Integrand size = 11, antiderivative size = 58 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=-\frac {\text {arctanh}(\cos (x))}{a^3}+\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \] Output:
-arctanh(cos(x))/a^3+1/5*cos(x)/(a+a*sin(x))^3+7/15*cos(x)/a/(a+a*sin(x))^ 2+22*cos(x)/(15*a^3+15*a^3*sin(x))
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {-15 \text {arctanh}(\cos (x)) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^6+\cos (x) \left (32+51 \sin (x)+22 \sin ^2(x)\right )}{15 a^3 (1+\sin (x))^3} \] Input:
Integrate[Csc[x]/(a + a*Sin[x])^3,x]
Output:
(-15*ArcTanh[Cos[x]]*(Cos[x/2] + Sin[x/2])^6 + Cos[x]*(32 + 51*Sin[x] + 22 *Sin[x]^2))/(15*a^3*(1 + Sin[x])^3)
Time = 0.54 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 3245, 3042, 3457, 3042, 3457, 27, 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.
| \(\displaystyle \int \frac {\csc (x)}{(a \sin (x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (x) (a \sin (x)+a)^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle \frac {\int \frac {\csc (x) (5 a-2 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a-2 a \sin (x)}{\sin (x) (\sin (x) a+a)^2}dx}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\csc (x) \left (15 a^2-7 a^2 \sin (x)\right )}{\sin (x) a+a}dx}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {15 a^2-7 a^2 \sin (x)}{\sin (x) (\sin (x) a+a)}dx}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int 15 a^3 \csc (x)dx}{a^2}+\frac {22 a^2 \cos (x)}{a \sin (x)+a}}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {15 a \int \csc (x)dx+\frac {22 a^2 \cos (x)}{a \sin (x)+a}}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {15 a \int \csc (x)dx+\frac {22 a^2 \cos (x)}{a \sin (x)+a}}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {22 a^2 \cos (x)}{a \sin (x)+a}-15 a \text {arctanh}(\cos (x))}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3}\) |
Input:
Int[Csc[x]/(a + a*Sin[x])^3,x]
Output:
Cos[x]/(5*(a + a*Sin[x])^3) + ((7*a*Cos[x])/(3*(a + a*Sin[x])^2) + (-15*a* ArcTanh[Cos[x]] + (22*a^2*Cos[x])/(a + a*Sin[x]))/(3*a^2))/(5*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {20}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {6}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {6}{\tan \left (\frac {x}{2}\right )+1}+\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(61\) |
| parallelrisch | \(\frac {15 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5} \ln \left (\tan \left (\frac {x}{2}\right )\right )+90 \tan \left (\frac {x}{2}\right )^{4}+270 \tan \left (\frac {x}{2}\right )^{3}+370 \tan \left (\frac {x}{2}\right )^{2}+230 \tan \left (\frac {x}{2}\right )+64}{15 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(61\) |
| norman | \(\frac {\frac {6 \tan \left (\frac {x}{2}\right )^{4}}{a}+\frac {64}{15 a}+\frac {18 \tan \left (\frac {x}{2}\right )^{3}}{a}+\frac {74 \tan \left (\frac {x}{2}\right )^{2}}{3 a}+\frac {46 \tan \left (\frac {x}{2}\right )}{3 a}}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(71\) |
| risch | \(\frac {10 i {\mathrm e}^{3 i x}+2 \,{\mathrm e}^{4 i x}-\frac {38 i {\mathrm e}^{i x}}{3}-\frac {58 \,{\mathrm e}^{2 i x}}{3}+\frac {44}{15}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{3}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{3}}\) | \(74\) |
Input:
int(csc(x)/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
1/a^3*(8/5/(tan(1/2*x)+1)^5-4/(tan(1/2*x)+1)^4+20/3/(tan(1/2*x)+1)^3-6/(ta n(1/2*x)+1)^2+6/(tan(1/2*x)+1)+ln(tan(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (52) = 104\).
