Integrand size = 29, antiderivative size = 97 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{2 a^2}-\frac {3 \text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^2 d} \] Output:
-1/2*x/a^2-3*arctanh(cos(d*x+c))/a^2/d+2*cos(d*x+c)/a^2/d-1/3*cot(d*x+c)^3 /a^2/d+cot(d*x+c)*csc(d*x+c)/a^2/d-1/2*cos(d*x+c)*sin(d*x+c)/a^2/d
Time = 4.11 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.90 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )^4 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (30 \cos (c+d x)-\cos (3 (c+d x))+3 \left (\cos (5 (c+d x))+8 \left (c+d x-6 \cos (c+d x)+2 \cos (3 (c+d x))+6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\cos (2 (c+d x)) \left (c+d x+6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{768 a^2 d (1+\sin (c+d x))^2} \] Input:
Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]
Output:
-1/768*((1 + Cot[(c + d*x)/2])^4*Sec[(c + d*x)/2]^2*(30*Cos[c + d*x] - Cos [3*(c + d*x)] + 3*(Cos[5*(c + d*x)] + 8*(c + d*x - 6*Cos[c + d*x] + 2*Cos[ 3*(c + d*x)] + 6*Log[Cos[(c + d*x)/2]] - Cos[2*(c + d*x)]*(c + d*x + 6*Log [Cos[(c + d*x)/2]] - 6*Log[Sin[(c + d*x)/2]]) - 6*Log[Sin[(c + d*x)/2]])*S in[c + d*x]))*Tan[(c + d*x)/2])/(a^2*d*(1 + Sin[c + d*x])^2)
Time = 0.52 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 3042, 3188, 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.
\(\displaystyle \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^4 (a \sin (c+d x)+a)^2}dx\) |
\(\Big \downarrow \) 3354 |
\(\displaystyle \frac {\int \cot ^4(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^2}{\tan (c+d x)^4}dx}{a^4}\) |
\(\Big \downarrow \) 3188 |
\(\displaystyle \frac {\int \left (\csc ^4(c+d x) a^6-2 \csc ^3(c+d x) a^6-\csc ^2(c+d x) a^6+\sin ^2(c+d x) a^6+4 \csc (c+d x) a^6-2 \sin (c+d x) a^6-a^6\right )dx}{a^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {3 a^6 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^6 \cos (c+d x)}{d}-\frac {a^6 \cot ^3(c+d x)}{3 d}-\frac {a^6 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a^6 \cot (c+d x) \csc (c+d x)}{d}-\frac {a^6 x}{2}}{a^8}\) |
Input:
Int[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]
Output:
(-1/2*(a^6*x) - (3*a^6*ArcTanh[Cos[c + d*x]])/d + (2*a^6*Cos[c + d*x])/d - (a^6*Cot[c + d*x]^3)/(3*d) + (a^6*Cot[c + d*x]*Csc[c + d*x])/d - (a^6*Cos [c + d*x]*Sin[c + d*x])/(2*d))/a^8
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 2.57 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.69
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16 \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-2\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}\) | \(164\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16 \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-2\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}\) | \(164\) |
risch | \(-\frac {x}{2 a^{2}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{2}}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{d \,a^{2}}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {2 \left (-3 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}\) | \(174\) |
Input:
int(cos(d*x+c)^4*cot(d*x+c)^4/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/8/d/a^2*(1/3*tan(1/2*d*x+1/2*c)^3-2*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2 *c)-1/3/tan(1/2*d*x+1/2*c)^3+2/tan(1/2*d*x+1/2*c)^2+1/tan(1/2*d*x+1/2*c)+2 4*ln(tan(1/2*d*x+1/2*c))-16*(-1/2*tan(1/2*d*x+1/2*c)^3-2*tan(1/2*d*x+1/2*c )^2+1/2*tan(1/2*d*x+1/2*c)-2)/(1+tan(1/2*d*x+1/2*c)^2)^2-8*arctan(tan(1/2* d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.66 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3} - 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (d x \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right )^{3} - d x + 6 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 3 \, \cos \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/6*(3*cos(d*x + c)^5 - 4*cos(d*x + c)^3 - 9*(cos(d*x + c)^2 - 1)*log(1/2* cos(d*x + c) + 1/2)*sin(d*x + c) + 9*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*(d*x*cos(d*x + c)^2 - 4*cos(d*x + c)^3 - d*x + 6*cos(d*x + c))*sin(d*x + c) + 3*cos(d*x + c))/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))
\[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cos ^{4}{\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cos(d*x+c)**4*cot(d*x+c)**4/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cos(c + d*x)**4*cot(c + d*x)**4/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (91) = 182\).
