\(\int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx\) [169]

Optimal result

Integrand size = 21, antiderivative size = 273 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx=-a^3 x+\frac {15}{2} a b^2 x-\frac {45 a^2 b \text {arctanh}(\cos (c+d x))}{8 d}+\frac {5 b^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^2 b \cos (c+d x)}{d}-\frac {2 b^3 \cos (c+d x)}{d}-\frac {b^3 \cos ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {6 a b^2 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {27 a^2 b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^2 b \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a b^2 \cos (c+d x) \sin (c+d x)}{2 d} \] Output:

-a^3*x+15/2*a*b^2*x-45/8*a^2*b*arctanh(cos(d*x+c))/d+5/2*b^3*arctanh(cos(d 
*x+c))/d+3*a^2*b*cos(d*x+c)/d-2*b^3*cos(d*x+c)/d-1/3*b^3*cos(d*x+c)^3/d-a^ 
3*cot(d*x+c)/d+6*a*b^2*cot(d*x+c)/d+1/3*a^3*cot(d*x+c)^3/d-a*b^2*cot(d*x+c 
)^3/d-1/5*a^3*cot(d*x+c)^5/d+27/8*a^2*b*cot(d*x+c)*csc(d*x+c)/d-1/2*b^3*co 
t(d*x+c)*csc(d*x+c)/d-3/4*a^2*b*cot(d*x+c)*csc(d*x+c)^3/d+3/2*a*b^2*cos(d* 
x+c)*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 2.49 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.27 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-600 a \left (2 a^2-15 b^2\right ) (c+d x) \csc ^4(c+d x)+1200 b \left (-9 a^2+4 b^2\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+5 \cot (c+d x) \csc ^4(c+d x) \left (-80 a^3+285 a b^2+12 b \left (60 a^2-29 b^2\right ) \sin (c+d x)\right )+\csc ^5(c+d x) \left (5 \left (40 a^3-489 a b^2\right ) \cos (3 (c+d x))+\left (-184 a^3+1065 a b^2\right ) \cos (5 (c+d x))+5 \left (-9 a b^2 \cos (7 (c+d x))+60 a \left (2 a^2-15 b^2\right ) (c+d x) \sin (3 (c+d x))-306 a^2 b \sin (4 (c+d x))+122 b^3 \sin (4 (c+d x))-24 a^3 c \sin (5 (c+d x))+180 a b^2 c \sin (5 (c+d x))-24 a^3 d x \sin (5 (c+d x))+180 a b^2 d x \sin (5 (c+d x))+36 a^2 b \sin (6 (c+d x))-22 b^3 \sin (6 (c+d x))-b^3 \sin (8 (c+d x))\right )\right )}{1920 d} \] Input:

Integrate[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^3,x]
 

Output:

(-600*a*(2*a^2 - 15*b^2)*(c + d*x)*Csc[c + d*x]^4 + 1200*b*(-9*a^2 + 4*b^2 
)*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + 5*Cot[c + d*x]*Csc[c + 
 d*x]^4*(-80*a^3 + 285*a*b^2 + 12*b*(60*a^2 - 29*b^2)*Sin[c + d*x]) + Csc[ 
c + d*x]^5*(5*(40*a^3 - 489*a*b^2)*Cos[3*(c + d*x)] + (-184*a^3 + 1065*a*b 
^2)*Cos[5*(c + d*x)] + 5*(-9*a*b^2*Cos[7*(c + d*x)] + 60*a*(2*a^2 - 15*b^2 
)*(c + d*x)*Sin[3*(c + d*x)] - 306*a^2*b*Sin[4*(c + d*x)] + 122*b^3*Sin[4* 
(c + d*x)] - 24*a^3*c*Sin[5*(c + d*x)] + 180*a*b^2*c*Sin[5*(c + d*x)] - 24 
*a^3*d*x*Sin[5*(c + d*x)] + 180*a*b^2*d*x*Sin[5*(c + d*x)] + 36*a^2*b*Sin[ 
6*(c + d*x)] - 22*b^3*Sin[6*(c + d*x)] - b^3*Sin[8*(c + d*x)])))/(1920*d)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3201, 2009}

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.

