\(\int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) [1321]

Optimal result

Integrand size = 29, antiderivative size = 174 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (2 a^2-5 b^2\right ) x}{2 b^3}+\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {\left (5 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {a \cos (c+d x)}{b^2 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b d} \] Output:

-1/2*(2*a^2-5*b^2)*x/b^3+2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c)) 
/(a^2-b^2)^(1/2))/a^3/b^3/d+1/2*(5*a^2-2*b^2)*arctanh(cos(d*x+c))/a^3/d-a* 
cos(d*x+c)/b^2/d+b*cot(d*x+c)/a^2/d-1/2*cot(d*x+c)*csc(d*x+c)/a/d+1/2*cos( 
d*x+c)*sin(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 3.52 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.49 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-8 a^5 c+20 a^3 b^2 c-8 a^5 d x+20 a^3 b^2 d x+16 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-8 a^4 b \cos (c+d x)+4 a b^4 \cot \left (\frac {1}{2} (c+d x)\right )-a^2 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+20 a^2 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8 b^5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-20 a^2 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 b^5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+2 a^3 b^2 \sin (2 (c+d x))-4 a b^4 \tan \left (\frac {1}{2} (c+d x)\right )}{8 a^3 b^3 d} \] Input:

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

Output:

(-8*a^5*c + 20*a^3*b^2*c - 8*a^5*d*x + 20*a^3*b^2*d*x + 16*(a^2 - b^2)^(5/ 
2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 8*a^4*b*Cos[c + d*x] 
 + 4*a*b^4*Cot[(c + d*x)/2] - a^2*b^3*Csc[(c + d*x)/2]^2 + 20*a^2*b^3*Log[ 
Cos[(c + d*x)/2]] - 8*b^5*Log[Cos[(c + d*x)/2]] - 20*a^2*b^3*Log[Sin[(c + 
d*x)/2]] + 8*b^5*Log[Sin[(c + d*x)/2]] + a^2*b^3*Sec[(c + d*x)/2]^2 + 2*a^ 
3*b^2*Sin[2*(c + d*x)] - 4*a*b^4*Tan[(c + d*x)/2])/(8*a^3*b^3*d)
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 3375, 27, 3042, 3536, 3042, 3139, 1083, 217, 4257}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx\)

\(\Big \downarrow \) 3375

\(\displaystyle \frac {\int -\frac {2 \csc (c+d x) \left (\left (5 a^2-2 b^2\right ) b^2+a \left (a^2-b^2\right ) \sin (c+d x) b+a^2 \left (2 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{4 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\csc (c+d x) \left (\left (5 a^2-2 b^2\right ) b^2+a \left (a^2-b^2\right ) \sin (c+d x) b+a^2 \left (2 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\left (5 a^2-2 b^2\right ) b^2+a \left (a^2-b^2\right ) \sin (c+d x) b+a^2 \left (2 a^2-5 b^2\right ) \sin (c+d x)^2}{\sin (c+d x) (a+b \sin (c+d x))}dx}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 3536

\(\displaystyle -\frac {-\frac {2 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a b}+\frac {b^2 \left (5 a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {a^2 x \left (2 a^2-5 b^2\right )}{b}}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {2 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a b}+\frac {b^2 \left (5 a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {a^2 x \left (2 a^2-5 b^2\right )}{b}}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 3139

\(\displaystyle -\frac {-\frac {4 \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a b d}+\frac {b^2 \left (5 a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {a^2 x \left (2 a^2-5 b^2\right )}{b}}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {\frac {8 \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a b d}+\frac {b^2 \left (5 a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}+\frac {a^2 x \left (2 a^2-5 b^2\right )}{b}}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {b^2 \left (5 a^2-2 b^2\right ) \int \csc (c+d x)dx}{a}-\frac {4 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a b d}+\frac {a^2 x \left (2 a^2-5 b^2\right )}{b}}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {-\frac {4 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a b d}-\frac {b^2 \left (5 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a d}+\frac {a^2 x \left (2 a^2-5 b^2\right )}{b}}{2 a^2 b^2}+\frac {b \cot (c+d x)}{a^2 d}-\frac {a \cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}\)

