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

Optimal. Leaf size=183 \[ -\frac {20 \cos ^3(c+d x)}{3003 a^3 d (a \sin (c+d x)+a)^5}-\frac {8 \cos ^3(c+d x)}{9009 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {8 \cos ^3(c+d x)}{3003 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {20 \cos ^3(c+d x)}{1287 a^2 d (a \sin (c+d x)+a)^6}-\frac {5 \cos ^3(c+d x)}{143 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8} \]

[Out]

-1/13*cos(d*x+c)^3/d/(a+a*sin(d*x+c))^8-5/143*cos(d*x+c)^3/a/d/(a+a*sin(d*x+c))^7-20/1287*cos(d*x+c)^3/a^2/d/(
a+a*sin(d*x+c))^6-20/3003*cos(d*x+c)^3/a^3/d/(a+a*sin(d*x+c))^5-8/3003*cos(d*x+c)^3/d/(a^2+a^2*sin(d*x+c))^4-8
/9009*cos(d*x+c)^3/a^2/d/(a^2+a^2*sin(d*x+c))^3

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Rubi [A]  time = 0.27, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2672, 2671} \[ -\frac {8 \cos ^3(c+d x)}{9009 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {8 \cos ^3(c+d x)}{3003 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {20 \cos ^3(c+d x)}{3003 a^3 d (a \sin (c+d x)+a)^5}-\frac {20 \cos ^3(c+d x)}{1287 a^2 d (a \sin (c+d x)+a)^6}-\frac {5 \cos ^3(c+d x)}{143 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[c + d*x]^3/(13*d*(a + a*Sin[c + d*x])^8) - (5*Cos[c + d*x]^3)/(143*a*d*(a + a*Sin[c + d*x])^7) - (20*Cos[
c + d*x]^3)/(1287*a^2*d*(a + a*Sin[c + d*x])^6) - (20*Cos[c + d*x]^3)/(3003*a^3*d*(a + a*Sin[c + d*x])^5) - (8
*Cos[c + d*x]^3)/(3003*d*(a^2 + a^2*Sin[c + d*x])^4) - (8*Cos[c + d*x]^3)/(9009*a^2*d*(a^2 + a^2*Sin[c + d*x])
^3)

Rule 2671

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)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rule 2672

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)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}+\frac {5 \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{13 a}\\ &=-\frac {\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac {5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}+\frac {20 \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{143 a^2}\\ &=-\frac {\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac {5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac {20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}+\frac {20 \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{429 a^3}\\ &=-\frac {\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac {5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac {20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}-\frac {20 \cos ^3(c+d x)}{3003 a^3 d (a+a \sin (c+d x))^5}+\frac {40 \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{3003 a^4}\\ &=-\frac {\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac {5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac {20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}-\frac {20 \cos ^3(c+d x)}{3003 a^3 d (a+a \sin (c+d x))^5}-\frac {8 \cos ^3(c+d x)}{3003 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {8 \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{3003 a^5}\\ &=-\frac {\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac {5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac {20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}-\frac {20 \cos ^3(c+d x)}{3003 a^3 d (a+a \sin (c+d x))^5}-\frac {8 \cos ^3(c+d x)}{9009 a^5 d (a+a \sin (c+d x))^3}-\frac {8 \cos ^3(c+d x)}{3003 d \left (a^2+a^2 \sin (c+d x)\right )^4}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 78, normalized size = 0.43 \[ -\frac {\left (8 \sin ^5(c+d x)+64 \sin ^4(c+d x)+236 \sin ^3(c+d x)+544 \sin ^2(c+d x)+911 \sin (c+d x)+1240\right ) \cos ^3(c+d x)}{9009 a^8 d (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9009*(Cos[c + d*x]^3*(1240 + 911*Sin[c + d*x] + 544*Sin[c + d*x]^2 + 236*Sin[c + d*x]^3 + 64*Sin[c + d*x]^4
+ 8*Sin[c + d*x]^5))/(a^8*d*(1 + Sin[c + d*x])^8)

