3.75 \(\int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=103 \[ \frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {7 x}{8 a^3}+\frac {2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2680, 2682, 2635, 8} \[ \frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {7 x}{8 a^3}+\frac {2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

(7*x)/(8*a^3) + (7*Cos[c + d*x]^5)/(15*a^3*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (7*Cos[c + d*x]^3*Si
n[c + d*x])/(12*a^3*d) + (2*Cos[c + d*x]^7)/(3*a*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 2680

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

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]

Rubi steps

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

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Mathematica [A]  time = 1.08, size = 141, normalized size = 1.37 \[ -\frac {\left (\sqrt {\sin (c+d x)+1} \left (24 \sin ^5(c+d x)-114 \sin ^4(c+d x)+202 \sin ^3(c+d x)-127 \sin ^2(c+d x)-121 \sin (c+d x)+136\right )-210 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}\right ) \cos ^9(c+d x)}{120 a^3 d (\sin (c+d x)-1)^5 (\sin (c+d x)+1)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

-1/120*(Cos[c + d*x]^9*(-210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c +
d*x]]*(136 - 121*Sin[c + d*x] - 127*Sin[c + d*x]^2 + 202*Sin[c + d*x]^3 - 114*Sin[c + d*x]^4 + 24*Sin[c + d*x]
^5)))/(a^3*d*(-1 + Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))

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fricas [A]  time = 0.54, size = 60, normalized size = 0.58 \[ -\frac {24 \, \cos \left (d x + c\right )^{5} - 160 \, \cos \left (d x + c\right )^{3} - 105 \, d x + 15 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/120*(24*cos(d*x + c)^5 - 160*cos(d*x + c)^3 - 105*d*x + 15*(6*cos(d*x + c)^3 - 7*cos(d*x + c))*sin(d*x + c)
)/(a^3*d)

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giac [A]  time = 0.39, size = 140, normalized size = 1.36 \[ \frac {\frac {105 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 136\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/120*(105*(d*x + c)/a^3 - 2*(15*tan(1/2*d*x + 1/2*c)^9 - 360*tan(1/2*d*x + 1/2*c)^8 + 390*tan(1/2*d*x + 1/2*c
)^7 - 960*tan(1/2*d*x + 1/2*c)^6 - 400*tan(1/2*d*x + 1/2*c)^4 - 390*tan(1/2*d*x + 1/2*c)^3 - 320*tan(1/2*d*x +
1/2*c)^2 - 15*tan(1/2*d*x + 1/2*c) - 136)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*a^3))/d

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maple [B]  time = 0.21, size = 313, normalized size = 3.04 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {6 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {16 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {20 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {34}{15 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8/(a+a*sin(d*x+c))^3,x)

[Out]

-1/4/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9+6/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*
c)^8-13/2/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7+16/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*
x+1/2*c)^6+20/3/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4+13/2/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*ta
n(1/2*d*x+1/2*c)^3+16/3/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2+1/4/a^3/d/(1+tan(1/2*d*x+1/2*c)^
2)^5*tan(1/2*d*x+1/2*c)+34/15/a^3/d/(1+tan(1/2*d*x+1/2*c)^2)^5+7/4/a^3/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.77, size = 310, normalized size = 3.01 \[ \frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {390 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 136}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/60*((15*sin(d*x + c)/(cos(d*x + c) + 1) + 320*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 390*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 + 400*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 960*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 390*sin(
d*x + c)^7/(cos(d*x + c) + 1)^7 + 360*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 15*sin(d*x + c)^9/(cos(d*x + c) +
1)^9 + 136)/(a^3 + 5*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10
*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^3*sin(d*x + c)^10/(co
s(d*x + c) + 1)^10) + 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

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mupad [B]  time = 4.72, size = 81, normalized size = 0.79 \[ \frac {7\,x}{8\,a^3}+\frac {4\,{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3\,d}-\frac {3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}+\frac {7\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^8/(a + a*sin(c + d*x))^3,x)

[Out]

(7*x)/(8*a^3) + (4*cos(c + d*x)^3)/(3*a^3*d) - cos(c + d*x)^5/(5*a^3*d) - (3*cos(c + d*x)^3*sin(c + d*x))/(4*a
^3*d) + (7*cos(c + d*x)*sin(c + d*x))/(8*a^3*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(cos(d*x+c)**8/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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