d x\right )-70 \sin \left (\frac {11 c}{2}+6 d x\right )-70 \sin \left (\frac {13 c}{2}+6 d x\right )+15 \sin \left (\frac {13 c}{2}+7 d x\right )-15 \sin \left (\frac {15 c}{2}+7 d x\right )+70 \cos \left (\frac {c}{2}\right ) (24 d x+7)+1155 \cos \left (\frac {c}{2}+d x\right )+1155 \cos \left (\frac {3 c}{2}+d x\right )+210 \cos \left (\frac {3 c}{2}+2 d x\right )-210 \cos \left (\frac {5 c}{2}+2 d x\right )+525 \cos \left (\frac {5 c}{2}+3 d x\right )+525 \cos \left (\frac {7 c}{2}+3 d x\right )-210 \cos \left (\frac {7 c}{2}+4 d x\right )+210 \cos \left (\frac {9 c}{2}+4 d x\right )+63 \cos \left (\frac {9 c}{2}+5 d x\right )+63 \cos \left (\frac {11 c}{2}+5 d x\right )-70 \cos \left (\frac {11 c}{2}+6 d x\right )+70 \cos \left (\frac {13 c}{2}+6 d x\right )-15 \cos \left (\frac {13 c}{2}+7 d x\right )-15 \cos \left (\frac {15 c}{2}+7 d x\right )-490 \sin \left (\frac {c}{2}\right )}{13440 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/13440*(70*(7 + 24*d*x)*Cos[c/2] + 1155*Cos[c/2 + d*x] + 1155*Cos[(3*c)/2 + d*x] + 210*Cos[(3*c)/2 + 2*d*x]
- 210*Cos[(5*c)/2 + 2*d*x] + 525*Cos[(5*c)/2 + 3*d*x] + 525*Cos[(7*c)/2 + 3*d*x] - 210*Cos[(7*c)/2 + 4*d*x] +
210*Cos[(9*c)/2 + 4*d*x] + 63*Cos[(9*c)/2 + 5*d*x] + 63*Cos[(11*c)/2 + 5*d*x] - 70*Cos[(11*c)/2 + 6*d*x] + 70*
Cos[(13*c)/2 + 6*d*x] - 15*Cos[(13*c)/2 + 7*d*x] - 15*Cos[(15*c)/2 + 7*d*x] - 490*Sin[c/2] + 1680*d*x*Sin[c/2]
- 1155*Sin[c/2 + d*x] + 1155*Sin[(3*c)/2 + d*x] + 210*Sin[(3*c)/2 + 2*d*x] + 210*Sin[(5*c)/2 + 2*d*x] - 525*S
in[(5*c)/2 + 3*d*x] + 525*Sin[(7*c)/2 + 3*d*x] - 210*Sin[(7*c)/2 + 4*d*x] - 210*Sin[(9*c)/2 + 4*d*x] - 63*Sin[
(9*c)/2 + 5*d*x] + 63*Sin[(11*c)/2 + 5*d*x] - 70*Sin[(11*c)/2 + 6*d*x] - 70*Sin[(13*c)/2 + 6*d*x] + 15*Sin[(13
*c)/2 + 7*d*x] - 15*Sin[(15*c)/2 + 7*d*x])/(a^2*d*(Cos[c/2] + Sin[c/2]))

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fricas [A]  time = 0.46, size = 70, normalized size = 0.56 \[ \frac {120 \, \cos \left (d x + c\right )^{7} - 336 \, \cos \left (d x + c\right )^{5} - 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(120*cos(d*x + c)^7 - 336*cos(d*x + c)^5 - 105*d*x + 35*(8*cos(d*x + c)^5 - 2*cos(d*x + c)^3 - 3*cos(d*x
+ c))*sin(d*x + c))/(a^2*d)

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giac [A]  time = 0.19, size = 192, normalized size = 1.55 \[ -\frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1176 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 216\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*(105*(d*x + c)/a^2 + 2*(105*tan(1/2*d*x + 1/2*c)^13 + 840*tan(1/2*d*x + 1/2*c)^12 - 1540*tan(1/2*d*x +
1/2*c)^11 + 3360*tan(1/2*d*x + 1/2*c)^10 + 1085*tan(1/2*d*x + 1/2*c)^9 + 840*tan(1/2*d*x + 1/2*c)^8 + 6720*tan
(1/2*d*x + 1/2*c)^6 - 1085*tan(1/2*d*x + 1/2*c)^5 + 1176*tan(1/2*d*x + 1/2*c)^4 + 1540*tan(1/2*d*x + 1/2*c)^3
+ 672*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 216)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*a^2))/d