\(\displaystyle \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3201

\(\displaystyle \int \left (a^3 \cot ^6(c+d x)+3 a^2 b \cos (c+d x) \cot ^5(c+d x)+3 a b^2 \cos ^2(c+d x) \cot ^4(c+d x)+b^3 \cos ^3(c+d x) \cot ^3(c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}-a^3 x-\frac {45 a^2 b \text {arctanh}(\cos (c+d x))}{8 d}+\frac {45 a^2 b \cos (c+d x)}{8 d}-\frac {3 a^2 b \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac {15 a^2 b \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac {5 a b^2 \cot ^3(c+d x)}{2 d}+\frac {15 a b^2 \cot (c+d x)}{2 d}+\frac {3 a b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac {15}{2} a b^2 x+\frac {5 b^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 b^3 \cos ^3(c+d x)}{6 d}-\frac {5 b^3 \cos (c+d x)}{2 d}-\frac {b^3 \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}\)

Input:

Int[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^3,x]
 

Output:

-(a^3*x) + (15*a*b^2*x)/2 - (45*a^2*b*ArcTanh[Cos[c + d*x]])/(8*d) + (5*b^ 
3*ArcTanh[Cos[c + d*x]])/(2*d) + (45*a^2*b*Cos[c + d*x])/(8*d) - (5*b^3*Co 
s[c + d*x])/(2*d) - (5*b^3*Cos[c + d*x]^3)/(6*d) - (a^3*Cot[c + d*x])/d + 
(15*a*b^2*Cot[c + d*x])/(2*d) + (15*a^2*b*Cos[c + d*x]*Cot[c + d*x]^2)/(8* 
d) - (b^3*Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*d) + (a^3*Cot[c + d*x]^3)/(3*d 
) - (5*a*b^2*Cot[c + d*x]^3)/(2*d) + (3*a*b^2*Cos[c + d*x]^2*Cot[c + d*x]^ 
3)/(2*d) - (3*a^2*b*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a^3*Cot[c + d*x] 
^5)/(5*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3201

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((g_.)*tan[(e_.) + (f_.)*( 
x_)])^(p_.), x_Symbol] :> Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Si 
n[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] 
&& IGtQ[m, 0]
 

Maple [A] (verified)

Time = 6.04 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(289\)
default	\(\frac {a^{3} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \cos \left (d x +c \right )^{7}}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{7}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{2}-\frac {5 \cos \left (d x +c \right )^{3}}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(289\)
risch	\(-a^{3} x +\frac {15 a \,b^{2} x}{2}-\frac {b^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 b \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 d}-\frac {9 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 b \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 d}-\frac {9 b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i a \,b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {b^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {720 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-184 i a^{3}-405 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+60 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+4800 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+840 i a \,b^{2}+450 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-120 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-3120 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+1080 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-360 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+560 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-450 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+120 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-3600 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-1120 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+405 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-60 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {45 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {45 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\)	\(511\)

Input:

int(cot(d*x+c)^6*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.51 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {360 \, a b^{2} \cos \left (d x + c\right )^{7} + 184 \, {\left (2 \, a^{3} - 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 280 \, {\left (2 \, a^{3} - 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 75 \, {\left ({\left (9 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 9 \, a^{2} b - 4 \, b^{3} - 2 \, {\left (9 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 75 \, {\left ({\left (9 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 9 \, a^{2} b - 4 \, b^{3} - 2 \, {\left (9 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 120 \, {\left (2 \, a^{3} - 15 \, a b^{2}\right )} \cos \left (d x + c\right ) + 10 \, {\left (8 \, b^{3} \cos \left (d x + c\right )^{7} + 12 \, {\left (2 \, a^{3} - 15 \, a b^{2}\right )} d x \cos \left (d x + c\right )^{4} - 8 \, {\left (9 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{5} - 24 \, {\left (2 \, a^{3} - 15 \, a b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 25 \, {\left (9 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left (2 \, a^{3} - 15 \, a b^{2}\right )} d x - 15 \, {\left (9 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:

integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^3,x, algorithm="fricas")
 

Output:

