\(\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx\) [530]

Optimal result

Integrand size = 27, antiderivative size = 145 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {4 a^4 \csc (c+d x)}{d}-\frac {2 a^4 \csc ^2(c+d x)}{d}-\frac {a^4 \csc ^3(c+d x)}{3 d}-\frac {4 a^4 \log (\sin (c+d x))}{d}-\frac {10 a^4 \sin (c+d x)}{d}-\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{d}+\frac {a^4 \sin ^5(c+d x)}{5 d} \]

[Out]

-4*a^4*csc(d*x+c)/d-2*a^4*csc(d*x+c)^2/d-1/3*a^4*csc(d*x+c)^3/d-4*a^4*ln(sin(d*x+c))/d-10*a^4*sin(d*x+c)/d-2*a
^4*sin(d*x+c)^2/d+4/3*a^4*sin(d*x+c)^3/d+a^4*sin(d*x+c)^4/d+1/5*a^4*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 90} \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^5(c+d x)}{5 d}+\frac {a^4 \sin ^4(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {2 a^4 \sin ^2(c+d x)}{d}-\frac {10 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc ^3(c+d x)}{3 d}-\frac {2 a^4 \csc ^2(c+d x)}{d}-\frac {4 a^4 \csc (c+d x)}{d}-\frac {4 a^4 \log (\sin (c+d x))}{d} \]

[In]

Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]

[Out]

(-4*a^4*Csc[c + d*x])/d - (2*a^4*Csc[c + d*x]^2)/d - (a^4*Csc[c + d*x]^3)/(3*d) - (4*a^4*Log[Sin[c + d*x]])/d
- (10*a^4*Sin[c + d*x])/d - (2*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/d +
(a^4*Sin[c + d*x]^5)/(5*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^4 (a-x)^2 (a+x)^6}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^2 (a+x)^6}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \left (-10 a^4+\frac {a^8}{x^4}+\frac {4 a^7}{x^3}+\frac {4 a^6}{x^2}-\frac {4 a^5}{x}-4 a^3 x+4 a^2 x^2+4 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = -\frac {4 a^4 \csc (c+d x)}{d}-\frac {2 a^4 \csc ^2(c+d x)}{d}-\frac {a^4 \csc ^3(c+d x)}{3 d}-\frac {4 a^4 \log (\sin (c+d x))}{d}-\frac {10 a^4 \sin (c+d x)}{d}-\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{d}+\frac {a^4 \sin ^5(c+d x)}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.66 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {a^4 \left (60 \csc (c+d x)+30 \csc ^2(c+d x)+5 \csc ^3(c+d x)+60 \log (\sin (c+d x))+150 \sin (c+d x)+30 \sin ^2(c+d x)-20 \sin ^3(c+d x)-15 \sin ^4(c+d x)-3 \sin ^5(c+d x)\right )}{15 d} \]

[In]

Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]

[Out]

-1/15*(a^4*(60*Csc[c + d*x] + 30*Csc[c + d*x]^2 + 5*Csc[c + d*x]^3 + 60*Log[Sin[c + d*x]] + 150*Sin[c + d*x] +
30*Sin[c + d*x]^2 - 20*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 - 3*Sin[c + d*x]^5))/d

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.38

method	result	size
parallelrisch	\(-\frac {a^{4} \left (-1920 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (3 d x +3 c \right )+1920 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (3 d x +3 c \right )+5760 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )-5760 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+1836 \cos \left (4 d x +4 c \right )-12072 \cos \left (2 d x +2 c \right )+1110 \sin \left (3 d x +3 c \right )+150 \sin \left (d x +c \right )-3 \cos \left (8 d x +8 c \right )+30 \sin \left (7 d x +7 c \right )+30 \sin \left (5 d x +5 c \right )+104 \cos \left (6 d x +6 c \right )+10775\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15360 d}\)	\(200\)
derivativedivides	\(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+4 a^{4} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\)	\(242\)
default	\(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+4 a^{4} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\)	\(242\)
risch	\(4 i a^{4} x -\frac {i a^{4} {\mathrm e}^{5 i \left (d x +c \right )}}{160 d}+\frac {a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{16 d}+\frac {19 i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{96 d}+\frac {a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {71 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {71 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}-\frac {19 i a^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{96 d}+\frac {a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{16 d}+\frac {i a^{4} {\mathrm e}^{-5 i \left (d x +c \right )}}{160 d}+\frac {8 i a^{4} c}{d}-\frac {8 i a^{4} \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-7 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}-3 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {4 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(295\)
norman	\(\frac {-\frac {a^{4}}{24 d}-\frac {a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {7 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {199 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {305 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {8111 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}-\frac {305 a^{4} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {199 a^{4} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {7 a^{4} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a^{4} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {19 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {19 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {4 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{4} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(342\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

