3.110 \(\int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=227 \[ \frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a b^2 \sin (c+d x) \cos (c+d x)}{2 d \left (a^2+b^2\right )^2}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d \left (a^2+b^2\right )}+\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {3 a x}{8 \left (a^2+b^2\right )}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {b^3 \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )^2} \]

[Out]

a*b^4*x/(a^2+b^2)^3+1/2*a*b^2*x/(a^2+b^2)^2+3/8*a*x/(a^2+b^2)+1/2*b^3*cos(d*x+c)^2/(a^2+b^2)^2/d+1/4*b*cos(d*x
+c)^4/(a^2+b^2)/d+b^5*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^3/d+1/2*a*b^2*cos(d*x+c)*sin(d*x+c)/(a^2+b^2)^2/
d+3/8*a*cos(d*x+c)*sin(d*x+c)/(a^2+b^2)/d+1/4*a*cos(d*x+c)^3*sin(d*x+c)/(a^2+b^2)/d

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3100, 2635, 8, 3098, 3133} \[ \frac {b^3 \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a b^2 \sin (c+d x) \cos (c+d x)}{2 d \left (a^2+b^2\right )^2}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d \left (a^2+b^2\right )}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {3 a x}{8 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

(a*b^4*x)/(a^2 + b^2)^3 + (a*b^2*x)/(2*(a^2 + b^2)^2) + (3*a*x)/(8*(a^2 + b^2)) + (b^3*Cos[c + d*x]^2)/(2*(a^2
+ b^2)^2*d) + (b*Cos[c + d*x]^4)/(4*(a^2 + b^2)*d) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^
3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*(a^2 + b^2)^2*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*(a^2 + b^2)*
d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*(a^2 + b^2)*d)

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 3098

Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(a*x)/(a^2 + b^2), x] + Dist[b/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c +
d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3100

Int[cos[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(b*Cos[c + d*x]^(m - 1))/(d*(a^2 + b^2)*(m - 1)), x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1), x]
, x] + Dist[b^2/(a^2 + b^2), Int[Cos[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a,
b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]

Rule 3133

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rubi steps

\begin {align*} \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx &=\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {a \int \cos ^4(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {\left (a b^2\right ) \int \cos ^2(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {(3 a) \int \cos ^2(c+d x) \, dx}{4 \left (a^2+b^2\right )}\\ &=\frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {a b^2 \cos (c+d x) \sin (c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {b^5 \int \frac {b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a b^2\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^2}+\frac {(3 a) \int 1 \, dx}{8 \left (a^2+b^2\right )}\\ &=\frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {3 a x}{8 \left (a^2+b^2\right )}+\frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a b^2 \cos (c+d x) \sin (c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.42, size = 218, normalized size = 0.96 \[ \frac {8 a^5 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+12 a^5 c+12 a^5 d x+24 a^3 b^2 \sin (2 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+40 a^3 b^2 c+40 a^3 b^2 d x+b \left (a^2+b^2\right )^2 \cos (4 (c+d x))+4 b \left (a^4+4 a^2 b^2+3 b^4\right ) \cos (2 (c+d x))+32 b^5 \log (a \cos (c+d x)+b \sin (c+d x))+16 a b^4 \sin (2 (c+d x))+a b^4 \sin (4 (c+d x))+60 a b^4 c+60 a b^4 d x}{32 d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

(12*a^5*c + 40*a^3*b^2*c + 60*a*b^4*c + 12*a^5*d*x + 40*a^3*b^2*d*x + 60*a*b^4*d*x + 4*b*(a^4 + 4*a^2*b^2 + 3*
b^4)*Cos[2*(c + d*x)] + b*(a^2 + b^2)^2*Cos[4*(c + d*x)] + 32*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + 8*a^5
*Sin[2*(c + d*x)] + 24*a^3*b^2*Sin[2*(c + d*x)] + 16*a*b^4*Sin[2*(c + d*x)] + a^5*Sin[4*(c + d*x)] + 2*a^3*b^2
*Sin[4*(c + d*x)] + a*b^4*Sin[4*(c + d*x)])/(32*(a^2 + b^2)^3*d)

