\(\int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx\) [713]

Optimal result

Integrand size = 29, antiderivative size = 146 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 x}{2 a}-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {5 \cos (c+d x)}{2 a d}+\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d} \]

[Out]

5/2*x/a-5/2*arctanh(cos(d*x+c))/a/d+5/2*cos(d*x+c)/a/d+5/6*cos(d*x+c)^3/a/d+5/2*cot(d*x+c)/a/d+1/2*cos(d*x+c)^
3*cot(d*x+c)^2/a/d-5/6*cot(d*x+c)^3/a/d+1/2*cos(d*x+c)^2*cot(d*x+c)^3/a/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2918, 2671, 294, 308, 209, 2672, 212} \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cos (c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}+\frac {5 x}{2 a} \]

[In]

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

[Out]

(5*x)/(2*a) - (5*ArcTanh[Cos[c + d*x]])/(2*a*d) + (5*Cos[c + d*x])/(2*a*d) + (5*Cos[c + d*x]^3)/(6*a*d) + (5*C
ot[c + d*x])/(2*a*d) + (Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*a*d) - (5*Cot[c + d*x]^3)/(6*a*d) + (Cos[c + d*x]^2*
Cot[c + d*x]^3)/(2*a*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2671

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[b*(ff/f), Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff
)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 2918

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cos ^3(c+d x) \cot ^3(c+d x) \, dx}{a}+\frac {\int \cos ^2(c+d x) \cot ^4(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{a d} \\ & = \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {5 \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 a d} \\ & = \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {5 \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 a d} \\ & = \frac {5 \cos (c+d x)}{2 a d}+\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {5 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 a d} \\ & = \frac {5 x}{2 a}-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {5 \cos (c+d x)}{2 a d}+\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^3(c+d x) \left (-30 \cos (c+d x)+65 \cos (3 (c+d x))-3 \cos (5 (c+d x))-180 c \sin (c+d x)-180 d x \sin (c+d x)+180 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-180 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-75 \sin (2 (c+d x))+60 c \sin (3 (c+d x))+60 d x \sin (3 (c+d x))-60 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+60 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+24 \sin (4 (c+d x))+\sin (6 (c+d x))\right )}{96 a d} \]

[In]

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

[Out]

-1/96*(Csc[c + d*x]^3*(-30*Cos[c + d*x] + 65*Cos[3*(c + d*x)] - 3*Cos[5*(c + d*x)] - 180*c*Sin[c + d*x] - 180*
d*x*Sin[c + d*x] + 180*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 180*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 75*Sin[2*
(c + d*x)] + 60*c*Sin[3*(c + d*x)] + 60*d*x*Sin[3*(c + d*x)] - 60*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 60*
Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 24*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(a*d)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14

method	result	size
parallelrisch	\(\frac {\left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-60 \left (-3 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 d x \sin \left (3 d x +3 c \right )+180 d x \sin \left (d x +c \right )-65 \sin \left (3 d x +3 c \right )-24 \sin \left (4 d x +4 c \right )-\sin \left (6 d x +6 c \right )+30 \cos \left (d x +c \right )-65 \cos \left (3 d x +3 c \right )+3 \cos \left (5 d x +5 c \right )+195 \sin \left (d x +c \right )+75 \sin \left (2 d x +2 c \right )\right )}{768 d a}\)	\(167\)
derivativedivides	\(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {9}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {-8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {112}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+40 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}\)	\(177\)
default	\(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {9}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {-8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {112}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+40 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}\)	\(177\)
risch	\(\frac {5 x}{2 a}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d a}+\frac {9 \,{\mathrm e}^{i \left (d x +c \right )}}{8 a d}+\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d a}-\frac {-18 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}+24 i {\mathrm e}^{2 i \left (d x +c \right )}-14 i-3 \,{\mathrm e}^{i \left (d x +c \right )}}{3 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}+\frac {\cos \left (3 d x +3 c \right )}{12 a d}\)	\(205\)
norman	\(\frac {\frac {10 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {15 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {15 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1}{24 a d}+\frac {5 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {19 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {13 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {13 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}+\frac {207 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {97 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {91 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {157 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {93 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {89 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {43 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {85 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}\)	\(503\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/768*csc(1/2*d*x+1/2*c)^3*sec(1/2*d*x+1/2*c)^3*(-60*(-3*sin(d*x+c)+sin(3*d*x+3*c))*ln(tan(1/2*d*x+1/2*c))-60*
d*x*sin(3*d*x+3*c)+180*d*x*sin(d*x+c)-65*sin(3*d*x+3*c)-24*sin(4*d*x+4*c)-sin(6*d*x+6*c)+30*cos(d*x+c)-65*cos(
3*d*x+3*c)+3*cos(5*d*x+5*c)+195*sin(d*x+c)+75*sin(2*d*x+2*c))/d/a