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fricas [A]  time = 0.73, size = 339, normalized size = 1.85 \[ \frac {8 \, \cos \left (d x + c\right )^{7} - 48 \, \cos \left (d x + c\right )^{6} - 196 \, \cos \left (d x + c\right )^{5} + 280 \, \cos \left (d x + c\right )^{4} + 735 \, \cos \left (d x + c\right )^{3} - 378 \, \cos \left (d x + c\right )^{2} - {\left (8 \, \cos \left (d x + c\right )^{6} + 56 \, \cos \left (d x + c\right )^{5} - 140 \, \cos \left (d x + c\right )^{4} - 420 \, \cos \left (d x + c\right )^{3} + 315 \, \cos \left (d x + c\right )^{2} + 693 \, \cos \left (d x + c\right ) + 1386\right )} \sin \left (d x + c\right ) + 693 \, \cos \left (d x + c\right ) + 1386}{9009 \, {\left (a^{8} d \cos \left (d x + c\right )^{7} + 7 \, a^{8} d \cos \left (d x + c\right )^{6} - 18 \, a^{8} d \cos \left (d x + c\right )^{5} - 56 \, a^{8} d \cos \left (d x + c\right )^{4} + 48 \, a^{8} d \cos \left (d x + c\right )^{3} + 112 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d \cos \left (d x + c\right ) - 64 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{6} - 6 \, a^{8} d \cos \left (d x + c\right )^{5} - 24 \, a^{8} d \cos \left (d x + c\right )^{4} + 32 \, a^{8} d \cos \left (d x + c\right )^{3} + 80 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d \cos \left (d x + c\right ) - 64 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9009*(8*cos(d*x + c)^7 - 48*cos(d*x + c)^6 - 196*cos(d*x + c)^5 + 280*cos(d*x + c)^4 + 735*cos(d*x + c)^3 -
378*cos(d*x + c)^2 - (8*cos(d*x + c)^6 + 56*cos(d*x + c)^5 - 140*cos(d*x + c)^4 - 420*cos(d*x + c)^3 + 315*cos
(d*x + c)^2 + 693*cos(d*x + c) + 1386)*sin(d*x + c) + 693*cos(d*x + c) + 1386)/(a^8*d*cos(d*x + c)^7 + 7*a^8*d
*cos(d*x + c)^6 - 18*a^8*d*cos(d*x + c)^5 - 56*a^8*d*cos(d*x + c)^4 + 48*a^8*d*cos(d*x + c)^3 + 112*a^8*d*cos(
d*x + c)^2 - 32*a^8*d*cos(d*x + c) - 64*a^8*d + (a^8*d*cos(d*x + c)^6 - 6*a^8*d*cos(d*x + c)^5 - 24*a^8*d*cos(
d*x + c)^4 + 32*a^8*d*cos(d*x + c)^3 + 80*a^8*d*cos(d*x + c)^2 - 32*a^8*d*cos(d*x + c) - 64*a^8*d)*sin(d*x + c
))

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giac [A]  time = 0.78, size = 177, normalized size = 0.97 \[ -\frac {2 \, {\left (9009 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 45045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 183183 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 435435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 810810 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1051050 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1076790 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 785070 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 451165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 171457 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 51675 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7111 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1240\right )}}{9009 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9009*(9009*tan(1/2*d*x + 1/2*c)^12 + 45045*tan(1/2*d*x + 1/2*c)^11 + 183183*tan(1/2*d*x + 1/2*c)^10 + 43543
5*tan(1/2*d*x + 1/2*c)^9 + 810810*tan(1/2*d*x + 1/2*c)^8 + 1051050*tan(1/2*d*x + 1/2*c)^7 + 1076790*tan(1/2*d*
x + 1/2*c)^6 + 785070*tan(1/2*d*x + 1/2*c)^5 + 451165*tan(1/2*d*x + 1/2*c)^4 + 171457*tan(1/2*d*x + 1/2*c)^3 +
51675*tan(1/2*d*x + 1/2*c)^2 + 7111*tan(1/2*d*x + 1/2*c) + 1240)/(a^8*d*(tan(1/2*d*x + 1/2*c) + 1)^13)