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.90 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {44 \, \cos \left (x\right )^{3} - 58 \, \cos \left (x\right )^{2} - 15 \, {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (22 \, \cos \left (x\right )^{2} + 51 \, \cos \left (x\right ) - 3\right )} \sin \left (x\right ) - 108 \, \cos \left (x\right ) - 6}{30 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(csc(x)/(a+a*sin(x))^3,x, algorithm="fricas")
Output:
1/30*(44*cos(x)^3 - 58*cos(x)^2 - 15*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*log(1/2*cos(x) + 1/2) + 15*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*log(-1/2*c os(x) + 1/2) - 2*(22*cos(x)^2 + 51*cos(x) - 3)*sin(x) - 108*cos(x) - 6)/(a ^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^ 3*cos(x) - 4*a^3)*sin(x))
\[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {\int \frac {\csc {\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin {\left (x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(csc(x)/(a+a*sin(x))**3,x)
Output:
Integral(csc(x)/(sin(x)**3 + 3*sin(x)**2 + 3*sin(x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (52) = 104\).
Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.47 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {2 \, {\left (\frac {115 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {185 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {135 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 32\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} + \frac {\log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \] Input:
integrate(csc(x)/(a+a*sin(x))^3,x, algorithm="maxima")
Output:
2/15*(115*sin(x)/(cos(x) + 1) + 185*sin(x)^2/(cos(x) + 1)^2 + 135*sin(x)^3 /(cos(x) + 1)^3 + 45*sin(x)^4/(cos(x) + 1)^4 + 32)/(a^3 + 5*a^3*sin(x)/(co s(x) + 1) + 10*a^3*sin(x)^2/(cos(x) + 1)^2 + 10*a^3*sin(x)^3/(cos(x) + 1)^ 3 + 5*a^3*sin(x)^4/(cos(x) + 1)^4 + a^3*sin(x)^5/(cos(x) + 1)^5) + log(sin (x)/(cos(x) + 1))/a^3
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (45 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 135 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 185 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 115 \, \tan \left (\frac {1}{2} \, x\right ) + 32\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \] Input:
integrate(csc(x)/(a+a*sin(x))^3,x, algorithm="giac")
Output:
log(abs(tan(1/2*x)))/a^3 + 2/15*(45*tan(1/2*x)^4 + 135*tan(1/2*x)^3 + 185* tan(1/2*x)^2 + 115*tan(1/2*x) + 32)/(a^3*(tan(1/2*x) + 1)^5)
Time = 17.39 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3}+\frac {6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+18\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {74\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+\frac {46\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {64}{15}}{a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(1/(sin(x)*(a + a*sin(x))^3),x)
Output:
log(tan(x/2))/a^3 + ((46*tan(x/2))/3 + (74*tan(x/2)^2)/3 + 18*tan(x/2)^3 + 6*tan(x/2)^4 + 64/15)/(a^3*(tan(x/2) + 1)^5)
Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.53 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {15 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2}+30 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )+15 \cos \left (x \right ) \mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right )-8 \cos \left (x \right ) \sin \left (x \right )^{2}-23 \cos \left (x \right ) \sin \left (x \right )-18 \cos \left (x \right )-15 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3}-45 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2}-45 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )-15 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right )-36 \sin \left (x \right )^{3}-71 \sin \left (x \right )^{2}-23 \sin \left (x \right )+18}{15 a^{3} \left (\cos \left (x \right ) \sin \left (x \right )^{2}+2 \cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{3}-3 \sin \left (x \right )^{2}-3 \sin \left (x \right )-1\right )} \] Input:
int(csc(x)/(a+a*sin(x))^3,x)
Output:
(15*cos(x)*log(tan(x/2))*sin(x)**2 + 30*cos(x)*log(tan(x/2))*sin(x) + 15*c os(x)*log(tan(x/2)) - 8*cos(x)*sin(x)**2 - 23*cos(x)*sin(x) - 18*cos(x) - 15*log(tan(x/2))*sin(x)**3 - 45*log(tan(x/2))*sin(x)**2 - 45*log(tan(x/2)) *sin(x) - 15*log(tan(x/2)) - 36*sin(x)**3 - 71*sin(x)**2 - 23*sin(x) + 18) /(15*a**3*(cos(x)*sin(x)**2 + 2*cos(x)*sin(x) + cos(x) - sin(x)**3 - 3*sin (x)**2 - 3*sin(x) - 1))