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 208, normalized size = 0.92 \[ \frac {4 \, b^{5} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 4 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/8*(4*b^5*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) + 2*(a^4*b + 2*a^2*b^3 + b^
5)*cos(d*x + c)^4 + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*d*x + 4*(a^2*b^3 + b^5)*cos(d*x + c)^2 + (2*(a^5 + 2*a^3*b
^2 + a*b^4)*cos(d*x + c)^3 + (3*a^5 + 10*a^3*b^2 + 7*a*b^4)*cos(d*x + c))*sin(d*x + c))/((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)*d)

________________________________________________________________________________________

giac [A]  time = 2.93, size = 322, normalized size = 1.42 \[ \frac {\frac {8 \, b^{6} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {4 \, b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {6 \, b^{5} \tan \left (d x + c\right )^{4} + 3 \, a^{5} \tan \left (d x + c\right )^{3} + 10 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 7 \, a b^{4} \tan \left (d x + c\right )^{3} + 4 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} + 16 \, b^{5} \tan \left (d x + c\right )^{2} + 5 \, a^{5} \tan \left (d x + c\right ) + 14 \, a^{3} b^{2} \tan \left (d x + c\right ) + 9 \, a b^{4} \tan \left (d x + c\right ) + 2 \, a^{4} b + 8 \, a^{2} b^{3} + 12 \, b^{5}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/8*(8*b^6*log(abs(b*tan(d*x + c) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - 4*b^5*log(tan(d*x + c)^2 + 1)/
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6) + (6*b^5*tan(d*x + c)^4 + 3*a^5*tan(d*x + c)^3 + 10*a^3*b^2*tan(d*x + c)^3 + 7*a*b^4*tan(d*x + c)^3 + 4*
a^2*b^3*tan(d*x + c)^2 + 16*b^5*tan(d*x + c)^2 + 5*a^5*tan(d*x + c) + 14*a^3*b^2*tan(d*x + c) + 9*a*b^4*tan(d*
x + c) + 2*a^4*b + 8*a^2*b^3 + 12*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(d*x + c)^2 + 1)^2))/d

________________________________________________________________________________________

maple [B]  time = 0.21, size = 524, normalized size = 2.31 \[ \frac {b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \left (\tan ^{3}\left (d x +c \right )\right ) a^{5}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {5 \left (\tan ^{3}\left (d x +c \right )\right ) a^{3} b^{2}}{4 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {7 \left (\tan ^{3}\left (d x +c \right )\right ) a \,b^{4}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {\left (\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{3}}{2 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {\left (\tan ^{2}\left (d x +c \right )\right ) b^{5}}{2 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {7 \tan \left (d x +c \right ) a^{3} b^{2}}{4 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {9 \tan \left (d x +c \right ) a \,b^{4}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {5 \tan \left (d x +c \right ) a^{5}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {a^{4} b}{4 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {a^{2} b^{3}}{d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}+\frac {3 b^{5}}{4 d \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\left (d x +c \right )+1\right )^{2}}-\frac {b^{5} \ln \left (\tan ^{2}\left (d x +c \right )+1\right )}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {15 \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{4}}{8 d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \arctan \left (\tan \left (d x +c \right )\right ) a^{5}}{8 d \left (a^{2}+b^{2}\right )^{3}}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right ) a^{3} b^{2}}{4 d \left (a^{2}+b^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a*cos(d*x+c)+b*sin(d*x+c)),x)

[Out]

1/d*b^5/(a^2+b^2)^3*ln(a+b*tan(d*x+c))+3/8/d/(a^2+b^2)^3/(tan(d*x+c)^2+1)^2*tan(d*x+c)^3*a^5+5/4/d/(a^2+b^2)^3
/(tan(d*x+c)^2+1)^2*tan(d*x+c)^3*a^3*b^2+7/8/d/(a^2+b^2)^3/(tan(d*x+c)^2+1)^2*tan(d*x+c)^3*a*b^4+1/2/d/(a^2+b^
2)^3/(tan(d*x+c)^2+1)^2*tan(d*x+c)^2*a^2*b^3+1/2/d/(a^2+b^2)^3/(tan(d*x+c)^2+1)^2*tan(d*x+c)^2*b^5+7/4/d/(a^2+
b^2)^3/(tan(d*x+c)^2+1)^2*tan(d*x+c)*a^3*b^2+9/8/d/(a^2+b^2)^3/(tan(d*x+c)^2+1)^2*tan(d*x+c)*a*b^4+5/8/d/(a^2+
b^2)^3/(tan(d*x+c)^2+1)^2*tan(d*x+c)*a^5+1/4/d/(a^2+b^2)^3/(tan(d*x+c)^2+1)^2*a^4*b+1/d/(a^2+b^2)^3/(tan(d*x+c
)^2+1)^2*a^2*b^3+3/4/d/(a^2+b^2)^3/(tan(d*x+c)^2+1)^2*b^5-1/2/d/(a^2+b^2)^3*b^5*ln(tan(d*x+c)^2+1)+15/8/d/(a^2
+b^2)^3*arctan(tan(d*x+c))*a*b^4+3/8/d/(a^2+b^2)^3*arctan(tan(d*x+c))*a^5+5/4/d/(a^2+b^2)^3*arctan(tan(d*x+c))
*a^3*b^2