-1/15360*a^4*(-1920*ln(tan(1/2*d*x+1/2*c))*sin(3*d*x+3*c)+1920*ln(sec(1/2*d*x+1/2*c)^2)*sin(3*d*x+3*c)+5760*ln
(tan(1/2*d*x+1/2*c))*sin(d*x+c)-5760*ln(sec(1/2*d*x+1/2*c)^2)*sin(d*x+c)+1836*cos(4*d*x+4*c)-12072*cos(2*d*x+2
*c)+1110*sin(3*d*x+3*c)+150*sin(d*x+c)-3*cos(8*d*x+8*c)+30*sin(7*d*x+7*c)+30*sin(5*d*x+5*c)+104*cos(6*d*x+6*c)
+10775)*sec(1/2*d*x+1/2*c)^3*csc(1/2*d*x+1/2*c)^3/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {24 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \, a^{4} \cos \left (d x + c\right )^{6} - 576 \, a^{4} \cos \left (d x + c\right )^{4} + 2304 \, a^{4} \cos \left (d x + c\right )^{2} - 1536 \, a^{4} + 480 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 15 \, {\left (8 \, a^{4} \cos \left (d x + c\right )^{6} - 8 \, a^{4} \cos \left (d x + c\right )^{4} - 3 \, a^{4} \cos \left (d x + c\right )^{2} + 19 \, a^{4}\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/120*(24*a^4*cos(d*x + c)^8 - 256*a^4*cos(d*x + c)^6 - 576*a^4*cos(d*x + c)^4 + 2304*a^4*cos(d*x + c)^2 - 15
36*a^4 + 480*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*sin(d*x + c))*sin(d*x + c) - 15*(8*a^4*cos(d*x + c)^6 - 8*a^4*
cos(d*x + c)^4 - 3*a^4*cos(d*x + c)^2 + 19*a^4)*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**4*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.82 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) - 150 \, a^{4} \sin \left (d x + c\right ) - \frac {5 \, {\left (12 \, a^{4} \sin \left (d x + c\right )^{2} + 6 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/15*(3*a^4*sin(d*x + c)^5 + 15*a^4*sin(d*x + c)^4 + 20*a^4*sin(d*x + c)^3 - 30*a^4*sin(d*x + c)^2 - 60*a^4*lo
g(sin(d*x + c)) - 150*a^4*sin(d*x + c) - 5*(12*a^4*sin(d*x + c)^2 + 6*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^3)/
d

Giac [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.93 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 150 \, a^{4} \sin \left (d x + c\right ) + \frac {5 \, {\left (22 \, a^{4} \sin \left (d x + c\right )^{3} - 12 \, a^{4} \sin \left (d x + c\right )^{2} - 6 \, a^{4} \sin \left (d x + c\right ) - a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/15*(3*a^4*sin(d*x + c)^5 + 15*a^4*sin(d*x + c)^4 + 20*a^4*sin(d*x + c)^3 - 30*a^4*sin(d*x + c)^2 - 60*a^4*lo
g(abs(sin(d*x + c))) - 150*a^4*sin(d*x + c) + 5*(22*a^4*sin(d*x + c)^3 - 12*a^4*sin(d*x + c)^2 - 6*a^4*sin(d*x
+ c) - a^4)/sin(d*x + c)^3)/d

Mupad [B] (verification not implemented)

Time = 9.73 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.61 \[ \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {4\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {4\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {177\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+68\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+640\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+84\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {4549\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}+104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+728\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {745\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {56\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {17\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d} \]

[In]

int((cos(c + d*x)^5*(a + a*sin(c + d*x))^4)/sin(c + d*x)^4,x)

[Out]

(4*a^4*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (a^4*tan(c/2 + (d*x)/2)^3)/(24*d) - (4*a^4*log(tan(c/2 + (d*x)/2)))/
d - ((56*a^4*tan(c/2 + (d*x)/2)^2)/3 + 20*a^4*tan(c/2 + (d*x)/2)^3 + (745*a^4*tan(c/2 + (d*x)/2)^4)/3 + 104*a^
4*tan(c/2 + (d*x)/2)^5 + 728*a^4*tan(c/2 + (d*x)/2)^6 + 104*a^4*tan(c/2 + (d*x)/2)^7 + (4549*a^4*tan(c/2 + (d*
x)/2)^8)/5 + 84*a^4*tan(c/2 + (d*x)/2)^9 + 640*a^4*tan(c/2 + (d*x)/2)^10 + 68*a^4*tan(c/2 + (d*x)/2)^11 + 177*
a^4*tan(c/2 + (d*x)/2)^12 + a^4/3 + 4*a^4*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 40*tan(c/2 + (d*x)/
2)^5 + 80*tan(c/2 + (d*x)/2)^7 + 80*tan(c/2 + (d*x)/2)^9 + 40*tan(c/2 + (d*x)/2)^11 + 8*tan(c/2 + (d*x)/2)^13)
) - (17*a^4*tan(c/2 + (d*x)/2))/(8*d) - (a^4*tan(c/2 + (d*x)/2)^2)/(2*d)