Integrand size = 21, antiderivative size = 145 \[ \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 b \csc (c+d x)}{a^4 d}-\frac {\csc ^2(c+d x)}{2 a^3 d}-\frac {\left (a^2-6 b^2\right ) \log (\sin (c+d x))}{a^5 d}+\frac {\left (a^2-6 b^2\right ) \log (a+b \sin (c+d x))}{a^5 d}-\frac {a^2-b^2}{2 a^3 d (a+b \sin (c+d x))^2}-\frac {a^2-3 b^2}{a^4 d (a+b \sin (c+d x))} \]
3*b*csc(d*x+c)/a^4/d-1/2*csc(d*x+c)^2/a^3/d-(a^2-6*b^2)*ln(sin(d*x+c))/a^5 /d+(a^2-6*b^2)*ln(a+b*sin(d*x+c))/a^5/d+1/2*(-a^2+b^2)/a^3/d/(a+b*sin(d*x+ c))^2+(-a^2+3*b^2)/a^4/d/(a+b*sin(d*x+c))
Time = 0.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {-6 a b \csc (c+d x)+a^2 \csc ^2(c+d x)+2 \left (a^2-6 b^2\right ) \log (\sin (c+d x))-2 \left (a^2-6 b^2\right ) \log (a+b \sin (c+d x))+\frac {a^2 (a-b) (a+b)}{(a+b \sin (c+d x))^2}+\frac {2 a \left (a^2-3 b^2\right )}{a+b \sin (c+d x)}}{2 a^5 d} \]
-1/2*(-6*a*b*Csc[c + d*x] + a^2*Csc[c + d*x]^2 + 2*(a^2 - 6*b^2)*Log[Sin[c + d*x]] - 2*(a^2 - 6*b^2)*Log[a + b*Sin[c + d*x]] + (a^2*(a - b)*(a + b)) /(a + b*Sin[c + d*x])^2 + (2*a*(a^2 - 3*b^2))/(a + b*Sin[c + d*x]))/(a^5*d )
Time = 0.33 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3200, 522, 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.
| \(\displaystyle \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^3 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )}{b^3 (a+b \sin (c+d x))^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {\int \left (\frac {\csc ^3(c+d x)}{a^3 b}-\frac {3 \csc ^2(c+d x)}{a^4}+\frac {\left (6 b^2-a^2\right ) \csc (c+d x)}{a^5 b}+\frac {a^2-6 b^2}{a^5 (a+b \sin (c+d x))}+\frac {a^2-3 b^2}{a^4 (a+b \sin (c+d x))^2}+\frac {a^2-b^2}{a^3 (a+b \sin (c+d x))^3}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 b \csc (c+d x)}{a^4}-\frac {\csc ^2(c+d x)}{2 a^3}-\frac {\left (a^2-6 b^2\right ) \log (b \sin (c+d x))}{a^5}+\frac {\left (a^2-6 b^2\right ) \log (a+b \sin (c+d x))}{a^5}-\frac {a^2-3 b^2}{a^4 (a+b \sin (c+d x))}-\frac {a^2-b^2}{2 a^3 (a+b \sin (c+d x))^2}}{d}\) |
((3*b*Csc[c + d*x])/a^4 - Csc[c + d*x]^2/(2*a^3) - ((a^2 - 6*b^2)*Log[b*Si n[c + d*x]])/a^5 + ((a^2 - 6*b^2)*Log[a + b*Sin[c + d*x]])/a^5 - (a^2 - b^ 2)/(2*a^3*(a + b*Sin[c + d*x])^2) - (a^2 - 3*b^2)/(a^4*(a + b*Sin[c + d*x] )))/d
3.2.96.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 5.47 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (a^{2}-6 b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{5}}-\frac {a^{2}-3 b^{2}}{a^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{2}-b^{2}}{2 a^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {1}{2 a^{3} \sin \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+6 b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{5}}+\frac {3 b}{a^{4} \sin \left (d x +c \right )}}{d}\) | \(131\) |
| default | \(\frac {\frac {\left (a^{2}-6 b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{5}}-\frac {a^{2}-3 b^{2}}{a^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{2}-b^{2}}{2 a^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {1}{2 a^{3} \sin \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+6 b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{5}}+\frac {3 b}{a^{4} \sin \left (d x +c \right )}}{d}\) | \(131\) |
| risch | \(-\frac {2 i \left (3 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-18 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-6 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-10 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+36 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+5 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+18 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-18 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-5 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-18 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{5} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{a^{3} d}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right ) b^{2}}{a^{5} d}\) | \(380\) |
1/d*((a^2-6*b^2)/a^5*ln(a+b*sin(d*x+c))-(a^2-3*b^2)/a^4/(a+b*sin(d*x+c))-1 /2*(a^2-b^2)/a^3/(a+b*sin(d*x+c))^2-1/2/a^3/sin(d*x+c)^2+(-a^2+6*b^2)/a^5* ln(sin(d*x+c))+3/a^4*b/sin(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (141) = 282\).