________________________________________________________________________________________

maxima [B]  time = 1.67, size = 564, normalized size = 2.48 \[ \frac {\frac {4 \, b^{5} \log \left (-a - \frac {2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, b^{5} \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {16 \, b^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (3 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {{\left (3 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(4*b^5*log(-a - 2*b*sin(d*x + c)/(cos(d*x + c) + 1) + a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6) - 4*b^5*log(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
) + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
) - (16*b^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - (5*a^3 + 9*a*b^2)*sin(d*x + c)/(cos(d*x + c) + 1) + 8*(a^2*b
+ 2*b^3)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + (3*a^3 - a*b^2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - (3*a^3 -
a*b^2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 8*(a^2*b + 2*b^3)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + (5*a^3 +
9*a*b^2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 + 6*(a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*(a^4 + 2*a^2*b^2 +
b^4)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + (a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8))/d

________________________________________________________________________________________

mupad [B]  time = 11.16, size = 6099, normalized size = 26.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(a*cos(c + d*x) + b*sin(c + d*x)),x)

[Out]

(b^5*log(a + 2*b*tan(c/2 + (d*x)/2) - a*tan(c/2 + (d*x)/2)^2))/(d*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (64*b
^5*log(1/(cos(c + d*x) + 1)))/(d*(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)) - ((4*b^3*tan(c/2 + (d*x)/2)^4
)/(a^4 + b^4 + 2*a^2*b^2) - (tan(c/2 + (d*x)/2)*(9*a*b^2 + 5*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c/2 + (d
*x)/2)^3*(a*b^2 - 3*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^5*(a*b^2 - 3*a^3))/(4*(a^4 + b^4 +
2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^7*(9*a*b^2 + 5*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (2*b*tan(c/2 + (d*x)/2)^
2*(a^2 + 2*b^2))/(a^4 + b^4 + 2*a^2*b^2) + (2*b*tan(c/2 + (d*x)/2)^6*(a^2 + 2*b^2))/(a^4 + b^4 + 2*a^2*b^2))/(
d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) - (a*
atan((tan(c/2 + (d*x)/2)*((((64*b^5*((a*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13 + 2016*a^5*b^11 + 3152*a^7*b^9
+ 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8
*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 403
2*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^12 + b^12 + 6*a^
2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3
*a^2*b^4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 +
6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4
+ 192*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^
8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a*((9*a^15 - 192*a*b^14 - 222*a^3*b^12
+ 475*a^5*b^10 + 1089*a^7*b^8 + 894*a^9*b^6 + 388*a^11*b^4 + 87*a^13*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a
^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13 + 2016*a^5*b^
11 + 3152*a^7*b^9 + 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 2
0*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 +
6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(
a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^
4 + 192*a^4*b^2))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a^3*(3*a^4 + 15*b^
4 + 10*a^2*b^2)^3*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 +
1344*a^13*b^4 + 192*a^15*b^2))/(1024*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4
*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(9*a^12 + 256*b^12 - 1441*a^2*b^10 - 715*a^4*b^8 - 82*a^6*b^6 +
130*a^8*b^4 + 51*a^10*b^2))/(9*a^12 + 256*b^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490*a^6*b^6 + 250*a^8*b^4 + 69*
a^10*b^2)^2 - (2*a*b*((64*a*b^13 + 210*a^3*b^11 + 181*a^5*b^9 + 60*a^7*b^7 + 9*a^9*b^5)/(2*(a^12 + b^12 + 6*a^
2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) + (64*b^5*((9*a^15 - 192*a*b^14 - 222*a^3*b^12 +
475*a^5*b^10 + 1089*a^7*b^8 + 894*a^9*b^6 + 388*a^11*b^4 + 87*a^13*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*
b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13 + 2016*a^5*b^11
+ 3152*a^7*b^9 + 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a
^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 672
0*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^1
2 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 +
192*a^4*b^2)))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) + (a*((a*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13
+ 2016*a^5*b^11 + 3152*a^7*b^9 + 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 1