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maple [A]  time = 0.30, size = 205, normalized size = 1.12 \[ \frac {-\frac {480}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {864}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}+\frac {1472}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {4544}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}-\frac {11680}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {9056}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {2672}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {200}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {188}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {256}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}}{d \,a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^8*(-240/(tan(1/2*d*x+1/2*c)+1)^5+432/(tan(1/2*d*x+1/2*c)+1)^10+736/(tan(1/2*d*x+1/2*c)+1)^8+64/(tan(1/2*
d*x+1/2*c)+1)^12-2272/11/(tan(1/2*d*x+1/2*c)+1)^11-5840/9/(tan(1/2*d*x+1/2*c)+1)^9-4528/7/(tan(1/2*d*x+1/2*c)+
1)^7+1336/3/(tan(1/2*d*x+1/2*c)+1)^6+7/(tan(1/2*d*x+1/2*c)+1)^2-1/(tan(1/2*d*x+1/2*c)+1)+100/(tan(1/2*d*x+1/2*
c)+1)^4-94/3/(tan(1/2*d*x+1/2*c)+1)^3-128/13/(tan(1/2*d*x+1/2*c)+1)^13)

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maxima [B]  time = 0.52, size = 547, normalized size = 2.99 \[ -\frac {2 \, {\left (\frac {7111 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {51675 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {171457 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {451165 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {785070 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1076790 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1051050 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {810810 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {435435 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {183183 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {45045 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {9009 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + 1240\right )}}{9009 \, {\left (a^{8} + \frac {13 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {78 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {286 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {715 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1287 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1716 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1716 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1287 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {715 \, a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {286 \, a^{8} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {78 \, a^{8} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {13 \, a^{8} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{8} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9009*(7111*sin(d*x + c)/(cos(d*x + c) + 1) + 51675*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 171457*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 + 451165*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 785070*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 1076790*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1051050*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 810810*sin(
d*x + c)^8/(cos(d*x + c) + 1)^8 + 435435*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 183183*sin(d*x + c)^10/(cos(d*x
+ c) + 1)^10 + 45045*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 9009*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 124
0)/((a^8 + 13*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 78*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 286*a^8*sin(d
*x + c)^3/(cos(d*x + c) + 1)^3 + 715*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1287*a^8*sin(d*x + c)^5/(cos(d*
x + c) + 1)^5 + 1716*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1716*a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 +
1287*a^8*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 715*a^8*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 286*a^8*sin(d*x +
c)^10/(cos(d*x + c) + 1)^10 + 78*a^8*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 13*a^8*sin(d*x + c)^12/(cos(d*x
+ c) + 1)^12 + a^8*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)*d)

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mupad [B]  time = 8.12, size = 162, normalized size = 0.89 \[ \frac {\sqrt {2}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {14983\,\cos \left (c+d\,x\right )}{2}-\frac {63921\,\sin \left (c+d\,x\right )}{2}+17605\,\cos \left (2\,c+2\,d\,x\right )-\frac {15365\,\cos \left (3\,c+3\,d\,x\right )}{4}-\frac {6943\,\cos \left (4\,c+4\,d\,x\right )}{4}+\frac {937\,\cos \left (5\,c+5\,d\,x\right )}{4}+\frac {77\,\cos \left (6\,c+6\,d\,x\right )}{4}+\frac {28743\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {27027\,\sin \left (3\,c+3\,d\,x\right )}{4}-\frac {5005\,\sin \left (4\,c+4\,d\,x\right )}{4}-\frac {1079\,\sin \left (5\,c+5\,d\,x\right )}{4}+\frac {39\,\sin \left (6\,c+6\,d\,x\right )}{2}-21013\right )}{576576\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*cos(c/2 + (d*x)/2)*((14983*cos(c + d*x))/2 - (63921*sin(c + d*x))/2 + 17605*cos(2*c + 2*d*x) - (15365
*cos(3*c + 3*d*x))/4 - (6943*cos(4*c + 4*d*x))/4 + (937*cos(5*c + 5*d*x))/4 + (77*cos(6*c + 6*d*x))/4 + (28743
*sin(2*c + 2*d*x))/4 + (27027*sin(3*c + 3*d*x))/4 - (5005*sin(4*c + 4*d*x))/4 - (1079*sin(5*c + 5*d*x))/4 + (3
9*sin(6*c + 6*d*x))/2 - 21013))/(576576*a^8*d*cos(c/2 - pi/4 + (d*x)/2)^13)

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

[Out]

Timed out

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