Integrand size = 27, antiderivative size = 60 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 x}{2 a^3}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d} \] Output:
-7/2*x/a^3-arctanh(cos(d*x+c))/a^3/d-3*cos(d*x+c)/a^3/d+1/2*cos(d*x+c)*sin (d*x+c)/a^3/d
Time = 1.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-12 \cos (c+d x)-2 \left (7 c+7 d x+2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sin (2 (c+d x))}{4 a^3 d} \] Input:
Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
(-12*Cos[c + d*x] - 2*(7*c + 7*d*x + 2*Log[Cos[(c + d*x)/2]] - 2*Log[Sin[( c + d*x)/2]]) + Sin[2*(c + d*x)])/(4*a^3*d)
Time = 0.39 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3348, 3042, 3236, 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.
\(\displaystyle \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x) (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3348 |
\(\displaystyle \frac {\int \csc (c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\sin (c+d x)}dx}{a^6}\) |
\(\Big \downarrow \) 3236 |
\(\displaystyle \frac {\int \left (-\sin ^2(c+d x) a^3+\csc (c+d x) a^3+3 \sin (c+d x) a^3-3 a^3\right )dx}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {7 a^3 x}{2}}{a^6}\) |
Input:
Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
((-7*a^3*x)/2 - (a^3*ArcTanh[Cos[c + d*x]])/d - (3*a^3*Cos[c + d*x])/d + ( a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d))/a^6
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m) Int[(d* Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
Time = 2.78 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.45
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(87\) |
default | \(\frac {-\frac {2 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(87\) |
risch | \(-\frac {7 x}{2 a^{3}}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}+\frac {\sin \left (2 d x +2 c \right )}{4 d \,a^{3}}\) | \(98\) |
Input:
int(cos(d*x+c)^5*cot(d*x+c)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(-2*(1/2*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2-1/2*tan(1/2*d *x+1/2*c)+3)/(1+tan(1/2*d*x+1/2*c)^2)^2-7*arctan(tan(1/2*d*x+1/2*c))+ln(ta n(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 \, d x - \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right ) + \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, a^{3} d} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/2*(7*d*x - cos(d*x + c)*sin(d*x + c) + 6*cos(d*x + c) + log(1/2*cos(d*x + c) + 1/2) - log(-1/2*cos(d*x + c) + 1/2))/(a^3*d)
Timed out. \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5*cot(d*x+c)/(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.68 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 6}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {7 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{d} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
((sin(d*x + c)/(cos(d*x + c) + 1) - 6*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 6)/(a^3 + 2*a^3*sin(d*x + c)^2/(co s(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 7*arctan(si n(d*x + c)/(cos(d*x + c) + 1))/a^3 + log(sin(d*x + c)/(cos(d*x + c) + 1))/ a^3)/d
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {7 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}}{2 \, d} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/2*(7*(d*x + c)/a^3 - 2*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + 2*(tan(1/2* d*x + 1/2*c)^3 + 6*tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c) + 6)/((ta n(1/2*d*x + 1/2*c)^2 + 1)^2*a^3))/d
Time = 34.64 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.50 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {7\,\mathrm {atan}\left (\frac {49}{49\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+14}-\frac {14\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{49\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+14}\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )} \] Input:
int((cos(c + d*x)^5*cot(c + d*x))/(a + a*sin(c + d*x))^3,x)
Output:
(7*atan(49/(49*tan(c/2 + (d*x)/2) + 14) - (14*tan(c/2 + (d*x)/2))/(49*tan( c/2 + (d*x)/2) + 14)))/(a^3*d) + log(tan(c/2 + (d*x)/2))/(a^3*d) - (6*tan( c/2 + (d*x)/2)^2 - tan(c/2 + (d*x)/2) + tan(c/2 + (d*x)/2)^3 + 6)/(d*(2*a^ 3*tan(c/2 + (d*x)/2)^2 + a^3*tan(c/2 + (d*x)/2)^4 + a^3))
Time = 0.45 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )-6 \cos \left (d x +c \right )+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7 c -7 d x +6}{2 a^{3} d} \] Input:
int(cos(d*x+c)^5*cot(d*x+c)/(a+a*sin(d*x+c))^3,x)
Output:
(cos(c + d*x)*sin(c + d*x) - 6*cos(c + d*x) + 2*log(tan((c + d*x)/2)) - 7* c - 7*d*x + 6)/(2*a**3*d)