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {6 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right )^{5} + 15 \, d x \cos \left (d x + c\right )^{2} + 10 \, \cos \left (d x + c\right )^{3} - 15 \, d x - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 30 \, \cos \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/12*(6*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*(cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c)
- 15*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2*(2*cos(d*x + c)^5 + 15*d*x*cos(d*x +
c)^2 + 10*cos(d*x + c)^3 - 15*d*x - 15*cos(d*x + c))*sin(d*x + c) + 30*cos(d*x + c))/((a*d*cos(d*x + c)^2 - a*
d)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (130) = 260\).

Time = 0.30 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.48 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a} - \frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {121 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {102 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {201 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {80 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {147 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 1}{\frac {a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{24 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/24*((27*sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - sin(d*x + c)^3/(cos(d*x +
c) + 1)^3)/a - (3*sin(d*x + c)/(cos(d*x + c) + 1) + 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 121*sin(d*x + c)
^3/(cos(d*x + c) + 1)^3 + 102*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 201*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 +
80*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 147*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8 - 1)/(a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3*a*sin(d*x
+ c)^7/(cos(d*x + c) + 1)^7 + a*sin(d*x + c)^9/(cos(d*x + c) + 1)^9) - 120*arctan(sin(d*x + c)/(cos(d*x + c)
+ 1))/a - 60*log(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {180 \, {\left (d x + c\right )}}{a} + \frac {180 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {3 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{3}} - \frac {110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 111 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 273 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 306 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 253 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3} a}}{72 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/72*(180*(d*x + c)/a + 180*log(abs(tan(1/2*d*x + 1/2*c)))/a + 3*(a^2*tan(1/2*d*x + 1/2*c)^3 - 3*a^2*tan(1/2*d
*x + 1/2*c)^2 - 27*a^2*tan(1/2*d*x + 1/2*c))/a^3 - (110*tan(1/2*d*x + 1/2*c)^9 - 9*tan(1/2*d*x + 1/2*c)^8 - 11
1*tan(1/2*d*x + 1/2*c)^7 - 240*tan(1/2*d*x + 1/2*c)^6 - 273*tan(1/2*d*x + 1/2*c)^5 - 306*tan(1/2*d*x + 1/2*c)^
4 - 253*tan(1/2*d*x + 1/2*c)^3 - 72*tan(1/2*d*x + 1/2*c)^2 - 9*tan(1/2*d*x + 1/2*c) + 3)/((tan(1/2*d*x + 1/2*c
)^3 + tan(1/2*d*x + 1/2*c))^3*a))/d

Mupad [B] (verification not implemented)

Time = 10.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.99 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {5\,\mathrm {atan}\left (\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-25}+\frac {25}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-25}\right )}{a\,d}+\frac {5\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+49\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+67\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {121\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}}{d\,\left (8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+24\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^4*(a + a*sin(c + d*x))),x)

[Out]

tan(c/2 + (d*x)/2)^3/(24*a*d) - tan(c/2 + (d*x)/2)^2/(8*a*d) - (5*atan((25*tan(c/2 + (d*x)/2))/(25*tan(c/2 + (
d*x)/2) - 25) + 25/(25*tan(c/2 + (d*x)/2) - 25)))/(a*d) + (5*log(tan(c/2 + (d*x)/2)))/(2*a*d) + (tan(c/2 + (d*
x)/2) + 8*tan(c/2 + (d*x)/2)^2 + (121*tan(c/2 + (d*x)/2)^3)/3 + 34*tan(c/2 + (d*x)/2)^4 + 67*tan(c/2 + (d*x)/2
)^5 + (80*tan(c/2 + (d*x)/2)^6)/3 + 49*tan(c/2 + (d*x)/2)^7 + tan(c/2 + (d*x)/2)^8 - 1/3)/(d*(8*a*tan(c/2 + (d
*x)/2)^3 + 24*a*tan(c/2 + (d*x)/2)^5 + 24*a*tan(c/2 + (d*x)/2)^7 + 8*a*tan(c/2 + (d*x)/2)^9)) - (9*tan(c/2 + (
d*x)/2))/(8*a*d)