-1/240*(360*a*b^2*cos(d*x + c)^7 + 184*(2*a^3 - 15*a*b^2)*cos(d*x + c)^5 - 
 280*(2*a^3 - 15*a*b^2)*cos(d*x + c)^3 + 75*((9*a^2*b - 4*b^3)*cos(d*x + c 
)^4 + 9*a^2*b - 4*b^3 - 2*(9*a^2*b - 4*b^3)*cos(d*x + c)^2)*log(1/2*cos(d* 
x + c) + 1/2)*sin(d*x + c) - 75*((9*a^2*b - 4*b^3)*cos(d*x + c)^4 + 9*a^2* 
b - 4*b^3 - 2*(9*a^2*b - 4*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/ 
2)*sin(d*x + c) + 120*(2*a^3 - 15*a*b^2)*cos(d*x + c) + 10*(8*b^3*cos(d*x 
+ c)^7 + 12*(2*a^3 - 15*a*b^2)*d*x*cos(d*x + c)^4 - 8*(9*a^2*b - 4*b^3)*co 
s(d*x + c)^5 - 24*(2*a^3 - 15*a*b^2)*d*x*cos(d*x + c)^2 + 25*(9*a^2*b - 4* 
b^3)*cos(d*x + c)^3 + 12*(2*a^3 - 15*a*b^2)*d*x - 15*(9*a^2*b - 4*b^3)*cos 
(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin( 
d*x + c))
 

Sympy [F]

\[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \cot ^{6}{\left (c + d x \right )}\, dx \] Input:

integrate(cot(d*x+c)**6*(a+b*sin(d*x+c))**3,x)
 

Output:

Integral((a + b*sin(c + d*x))**3*cot(c + d*x)**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.92 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {16 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} - 120 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a b^{2} + 20 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} b^{3} + 45 \, a^{2} b {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \] Input:

integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^3,x, algorithm="maxima")
 

Output:

-1/240*(16*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan 
(d*x + c)^5)*a^3 - 120*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + 
c)^2 - 2)/(tan(d*x + c)^5 + tan(d*x + c)^3))*a*b^2 + 20*(4*cos(d*x + c)^3 
- 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos(d*x + c) - 15*log(cos(d*x + 
 c) + 1) + 15*log(cos(d*x + c) - 1))*b^3 + 45*a^2*b*(2*(9*cos(d*x + c)^3 - 
 7*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) 
 + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.73 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^3,x, algorithm="giac")
 

Output:

1/960*(6*a^3*tan(1/2*d*x + 1/2*c)^5 + 45*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 70 
*a^3*tan(1/2*d*x + 1/2*c)^3 + 120*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 720*a^2*b 
*tan(1/2*d*x + 1/2*c)^2 + 120*b^3*tan(1/2*d*x + 1/2*c)^2 + 660*a^3*tan(1/2 
*d*x + 1/2*c) - 3240*a*b^2*tan(1/2*d*x + 1/2*c) - 480*(2*a^3 - 15*a*b^2)*( 
d*x + c) + 600*(9*a^2*b - 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - 320*(9*a 
*b^2*tan(1/2*d*x + 1/2*c)^5 - 18*a^2*b*tan(1/2*d*x + 1/2*c)^4 + 18*b^3*tan 
(1/2*d*x + 1/2*c)^4 - 36*a^2*b*tan(1/2*d*x + 1/2*c)^2 + 24*b^3*tan(1/2*d*x 
 + 1/2*c)^2 - 9*a*b^2*tan(1/2*d*x + 1/2*c) - 18*a^2*b + 14*b^3)/(tan(1/2*d 
*x + 1/2*c)^2 + 1)^3 - (12330*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 5480*b^3*tan( 
1/2*d*x + 1/2*c)^5 + 660*a^3*tan(1/2*d*x + 1/2*c)^4 - 3240*a*b^2*tan(1/2*d 
*x + 1/2*c)^4 - 720*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 120*b^3*tan(1/2*d*x + 1 
/2*c)^3 - 70*a^3*tan(1/2*d*x + 1/2*c)^2 + 120*a*b^2*tan(1/2*d*x + 1/2*c)^2 
 + 45*a^2*b*tan(1/2*d*x + 1/2*c) + 6*a^3)/tan(1/2*d*x + 1/2*c)^5)/d
 

Mupad [B] (verification not implemented)

Time = 18.06 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.86 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:

int(cot(c + d*x)^6*(a + b*sin(c + d*x))^3,x)
 

Output:

