Integrand size = 27, antiderivative size = 82 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{8 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \] Output:
1/8*arctanh(cos(d*x+c))/a/d+1/3*cot(d*x+c)^3/a/d+1/8*cot(d*x+c)*csc(d*x+c) /a/d-1/4*cot(d*x+c)*csc(d*x+c)^3/a/d
Time = 1.50 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^4(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-42 \cos (c+d x)+2 \cos (3 (c+d x)) (-3+8 \sin (c+d x))+24 \left (\left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^4(c+d x)+\sin (2 (c+d x))\right )\right )}{192 a d (1+\sin (c+d x))} \] Input:
Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-42*Cos[c + d*x] + 2*Cos[3*(c + d*x)]*(-3 + 8*Sin[c + d*x]) + 24*((Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + Sin[2*(c + d*x)])))/(192*a*d*(1 + Sin[c + d*x]))
Time = 0.60 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3318, 3042, 3087, 15, 3091, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^5 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) \csc ^3(c+d x)dx}{a}-\frac {\int \cot ^2(c+d x) \csc ^2(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\int \cot ^2(c+d x)d(-\cot (c+d x))}{a d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}+\frac {\cot ^3(c+d x)}{3 a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{4} \int \csc ^3(c+d x)dx-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}}{a}+\frac {\cot ^3(c+d x)}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{4} \int \csc (c+d x)^3dx-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}}{a}+\frac {\cot ^3(c+d x)}{3 a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\cot (c+d x) \csc (c+d x)}{2 d}-\frac {1}{2} \int \csc (c+d x)dx\right )-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}}{a}+\frac {\cot ^3(c+d x)}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\cot (c+d x) \csc (c+d x)}{2 d}-\frac {1}{2} \int \csc (c+d x)dx\right )-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}}{a}+\frac {\cot ^3(c+d x)}{3 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\text {arctanh}(\cos (c+d x))}{2 d}+\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}}{a}+\frac {\cot ^3(c+d x)}{3 a d}\) |
Input:
Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
Cot[c + d*x]^3/(3*a*d) + (-1/4*(Cot[c + d*x]*Csc[c + d*x]^3)/d + (ArcTanh[ Cos[c + d*x]]/(2*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*d))/4)/a
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.55 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\) | \(98\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\) | \(98\) |
| risch | \(-\frac {24 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}-24 i {\mathrm e}^{4 i \left (d x +c \right )}+21 \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i {\mathrm e}^{2 i \left (d x +c \right )}+21 \,{\mathrm e}^{3 i \left (d x +c \right )}-8 i+3 \,{\mathrm e}^{i \left (d x +c \right )}}{12 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}\) | \(146\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/16/d/a*(1/4*tan(1/2*d*x+1/2*c)^4-2/3*tan(1/2*d*x+1/2*c)^3+2*tan(1/2*d*x+ 1/2*c)+2/3/tan(1/2*d*x+1/2*c)^3-1/4/tan(1/2*d*x+1/2*c)^4-2/tan(1/2*d*x+1/2 *c)-2*ln(tan(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.61 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {16 \, \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 6 \, \cos \left (d x + c\right )^{3} + 3 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/48*(16*cos(d*x + c)^3*sin(d*x + c) - 6*cos(d*x + c)^3 + 3*(cos(d*x + c)^ 4 - 2*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 3*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2) - 6*cos(d*x + c))/(a* d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)
\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cot ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)**4*csc(c + d*x)/(sin(c + d*x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (74) = 148\).
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.88 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a} - \frac {24 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a \sin \left (d x + c\right )^{4}}}{192 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
1/192*((24*sin(d*x + c)/(cos(d*x + c) + 1) - 8*sin(d*x + c)^3/(cos(d*x + c ) + 1)^3 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/a - 24*log(sin(d*x + c)/ (cos(d*x + c) + 1))/a + (8*sin(d*x + c)/(cos(d*x + c) + 1) - 24*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3)*(cos(d*x + c) + 1)^4/(a*sin(d*x + c)^4))/d
Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.57 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {24 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {50 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/192*(24*log(abs(tan(1/2*d*x + 1/2*c)))/a - (3*a^3*tan(1/2*d*x + 1/2*c)^ 4 - 8*a^3*tan(1/2*d*x + 1/2*c)^3 + 24*a^3*tan(1/2*d*x + 1/2*c))/a^4 - (50* tan(1/2*d*x + 1/2*c)^4 - 24*tan(1/2*d*x + 1/2*c)^3 + 8*tan(1/2*d*x + 1/2*c ) - 3)/(a*tan(1/2*d*x + 1/2*c)^4))/d
Time = 17.63 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.45 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{4}\right )}{16\,a\,d} \] Input:
int(cot(c + d*x)^4/(sin(c + d*x)*(a + a*sin(c + d*x))),x)
Output:
tan(c/2 + (d*x)/2)^4/(64*a*d) - tan(c/2 + (d*x)/2)^3/(24*a*d) - log(tan(c/ 2 + (d*x)/2))/(8*a*d) + tan(c/2 + (d*x)/2)/(8*a*d) - (cot(c/2 + (d*x)/2)^4 *(2*tan(c/2 + (d*x)/2)^3 - (2*tan(c/2 + (d*x)/2))/3 + 1/4))/(16*a*d)
Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.11 \[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )-6 \cos \left (d x +c \right )-3 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}}{24 \sin \left (d x +c \right )^{4} a d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
( - 8*cos(c + d*x)*sin(c + d*x)**3 + 3*cos(c + d*x)*sin(c + d*x)**2 + 8*co s(c + d*x)*sin(c + d*x) - 6*cos(c + d*x) - 3*log(tan((c + d*x)/2))*sin(c + d*x)**4)/(24*sin(c + d*x)**4*a*d)