\(\int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx\) [77]

Optimal result

Integrand size = 21, antiderivative size = 123 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {1}{16 a^2 d (1-\cos (c+d x))}-\frac {1}{12 a^2 d (1+\cos (c+d x))^3}+\frac {1}{2 a^2 d (1+\cos (c+d x))^2}-\frac {23}{16 a^2 d (1+\cos (c+d x))}-\frac {3 \log (1-\cos (c+d x))}{16 a^2 d}-\frac {13 \log (1+\cos (c+d x))}{16 a^2 d} \] Output:

-1/16/a^2/d/(1-cos(d*x+c))-1/12/a^2/d/(1+cos(d*x+c))^3+1/2/a^2/d/(1+cos(d* 
x+c))^2-23/16/a^2/d/(1+cos(d*x+c))-3/16*ln(1-cos(d*x+c))/a^2/d-13/16*ln(1+ 
cos(d*x+c))/a^2/d
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+156 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+36 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+69 \sec ^2\left (\frac {1}{2} (c+d x)\right )-12 \sec ^4\left (\frac {1}{2} (c+d x)\right )+\sec ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x)}{24 a^2 d (1+\sec (c+d x))^2} \] Input:

Integrate[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]
 

Output:

-1/24*(Cos[(c + d*x)/2]^4*(3*Csc[(c + d*x)/2]^2 + 156*Log[Cos[(c + d*x)/2] 
] + 36*Log[Sin[(c + d*x)/2]] + 69*Sec[(c + d*x)/2]^2 - 12*Sec[(c + d*x)/2] 
^4 + Sec[(c + d*x)/2]^6)*Sec[c + d*x]^2)/(a^2*d*(1 + Sec[c + d*x])^2)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 27, 99, 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.

\(\displaystyle \int \frac {\cot ^3(c+d x)}{(a \sec (c+d x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4367

\(\displaystyle -\frac {a^4 \int \frac {\cos ^5(c+d x)}{a^6 (1-\cos (c+d x))^2 (\cos (c+d x)+1)^4}d\cos (c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\cos ^5(c+d x)}{(1-\cos (c+d x))^2 (\cos (c+d x)+1)^4}d\cos (c+d x)}{a^2 d}\)

\(\Big \downarrow \) 99

\(\displaystyle -\frac {\int \left (\frac {13}{16 (\cos (c+d x)+1)}-\frac {23}{16 (\cos (c+d x)+1)^2}+\frac {1}{(\cos (c+d x)+1)^3}-\frac {1}{4 (\cos (c+d x)+1)^4}+\frac {3}{16 (\cos (c+d x)-1)}+\frac {1}{16 (\cos (c+d x)-1)^2}\right )d\cos (c+d x)}{a^2 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {1}{16 (1-\cos (c+d x))}+\frac {23}{16 (\cos (c+d x)+1)}-\frac {1}{2 (\cos (c+d x)+1)^2}+\frac {1}{12 (\cos (c+d x)+1)^3}+\frac {3}{16} \log (1-\cos (c+d x))+\frac {13}{16} \log (\cos (c+d x)+1)}{a^2 d}\)

Input:

Int[Cot[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]
 

Output:

-((1/(16*(1 - Cos[c + d*x])) + 1/(12*(1 + Cos[c + d*x])^3) - 1/(2*(1 + Cos 
[c + d*x])^2) + 23/(16*(1 + Cos[c + d*x])) + (3*Log[1 - Cos[c + d*x]])/16 
+ (13*Log[1 + Cos[c + d*x]])/16)/(a^2*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(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 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 4367

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n 
_.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d)   Subst[Int[(a - b*x)^((m - 
1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr 
eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer 
Q[n]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(\frac {\frac {1}{-16+16 \cos \left (d x +c \right )}-\frac {3 \ln \left (-1+\cos \left (d x +c \right )\right )}{16}-\frac {1}{12 \left (1+\cos \left (d x +c \right )\right )^{3}}+\frac {1}{2 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {23}{16 \left (1+\cos \left (d x +c \right )\right )}-\frac {13 \ln \left (1+\cos \left (d x +c \right )\right )}{16}}{d \,a^{2}}\)	\(79\)
default	\(\frac {\frac {1}{-16+16 \cos \left (d x +c \right )}-\frac {3 \ln \left (-1+\cos \left (d x +c \right )\right )}{16}-\frac {1}{12 \left (1+\cos \left (d x +c \right )\right )^{3}}+\frac {1}{2 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {23}{16 \left (1+\cos \left (d x +c \right )\right )}-\frac {13 \ln \left (1+\cos \left (d x +c \right )\right )}{16}}{d \,a^{2}}\)	\(79\)
risch	\(\frac {i x}{a^{2}}+\frac {2 i c}{d \,a^{2}}-\frac {33 \,{\mathrm e}^{7 i \left (d x +c \right )}+36 \,{\mathrm e}^{6 i \left (d x +c \right )}-49 \,{\mathrm e}^{5 i \left (d x +c \right )}-136 \,{\mathrm e}^{4 i \left (d x +c \right )}-49 \,{\mathrm e}^{3 i \left (d x +c \right )}+36 \,{\mathrm e}^{2 i \left (d x +c \right )}+33 \,{\mathrm e}^{i \left (d x +c \right )}}{12 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}-\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{2}}\)	\(171\)