(a^3*tan(c/2 + (d*x)/2)^5)/(160*d) + (tan(c/2 + (d*x)/2)^10*(12*a*b^2 - 22 
*a^3) - tan(c/2 + (d*x)/2)^2*(4*a*b^2 - (26*a^3)/15) + tan(c/2 + (d*x)/2)^ 
4*(96*a*b^2 - (78*a^3)/5) + tan(c/2 + (d*x)/2)^8*(320*a*b^2 - (191*a^3)/3) 
 + tan(c/2 + (d*x)/2)^6*(408*a*b^2 - (296*a^3)/5) + tan(c/2 + (d*x)/2)^3*( 
(39*a^2*b)/2 - 4*b^3) + tan(c/2 + (d*x)/2)^9*(216*a^2*b - 196*b^3) + tan(c 
/2 + (d*x)/2)^5*((519*a^2*b)/2 - (484*b^3)/3) + tan(c/2 + (d*x)/2)^7*((909 
*a^2*b)/2 - 268*b^3) - a^3/5 - (3*a^2*b*tan(c/2 + (d*x)/2))/2)/(d*(32*tan( 
c/2 + (d*x)/2)^5 + 96*tan(c/2 + (d*x)/2)^7 + 96*tan(c/2 + (d*x)/2)^9 + 32* 
tan(c/2 + (d*x)/2)^11)) + (tan(c/2 + (d*x)/2)^3*((a*b^2)/8 - (7*a^3)/96))/ 
d - (tan(c/2 + (d*x)/2)^2*((3*a^2*b)/4 - b^3/8))/d + (log(tan(c/2 + (d*x)/ 
2))*((45*a^2*b)/8 - (5*b^3)/2))/d - (log(tan(c/2 + (d*x)/2) - 1i)*((a*b^2* 
15i)/2 - a^3*1i))/d - (tan(c/2 + (d*x)/2)*((27*a*b^2)/8 - (11*a^3)/16))/d 
+ (3*a^2*b*tan(c/2 + (d*x)/2)^4)/(64*d) - (a*log(tan(c/2 + (d*x)/2) + 1i)* 
(2*a^2 - 15*b^2)*1i)/(2*d)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.26 \[ \int \cot ^6(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} b^{3}+1440 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a \,b^{2}+2880 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{2} b -2240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3}-1472 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3}+6720 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{2}+3240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b -480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+704 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3}-960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{2}-720 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b -192 \cos \left (d x +c \right ) a^{3}+5400 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} a^{2} b -2400 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} b^{3}-960 \sin \left (d x +c \right )^{5} a^{3} d x -4905 \sin \left (d x +c \right )^{5} a^{2} b +7200 \sin \left (d x +c \right )^{5} a \,b^{2} d x +2600 \sin \left (d x +c \right )^{5} b^{3}}{960 \sin \left (d x +c \right )^{5} d} \] Input:

int(cot(d*x+c)^6*(a+b*sin(d*x+c))^3,x)
 

Output:

(320*cos(c + d*x)*sin(c + d*x)**7*b**3 + 1440*cos(c + d*x)*sin(c + d*x)**6 
*a*b**2 + 2880*cos(c + d*x)*sin(c + d*x)**5*a**2*b - 2240*cos(c + d*x)*sin 
(c + d*x)**5*b**3 - 1472*cos(c + d*x)*sin(c + d*x)**4*a**3 + 6720*cos(c + 
d*x)*sin(c + d*x)**4*a*b**2 + 3240*cos(c + d*x)*sin(c + d*x)**3*a**2*b - 4 
80*cos(c + d*x)*sin(c + d*x)**3*b**3 + 704*cos(c + d*x)*sin(c + d*x)**2*a* 
*3 - 960*cos(c + d*x)*sin(c + d*x)**2*a*b**2 - 720*cos(c + d*x)*sin(c + d* 
x)*a**2*b - 192*cos(c + d*x)*a**3 + 5400*log(tan((c + d*x)/2))*sin(c + d*x 
)**5*a**2*b - 2400*log(tan((c + d*x)/2))*sin(c + d*x)**5*b**3 - 960*sin(c 
+ d*x)**5*a**3*d*x - 4905*sin(c + d*x)**5*a**2*b + 7200*sin(c + d*x)**5*a* 
b**2*d*x + 2600*sin(c + d*x)**5*b**3)/(960*sin(c + d*x)**5*d)