Input:

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

Output:

-1/2*((a^2*(2*a^2 - 5*b^2)*x)/b - (4*(a^2 - b^2)^(5/2)*ArcTan[(2*b + 2*a*T 
an[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*b*d) - (b^2*(5*a^2 - 2*b^2)*ArcT 
anh[Cos[c + d*x]])/(a*d))/(a^2*b^2) - (a*Cos[c + d*x])/(b^2*d) + (b*Cot[c 
+ d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d) + (Cos[c + d*x]*Sin[ 
c + d*x])/(2*b*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 3375

Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[ 
e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si 
mp[b*(m + n + 2)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x] 
)^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f*x]*(d 
*Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5) 
*(m + n + 6))), x] + Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin 
[e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x] + Simp[1/(a^2*b^2*d^2*(n + 1) 
*(n + 2)*(m + n + 5)*(m + n + 6))   Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin 
[e + f*x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2* 
n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m 
 + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e 
 + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4 
)*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m 
+ 13))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && Ne 
Q[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m 
 + n + 5, 0] && NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]
 

rule 3536

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) 
*(x_)])), x_Symbol] :> Simp[C*(x/(b*d)), x] + (Simp[(A*b^2 - a*b*B + a^2*C) 
/(b*(b*c - a*d))   Int[1/(a + b*Sin[e + f*x]), x], x] - Simp[(c^2*C - B*c*d 
 + A*d^2)/(d*(b*c - a*d))   Int[1/(c + d*Sin[e + f*x]), x], x]) /; FreeQ[{a 
, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 5.33 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (8 a^{6}-24 a^{4} b^{2}+24 a^{2} b^{4}-8 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 a^{3} b^{3} \sqrt {a^{2}-b^{2}}}-\frac {2 \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{2}}{2}+a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+a b}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (2 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{3}}}{d}\)	\(266\)
default	\(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (8 a^{6}-24 a^{4} b^{2}+24 a^{2} b^{4}-8 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 a^{3} b^{3} \sqrt {a^{2}-b^{2}}}-\frac {2 \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{2}}{2}+a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+a b}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (2 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{3}}}{d}\)	\(266\)
risch	\(-\frac {x \,a^{2}}{b^{3}}+\frac {5 x}{2 b}-\frac {2 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d b a}-\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}+\frac {i \left (-i a \,{\mathrm e}^{3 i \left (d x +c \right )}-i a \,{\mathrm e}^{i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \right )}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d \,a^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{3}}+\frac {2 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d b a}+\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d \,a^{3}}\)	\(554\)

Input:

int(cos(d*x+c)^3*cot(d*x+c)^3/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/4/a^2*(1/2*tan(1/2*d*x+1/2*c)^2*a-2*b*tan(1/2*d*x+1/2*c))-1/8/a/tan 
(1/2*d*x+1/2*c)^2+1/4/a^3*(-10*a^2+4*b^2)*ln(tan(1/2*d*x+1/2*c))+1/2/a^2*b 
/tan(1/2*d*x+1/2*c)+1/4*(8*a^6-24*a^4*b^2+24*a^2*b^4-8*b^6)/a^3/b^3/(a^2-b 
^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-2/b^3*( 
(1/2*tan(1/2*d*x+1/2*c)^3*b^2+a*b*tan(1/2*d*x+1/2*c)^2-1/2*b^2*tan(1/2*d*x 
+1/2*c)+a*b)/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(2*a^2-5*b^2)*arctan(tan(1/2*d 
*x+1/2*c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (161) = 322\).