Input:

int(cot(d*x+c)^3/(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d/a^2*(1/16/(-1+cos(d*x+c))-3/16*ln(-1+cos(d*x+c))-1/12/(1+cos(d*x+c))^3 
+1/2/(1+cos(d*x+c))^2-23/16/(1+cos(d*x+c))-13/16*ln(1+cos(d*x+c)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {66 \, \cos \left (d x + c\right )^{3} + 36 \, \cos \left (d x + c\right )^{2} + 39 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 74 \, \cos \left (d x + c\right ) - 52}{48 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )}} \] Input:

integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
 

Output:

-1/48*(66*cos(d*x + c)^3 + 36*cos(d*x + c)^2 + 39*(cos(d*x + c)^4 + 2*cos( 
d*x + c)^3 - 2*cos(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) + 9*(cos(d*x 
+ c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/ 
2) - 74*cos(d*x + c) - 52)/(a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 
- 2*a^2*d*cos(d*x + c) - a^2*d)
 

Sympy [F]

\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:

integrate(cot(d*x+c)**3/(a+a*sec(d*x+c))**2,x)
 

Output:

Integral(cot(c + d*x)**3/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x)/a**2
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{3} + 18 \, \cos \left (d x + c\right )^{2} - 37 \, \cos \left (d x + c\right ) - 26\right )}}{a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right ) - a^{2}} + \frac {39 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {9 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{48 \, d} \] Input:

integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^2,x, algorithm="maxima")
 

Output:

-1/48*(2*(33*cos(d*x + c)^3 + 18*cos(d*x + c)^2 - 37*cos(d*x + c) - 26)/(a 
^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c) - a^2) + 39* 
log(cos(d*x + c) + 1)/a^2 + 9*log(cos(d*x + c) - 1)/a^2)/d
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {13 \, \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{16 \, a^{2} d} - \frac {3 \, \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{16 \, a^{2} d} - \frac {33 \, \cos \left (d x + c\right )^{3} + 18 \, \cos \left (d x + c\right )^{2} - 37 \, \cos \left (d x + c\right ) - 26}{24 \, a^{2} d {\left (\cos \left (d x + c\right ) + 1\right )}^{3} {\left (\cos \left (d x + c\right ) - 1\right )}} \] Input:

integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^2,x, algorithm="giac")
 

Output:

-13/16*log(abs(cos(d*x + c) + 1))/(a^2*d) - 3/16*log(abs(cos(d*x + c) - 1) 
)/(a^2*d) - 1/24*(33*cos(d*x + c)^3 + 18*cos(d*x + c)^2 - 37*cos(d*x + c) 
- 26)/(a^2*d*(cos(d*x + c) + 1)^3*(cos(d*x + c) - 1))
 

Mupad [B] (verification not implemented)

Time = 11.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{96}}{a^2\,d} \] Input:

int(cot(c + d*x)^3/(a + a/cos(c + d*x))^2,x)
 

Output:

-((3*log(tan(c/2 + (d*x)/2)))/8 - log(tan(c/2 + (d*x)/2)^2 + 1) + cot(c/2 
+ (d*x)/2)^2/32 + tan(c/2 + (d*x)/2)^2/2 - (3*tan(c/2 + (d*x)/2)^4)/32 + t 
an(c/2 + (d*x)/2)^6/96)/(a^2*d)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-36 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-3}{96 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2} d} \] Input:

int(cot(d*x+c)^3/(a+a*sec(d*x+c))^2,x)
 

Output:

(96*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)**2 - 36*log(tan((c + d*x 
)/2))*tan((c + d*x)/2)**2 - tan((c + d*x)/2)**8 + 9*tan((c + d*x)/2)**6 - 
48*tan((c + d*x)/2)**4 - 3)/(96*tan((c + d*x)/2)**2*a**2*d)