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maple [B]  time = 0.31, size = 449, normalized size = 3.62 \[ -\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {2 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {8 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {31 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {16 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {31 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {14 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {18}{35 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13-2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2
*c)^12+11/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^11-8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*
d*x+1/2*c)^10-31/12/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9-2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*t
an(1/2*d*x+1/2*c)^8-16/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6+31/12/d/a^2/(1+tan(1/2*d*x+1/2*c)
^2)^7*tan(1/2*d*x+1/2*c)^5-14/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4-11/3/d/a^2/(1+tan(1/2*d*
x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3-8/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2+1/4/d/a^2/(1+tan(
1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)-18/35/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7-1/4/d/a^2*arctan(tan(1/2*d*x+1/2
*c))

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maxima [B]  time = 0.46, size = 436, normalized size = 3.52 \[ \frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {672 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1176 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1085 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {840 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1085 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3360 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1540 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {840 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 216}{a^{2} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 672*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1540*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 1176*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1085*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 672
0*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 840*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 1085*sin(d*x + c)^9/(cos(d*x
+ c) + 1)^9 - 3360*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1540*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 840*s
in(d*x + c)^12/(cos(d*x + c) + 1)^12 - 105*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 216)/(a^2 + 7*a^2*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 + 21*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 35*a^2*sin(d*x + c)^6/(cos(d*x + c)
+ 1)^6 + 35*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 21*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 7*a^2*sin
(d*x + c)^12/(cos(d*x + c) + 1)^12 + a^2*sin(d*x + c)^14/(cos(d*x + c) + 1)^14) - 105*arctan(sin(d*x + c)/(cos
(d*x + c) + 1))/a^2)/d

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mupad [B]  time = 12.65, size = 186, normalized size = 1.50 \[ -\frac {x}{8\,a^2}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {18}{35}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- x/(8*a^2) - ((8*tan(c/2 + (d*x)/2)^2)/5 - tan(c/2 + (d*x)/2)/4 + (11*tan(c/2 + (d*x)/2)^3)/3 + (14*tan(c/2 +
(d*x)/2)^4)/5 - (31*tan(c/2 + (d*x)/2)^5)/12 + 16*tan(c/2 + (d*x)/2)^6 + 2*tan(c/2 + (d*x)/2)^8 + (31*tan(c/2
+ (d*x)/2)^9)/12 + 8*tan(c/2 + (d*x)/2)^10 - (11*tan(c/2 + (d*x)/2)^11)/3 + 2*tan(c/2 + (d*x)/2)^12 + tan(c/2
+ (d*x)/2)^13/4 + 18/35)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^7)

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sympy [A]  time = 172.04, size = 3196, normalized size = 25.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-105*d*x*tan(c/2 + d*x/2)**14/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 +
17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 176
40*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 735*d*x*tan(c/2 + d*x/2)**12/(
840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400
*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*
d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 2205*d*x*tan(c/2 + d*x/2)**10/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a*
*2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*
d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 367
5*d*x*tan(c/2 + d*x/2)**8/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*t
an(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c
/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 3675*d*x*tan(c/2 + d*x/2)**6/(840*a**2*d*tan(
c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2
+ d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x
/2)**2 + 840*a**2*d) - 2205*d*x*tan(c/2 + d*x/2)**4/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d
*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/
2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 735*d*x*tan(c/2 + d
*x/2)**2/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**
10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 +
5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 105*d*x/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2
+ d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 +
d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 210*tan(c/2 + d
*x/2)**13/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)*
*10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 +
5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 1680*tan(c/2 + d*x/2)**12/(840*a**2*d*tan(c/2 + d*x/2)**14 +
5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 2940
0*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d
) + 3080*tan(c/2 + d*x/2)**11/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2
*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*t
an(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 6720*tan(c/2 + d*x/2)**10/(840*a**2*d*tan
(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/
2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*
x/2)**2 + 840*a**2*d) - 2170*tan(c/2 + d*x/2)**9/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/
2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)*
*6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 1680*tan(c/2 + d*x/2)*
*8/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 2
9400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a
**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 13440*tan(c/2 + d*x/2)**6/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a*
*2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*
d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) + 217
0*tan(c/2 + d*x/2)**5/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c
/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 +
d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 2352*tan(c/2 + d*x/2)**4/(840*a**2*d*tan(c/2 + d*
x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2
)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 +
840*a**2*d) - 3080*tan(c/2 + d*x/2)**3/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 +
17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 1764
0*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 1344*tan(c/2 + d*x/2)**2/(840*a
**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2
*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan
(c/2 + d*x/2)**2 + 840*a**2*d) + 210*tan(c/2 + d*x/2)/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 +
d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*
x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 432/(840*a**2*d*t
an(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(
c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 +
d*x/2)**2 + 840*a**2*d), Ne(d, 0)), (x*sin(c)*cos(c)**8/(a*sin(c) + a)**2, True))

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