Integrand size = 29, antiderivative size = 49 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 x}{a^3}+\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{a^3 d}-\frac {\cot (c+d x)}{a^3 d} \] Output:
3*x/a^3+3*arctanh(cos(d*x+c))/a^3/d+cos(d*x+c)/a^3/d-cot(d*x+c)/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(106\) vs. \(2(49)=98\).
Time = 1.94 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.16 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (6 (c+d x)+2 \cos (c+d x)-\cot \left (\frac {1}{2} (c+d x)\right )+6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 d (a+a \sin (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(6*(c + d*x) + 2*Cos[c + d*x] - C ot[(c + d*x)/2] + 6*Log[Cos[(c + d*x)/2]] - 6*Log[Sin[(c + d*x)/2]] + Tan[ (c + d*x)/2]))/(2*d*(a + a*Sin[c + d*x])^3)
Time = 0.42 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.
\(\displaystyle \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^2 (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3348 |
\(\displaystyle \frac {\int \csc ^2(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\sin (c+d x)^2}dx}{a^6}\) |
\(\Big \downarrow \) 3236 |
\(\displaystyle \frac {\int \left (\csc ^2(c+d x) a^3-3 \csc (c+d x) a^3-\sin (c+d x) a^3+3 a^3\right )dx}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+3 a^3 x}{a^6}\) |
Input:
Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
Output:
(3*a^3*x + (3*a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d - (a^3*C ot[c + d*x])/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 = 2.71 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.49
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+12 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(73\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+12 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(73\) |
risch | \(\frac {3 x}{a^{3}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {2 i}{a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\) | \(104\) |
Input:
int(cos(d*x+c)^4*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/2/d/a^3*(tan(1/2*d*x+1/2*c)+4/(1+tan(1/2*d*x+1/2*c)^2)+12*arctan(tan(1/2 *d*x+1/2*c))-1/tan(1/2*d*x+1/2*c)-6*ln(tan(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, {\left (3 \, d x + \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 3 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )}{2 \, a^{3} d \sin \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/2*(2*(3*d*x + cos(d*x + c))*sin(d*x + c) + 3*log(1/2*cos(d*x + c) + 1/2) *sin(d*x + c) - 3*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2*cos(d*x + c))/(a^3*d*sin(d*x + c))
\[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{4}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cos(d*x+c)**4*cot(d*x+c)**2/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cos(c + d*x)**4*cot(c + d*x)**2/(sin(c + d*x)**3 + 3*sin(c + d*x) **2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (49) = 98\).
Time = 0.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 3.22 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {\sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/2*((4*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1 )^2 - 1)/(a^3*sin(d*x + c)/(cos(d*x + c) + 1) + a^3*sin(d*x + c)^3/(cos(d* x + c) + 1)^3) + 12*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - 6*log(si n(d*x + c)/(cos(d*x + c) + 1))/a^3 + sin(d*x + c)/(a^3*(cos(d*x + c) + 1)) )/d
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (49) = 98\).
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.27 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {6 \, {\left (d x + c\right )}}{a^{3}} - \frac {6 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} + \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{3}}}{2 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/2*(6*(d*x + c)/a^3 - 6*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + tan(1/2*d*x + 1/2*c)/a^3 + (2*tan(1/2*d*x + 1/2*c)^3 - tan(1/2*d*x + 1/2*c)^2 + 6*tan( 1/2*d*x + 1/2*c) - 1)/((tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))*a^3 ))/d
Time = 34.50 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.08 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {6\,\mathrm {atan}\left (\frac {36}{36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+36}-\frac {36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+36}\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1}{d\,\left (2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \] Input:
int((cos(c + d*x)^4*cot(c + d*x)^2)/(a + a*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)/(2*a^3*d) - (3*log(tan(c/2 + (d*x)/2)))/(a^3*d) - (6*at an(36/(36*tan(c/2 + (d*x)/2) + 36) - (36*tan(c/2 + (d*x)/2))/(36*tan(c/2 + (d*x)/2) + 36)))/(a^3*d) - (tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2) + 1)/(d*(2*a^3*tan(c/2 + (d*x)/2)^3 + 2*a^3*tan(c/2 + (d*x)/2)))
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.49 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )-\cos \left (d x +c \right )-3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+3 \sin \left (d x +c \right ) d x -\sin \left (d x +c \right )}{\sin \left (d x +c \right ) a^{3} d} \] Input:
int(cos(d*x+c)^4*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x)
Output:
(cos(c + d*x)*sin(c + d*x) - cos(c + d*x) - 3*log(tan((c + d*x)/2))*sin(c + d*x) + 3*sin(c + d*x)*d*x - sin(c + d*x))/(sin(c + d*x)*a**3*d)