Time = 0.11 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.15 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {108 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {19 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {102 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {27 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1}{\frac {a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {72 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{24 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/24*((6*sin(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 108*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 19*sin(d*x + c)^4/(cos(d* x + c) + 1)^4 + 102*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 27*sin(d*x + c)^ 6/(cos(d*x + c) + 1)^6 - 1)/(a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2*a ^2*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + a^2*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) - (3*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)/a^2 - 24*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + 72*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 )/d
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (91) = 182\).
Time = 0.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.00 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {12 \, {\left (d x + c\right )}}{a^{2}} - \frac {72 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {24 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac {132 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{24 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-1/24*(12*(d*x + c)/a^2 - 72*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - 24*(tan( 1/2*d*x + 1/2*c)^3 + 4*tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c) + 4)/ ((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^2) + (132*tan(1/2*d*x + 1/2*c)^3 - 3*tan (1/2*d*x + 1/2*c)^2 - 6*tan(1/2*d*x + 1/2*c) + 1)/(a^2*tan(1/2*d*x + 1/2*c )^3) - (a^4*tan(1/2*d*x + 1/2*c)^3 - 6*a^4*tan(1/2*d*x + 1/2*c)^2 - 3*a^4* tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 31.03 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.61 \[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^2\,d}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}+\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}}{d\,\left (8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^2\,d}+\frac {\mathrm {atan}\left (\frac {1}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}-\frac {6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}\right )}{a^2\,d} \] Input:
int((cos(c + d*x)^4*cot(c + d*x)^4)/(a + a*sin(c + d*x))^2,x)
Output:
tan(c/2 + (d*x)/2)^3/(24*a^2*d) - tan(c/2 + (d*x)/2)^2/(4*a^2*d) + (3*log( tan(c/2 + (d*x)/2)))/(a^2*d) + (2*tan(c/2 + (d*x)/2) + tan(c/2 + (d*x)/2)^ 2/3 + 36*tan(c/2 + (d*x)/2)^3 - (19*tan(c/2 + (d*x)/2)^4)/3 + 34*tan(c/2 + (d*x)/2)^5 + 9*tan(c/2 + (d*x)/2)^6 - 1/3)/(d*(8*a^2*tan(c/2 + (d*x)/2)^3 + 16*a^2*tan(c/2 + (d*x)/2)^5 + 8*a^2*tan(c/2 + (d*x)/2)^7)) - tan(c/2 + (d*x)/2)/(8*a^2*d) + atan(1/(tan(c/2 + (d*x)/2) + 6) - (6*tan(c/2 + (d*x)/ 2))/(tan(c/2 + (d*x)/2) + 6))/(a^2*d)
\[ \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\int \frac {\cos \left (d x +c \right )^{4} \cot \left (d x +c \right )^{4}}{\left (\sin \left (d x +c \right ) a +a \right )^{2}}d x \] Input:
int(cos(d*x+c)^4*cot(d*x+c)^4/(a+a*sin(d*x+c))^2,x)
Output:
int(cos(d*x+c)^4*cot(d*x+c)^4/(a+a*sin(d*x+c))^2,x)