5*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 672
0*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 1
92*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 +
10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a*b^16 + 1
344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/(
(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4
+ 3*a^4*b^2)) - (a^2*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)^2*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^
7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/(2*(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192
*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b
^4 + 6*a^10*b^2)))*(9*a^10 - 496*b^10 + 305*a^2*b^8 + 412*a^4*b^6 + 238*a^6*b^4 + 60*a^8*b^2))/(9*a^12 + 256*b
^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490*a^6*b^6 + 250*a^8*b^4 + 69*a^10*b^2)^2)*(16*a^16 + 16*b^16 + 128*a^2*b^
14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12*b^4 + 128*a^14*b^2))/(3*a^6 + 15*a^2
*b^4 + 10*a^4*b^2) + (((64*b^5*((a*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^12 - 888*a^6*b^10 - 248*a^8*b^8 + 408*
a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 +
6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b
^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10
+ 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^
4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6
*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a
^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 +
6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a*((9*a^14*b + 48*a^2*b^13 + 193*a^4*b^11 + 3
33*a^6*b^9 + 330*a^8*b^7 + 202*a^10*b^5 + 69*a^12*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6
+ 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^12 - 888*a^6*b^10 - 248*a^8*b^8 + 4
08*a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4
+ 6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^1
0*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^
10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2))*(3*a
^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a^3*(3*a^4 + 15*b^4 + 10*a^2*b^2)^3*(192
*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a
^14*b^3))/(1024*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15
*a^8*b^4 + 6*a^10*b^2)))*(9*a^12 + 256*b^12 - 1441*a^2*b^10 - 715*a^4*b^8 - 82*a^6*b^6 + 130*a^8*b^4 + 51*a^10
*b^2)*(16*a^16 + 16*b^16 + 128*a^2*b^14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12
*b^4 + 128*a^14*b^2))/((3*a^6 + 15*a^2*b^4 + 10*a^4*b^2)*(9*a^12 + 256*b^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490
*a^6*b^6 + 250*a^8*b^4 + 69*a^10*b^2)^2) - (2*a*b*((56*a^2*b^12 + 73*a^4*b^10 + 42*a^6*b^8 + 9*a^8*b^6)/(2*(a^
12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) + (64*b^5*((9*a^14*b + 48*a^2*b^1
3 + 193*a^4*b^11 + 333*a^6*b^9 + 330*a^8*b^7 + 202*a^10*b^5 + 69*a^12*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a
^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^12 - 888*a^6*b^
10 - 248*a^8*b^8 + 408*a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*
a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 672
0*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^
12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4
+ 192*a^4*b^2)))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) + (a*((a*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^1
2 - 888*a^6*b^10 - 248*a^8*b^8 + 408*a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15
*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*
a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 19
2*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 +
10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a^16*b + 19
2*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((
64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 1
5*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 +
3*a^4*b^2)) - (a^2*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)^2*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*
b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/(2*(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*
a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^
4 + 6*a^10*b^2)))*(9*a^10 - 496*b^10 + 305*a^2*b^8 + 412*a^4*b^6 + 238*a^6*b^4 + 60*a^8*b^2)*(16*a^16 + 16*b^1
6 + 128*a^2*b^14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12*b^4 + 128*a^14*b^2))/(
(3*a^6 + 15*a^2*b^4 + 10*a^4*b^2)*(9*a^12 + 256*b^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490*a^6*b^6 + 250*a^8*b^4
+ 69*a^10*b^2)^2))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(4*d*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))

________________________________________________________________________________________

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)**5/(a*cos(d*x+c)+b*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________