Time = 0.40 (sec) , antiderivative size = 770, normalized size of antiderivative = 4.43 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*a^4*b*cos(d*x + c)^3 + 2*(2*a^5 - 5*a^3*b^2)*d*x*cos(d*x + c)^2 - 
 2*(2*a^5 - 5*a^3*b^2)*d*x + 2*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + 
 b^4)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 
- 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos( 
d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 
 - b^2)) - 2*(2*a^4*b + a^2*b^3)*cos(d*x + c) + (5*a^2*b^3 - 2*b^5 - (5*a^ 
2*b^3 - 2*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - (5*a^2*b^3 - 
2*b^5 - (5*a^2*b^3 - 2*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 
 2*(a^3*b^2*cos(d*x + c)^3 - (a^3*b^2 + 2*a*b^4)*cos(d*x + c))*sin(d*x + c 
))/(a^3*b^3*d*cos(d*x + c)^2 - a^3*b^3*d), -1/4*(4*a^4*b*cos(d*x + c)^3 + 
2*(2*a^5 - 5*a^3*b^2)*d*x*cos(d*x + c)^2 - 2*(2*a^5 - 5*a^3*b^2)*d*x - 4*( 
a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - 
 b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 2*(2* 
a^4*b + a^2*b^3)*cos(d*x + c) + (5*a^2*b^3 - 2*b^5 - (5*a^2*b^3 - 2*b^5)*c 
os(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - (5*a^2*b^3 - 2*b^5 - (5*a^2*b 
^3 - 2*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 2*(a^3*b^2*cos( 
d*x + c)^3 - (a^3*b^2 + 2*a*b^4)*cos(d*x + c))*sin(d*x + c))/(a^3*b^3*d*co 
s(d*x + c)^2 - a^3*b^3*d)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**3*cot(d*x+c)**3/(a+b*sin(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (161) = 322\).

Time = 0.35 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.48 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} - \frac {4 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} {\left (d x + c\right )}}{b^{3}} - \frac {4 \, {\left (5 \, a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {16 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{3} b^{3}} + \frac {10 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 19 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2} b^{2}}{{\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 )}^{2} a^{3} b^{2}}}{8 \, d} \] Input:

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

Output:

1/8*((a*tan(1/2*d*x + 1/2*c)^2 - 4*b*tan(1/2*d*x + 1/2*c))/a^2 - 4*(2*a^2 
- 5*b^2)*(d*x + c)/b^3 - 4*(5*a^2 - 2*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/ 
a^3 + 16*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(pi*floor(1/2*(d*x + c)/pi + 
1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt( 
a^2 - b^2)*a^3*b^3) + (10*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 - 4*b^4*tan(1/2*d 
*x + 1/2*c)^6 - 8*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 4*a*b^3*tan(1/2*d*x + 1/2 
*c)^5 - 16*a^4*tan(1/2*d*x + 1/2*c)^4 + 19*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 
- 8*b^4*tan(1/2*d*x + 1/2*c)^4 + 8*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 8*a*b^3* 
tan(1/2*d*x + 1/2*c)^3 - 16*a^4*tan(1/2*d*x + 1/2*c)^2 + 8*a^2*b^2*tan(1/2 
*d*x + 1/2*c)^2 - 4*b^4*tan(1/2*d*x + 1/2*c)^2 + 4*a*b^3*tan(1/2*d*x + 1/2 
*c) - a^2*b^2)/((tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))^2*a^3*b^2) 
)/d
 

Mupad [B] (verification not implemented)

Time = 20.44 (sec) , antiderivative size = 4898, normalized size of antiderivative = 28.15 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