Time = 0.33 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.79 \[ \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {4 \, a^{4} - 18 \, a^{2} b^{2} - 3 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left ({\left (a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} - 5 \, a^{2} b^{2} - 6 \, b^{4} - {\left (a^{4} - 4 \, a^{2} b^{2} - 12 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{3} b - 6 \, a b^{3} - {\left (a^{3} b - 6 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 2 \, {\left ({\left (a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} - 5 \, a^{2} b^{2} - 6 \, b^{4} - {\left (a^{4} - 4 \, a^{2} b^{2} - 12 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{3} b - 6 \, a b^{3} - {\left (a^{3} b - 6 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2 \, {\left (a^{3} b + 6 \, a b^{3} + {\left (a^{3} b - 6 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{5} b^{2} d \cos \left (d x + c\right )^{4} - {\left (a^{7} + 2 \, a^{5} b^{2}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{7} + a^{5} b^{2}\right )} d - 2 \, {\left (a^{6} b d \cos \left (d x + c\right )^{2} - a^{6} b d\right )} \sin \left (d x + c\right )\right )}} \]
-1/2*(4*a^4 - 18*a^2*b^2 - 3*(a^4 - 6*a^2*b^2)*cos(d*x + c)^2 - 2*((a^2*b^ 2 - 6*b^4)*cos(d*x + c)^4 + a^4 - 5*a^2*b^2 - 6*b^4 - (a^4 - 4*a^2*b^2 - 1 2*b^4)*cos(d*x + c)^2 + 2*(a^3*b - 6*a*b^3 - (a^3*b - 6*a*b^3)*cos(d*x + c )^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) + 2*((a^2*b^2 - 6*b^4)*cos(d*x + c)^4 + a^4 - 5*a^2*b^2 - 6*b^4 - (a^4 - 4*a^2*b^2 - 12*b^4)*cos(d*x + c) ^2 + 2*(a^3*b - 6*a*b^3 - (a^3*b - 6*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))* log(-1/2*sin(d*x + c)) - 2*(a^3*b + 6*a*b^3 + (a^3*b - 6*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^5*b^2*d*cos(d*x + c)^4 - (a^7 + 2*a^5*b^2)*d*cos(d* x + c)^2 + (a^7 + a^5*b^2)*d - 2*(a^6*b*d*cos(d*x + c)^2 - a^6*b*d)*sin(d* x + c))
\[ \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08 \[ \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {4 \, a^{2} b \sin \left (d x + c\right ) - 2 \, {\left (a^{2} b - 6 \, b^{3}\right )} \sin \left (d x + c\right )^{3} - a^{3} - 3 \, {\left (a^{3} - 6 \, a b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{4} b^{2} \sin \left (d x + c\right )^{4} + 2 \, a^{5} b \sin \left (d x + c\right )^{3} + a^{6} \sin \left (d x + c\right )^{2}} + \frac {2 \, {\left (a^{2} - 6 \, b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{5}} - \frac {2 \, {\left (a^{2} - 6 \, b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{5}}}{2 \, d} \]
1/2*((4*a^2*b*sin(d*x + c) - 2*(a^2*b - 6*b^3)*sin(d*x + c)^3 - a^3 - 3*(a ^3 - 6*a*b^2)*sin(d*x + c)^2)/(a^4*b^2*sin(d*x + c)^4 + 2*a^5*b*sin(d*x + c)^3 + a^6*sin(d*x + c)^2) + 2*(a^2 - 6*b^2)*log(b*sin(d*x + c) + a)/a^5 - 2*(a^2 - 6*b^2)*log(sin(d*x + c))/a^5)/d
Time = 0.40 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{5}} - \frac {2 \, {\left (a^{2} b - 6 \, b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{5} b} + \frac {2 \, a^{2} b \sin \left (d x + c\right )^{3} - 12 \, b^{3} \sin \left (d x + c\right )^{3} + 3 \, a^{3} \sin \left (d x + c\right )^{2} - 18 \, a b^{2} \sin \left (d x + c\right )^{2} - 4 \, a^{2} b \sin \left (d x + c\right ) + a^{3}}{{\left (b \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right )\right )}^{2} a^{4}}}{2 \, d} \]
-1/2*(2*(a^2 - 6*b^2)*log(abs(sin(d*x + c)))/a^5 - 2*(a^2*b - 6*b^3)*log(a bs(b*sin(d*x + c) + a))/(a^5*b) + (2*a^2*b*sin(d*x + c)^3 - 12*b^3*sin(d*x + c)^3 + 3*a^3*sin(d*x + c)^2 - 18*a*b^2*sin(d*x + c)^2 - 4*a^2*b*sin(d*x + c) + a^3)/((b*sin(d*x + c)^2 + a*sin(d*x + c))^2*a^4))/d
Time = 6.71 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.30 \[ \int \frac {\cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (22\,a\,b^2-a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (26\,a^2\,b-8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (22\,a^2\,b-32\,b^3\right )-\frac {a^3}{2}+4\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^4-96\,a^2\,b^2+112\,b^4\right )}{2\,a}}{d\,\left (4\,a^6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,a^6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (8\,a^6+16\,a^4\,b^2\right )+16\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+16\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2-6\,b^2\right )}{a^5\,d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^2-6\,b^2\right )}{a^5\,d} \]
(tan(c/2 + (d*x)/2)^2*(22*a*b^2 - a^3) + tan(c/2 + (d*x)/2)^3*(26*a^2*b - 8*b^3) + tan(c/2 + (d*x)/2)^5*(22*a^2*b - 32*b^3) - a^3/2 + 4*a^2*b*tan(c/ 2 + (d*x)/2) - (tan(c/2 + (d*x)/2)^4*(a^4 + 112*b^4 - 96*a^2*b^2))/(2*a))/ (d*(4*a^6*tan(c/2 + (d*x)/2)^2 + 4*a^6*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d *x)/2)^4*(8*a^6 + 16*a^4*b^2) + 16*a^5*b*tan(c/2 + (d*x)/2)^3 + 16*a^5*b*t an(c/2 + (d*x)/2)^5)) - tan(c/2 + (d*x)/2)^2/(8*a^3*d) + (3*b*tan(c/2 + (d *x)/2))/(2*a^4*d) - (log(tan(c/2 + (d*x)/2))*(a^2 - 6*b^2))/(a^5*d) + (log (a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c/2 + (d*x)/2)^2)*(a^2 - 6*b^2))/(a^5* d)