tan(c/2 + (d*x)/2)^2/(8*a*d) - (a/2 - 2*b*tan(c/2 + (d*x)/2) - (4*tan(c/2 
+ (d*x)/2)^3*(a^2 + b^2))/b + (tan(c/2 + (d*x)/2)^2*(a*b^2 + 8*a^3))/b^2 + 
 (tan(c/2 + (d*x)/2)^4*(a*b^2 + 16*a^3))/(2*b^2) + (2*tan(c/2 + (d*x)/2)^5 
*(2*a^2 - b^2))/b)/(d*(4*a^2*tan(c/2 + (d*x)/2)^2 + 8*a^2*tan(c/2 + (d*x)/ 
2)^4 + 4*a^2*tan(c/2 + (d*x)/2)^6)) - (b*tan(c/2 + (d*x)/2))/(2*a^2*d) + ( 
atan((((a^2*1i - (b^2*5i)/2)*((4*(20*a^3*b^12 - 28*a^15 + 272*a^5*b^10 - 1 
272*a^7*b^8 + 1709*a^9*b^6 - 998*a^11*b^4 + 270*a^13*b^2))/(a^6*b^5) + ((a 
^2*1i - (b^2*5i)/2)*(((a^2*1i - (b^2*5i)/2)*((4*(64*a^5*b^12 - 208*a^7*b^1 
0 + 160*a^9*b^8 - 28*a^11*b^6))/(a^6*b^5) + (((4*(32*a^8*b^10 - 24*a^10*b^ 
8))/(a^6*b^5) + (4*tan(c/2 + (d*x)/2)*(128*a^7*b^14 - 136*a^9*b^12 + 16*a^ 
11*b^10))/(a^6*b^8))*(a^2*1i - (b^2*5i)/2))/b^3 + (4*tan(c/2 + (d*x)/2)*(1 
28*a^4*b^16 - 456*a^6*b^14 + 484*a^8*b^12 - 200*a^10*b^10 + 32*a^12*b^8))/ 
(a^6*b^8)))/b^3 - (4*(184*a^4*b^12 - 32*a^2*b^14 - 360*a^6*b^10 + 78*a^8*b 
^8 + 240*a^10*b^6 - 152*a^12*b^4 + 24*a^14*b^2))/(a^6*b^5) + (4*tan(c/2 + 
(d*x)/2)*(8*a^3*b^16 - 160*a^5*b^14 + 1320*a^7*b^12 - 2128*a^9*b^10 + 1242 
*a^11*b^8 - 280*a^13*b^6 + 16*a^15*b^4))/(a^6*b^8)))/b^3 + (4*tan(c/2 + (d 
*x)/2)*(8*b^18 - 68*a^2*b^16 + 1040*a^4*b^14 - 3900*a^6*b^12 + 5738*a^8*b^ 
10 - 4201*a^10*b^8 + 1684*a^12*b^6 - 360*a^14*b^4 + 32*a^16*b^2))/(a^6*b^8 
))*1i)/b^3 + ((a^2*1i - (b^2*5i)/2)*((4*(20*a^3*b^12 - 28*a^15 + 272*a^5*b 
^10 - 1272*a^7*b^8 + 1709*a^9*b^6 - 998*a^11*b^4 + 270*a^13*b^2))/(a^6*...
 

Reduce [B] (verification not implemented)

Time = 94.71 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.06 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right )^{2} a^{4}-8 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right )^{2} a^{2} b^{2}+4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right )^{2} b^{4}+\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3} b^{2}-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{4} b +2 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{4}-\cos \left (d x +c \right ) a^{2} b^{3}-5 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a^{2} b^{3}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} b^{5}-2 \sin \left (d x +c \right )^{2} a^{5} c -2 \sin \left (d x +c \right )^{2} a^{5} d x +5 \sin \left (d x +c \right )^{2} a^{3} b^{2} c +5 \sin \left (d x +c \right )^{2} a^{3} b^{2} d x}{2 \sin \left (d x +c \right )^{2} a^{3} b^{3} d} \] Input:

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

Output:

(4*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin( 
c + d*x)**2*a**4 - 8*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt( 
a**2 - b**2))*sin(c + d*x)**2*a**2*b**2 + 4*sqrt(a**2 - b**2)*atan((tan((c 
 + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*b**4 + cos(c + d*x)*s 
in(c + d*x)**3*a**3*b**2 - 2*cos(c + d*x)*sin(c + d*x)**2*a**4*b + 2*cos(c 
 + d*x)*sin(c + d*x)*a*b**4 - cos(c + d*x)*a**2*b**3 - 5*log(tan((c + d*x) 
/2))*sin(c + d*x)**2*a**2*b**3 + 2*log(tan((c + d*x)/2))*sin(c + d*x)**2*b 
**5 - 2*sin(c + d*x)**2*a**5*c - 2*sin(c + d*x)**2*a**5*d*x + 5*sin(c + d* 
x)**2*a**3*b**2*c + 5*sin(c + d*x)**2*a**3*b**2*d*x)/(2*sin(c + d*x)**2*a* 
*3*b**3*d)