Integrand size = 29, antiderivative size = 131 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {\left (2 a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {\left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a^3 b^2 d (a+b \sin (c+d x))} \]
2*b*csc(d*x+c)/a^3/d-1/2*csc(d*x+c)^2/a^2/d-(2*a^2-3*b^2)*ln(sin(d*x+c))/a ^4/d+(a^4+2*a^2*b^2-3*b^4)*ln(a+b*sin(d*x+c))/a^4/b^2/d+(a^2-b^2)^2/a^3/b^ 2/d/(a+b*sin(d*x+c))
Time = 0.50 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {4 a b \csc (c+d x)-a^2 \csc ^2(c+d x)-2 \left (2 a^2-3 b^2\right ) \log (\sin (c+d x))+\frac {2 \left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{b^2}+\frac {2 a \left (a^2-b^2\right )^2}{b^2 (a+b \sin (c+d x))}}{2 a^4 d} \]
(4*a*b*Csc[c + d*x] - a^2*Csc[c + d*x]^2 - 2*(2*a^2 - 3*b^2)*Log[Sin[c + d *x]] + (2*(a^4 + 2*a^2*b^2 - 3*b^4)*Log[a + b*Sin[c + d*x]])/b^2 + (2*a*(a ^2 - b^2)^2)/(b^2*(a + b*Sin[c + d*x])))/(2*a^4*d)
Time = 0.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3316, 27, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5}{\sin (c+d x)^3 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{(a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b^3 (a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^2 d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {\int \left (\frac {b \csc ^3(c+d x)}{a^2}-\frac {2 b^2 \csc ^2(c+d x)}{a^3}+\frac {\left (3 b^4-2 a^2 b^2\right ) \csc (c+d x)}{a^4 b}+\frac {a^4+2 b^2 a^2-3 b^4}{a^4 (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right )^2}{a^3 (a+b \sin (c+d x))^2}\right )d(b \sin (c+d x))}{b^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 b^3 \csc (c+d x)}{a^3}-\frac {b^2 \csc ^2(c+d x)}{2 a^2}-\frac {b^2 \left (2 a^2-3 b^2\right ) \log (b \sin (c+d x))}{a^4}+\frac {\left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4}+\frac {\left (a^2-b^2\right )^2}{a^3 (a+b \sin (c+d x))}}{b^2 d}\) |
((2*b^3*Csc[c + d*x])/a^3 - (b^2*Csc[c + d*x]^2)/(2*a^2) - (b^2*(2*a^2 - 3 *b^2)*Log[b*Sin[c + d*x]])/a^4 + ((a^4 + 2*a^2*b^2 - 3*b^4)*Log[a + b*Sin[ c + d*x]])/a^4 + (a^2 - b^2)^2/(a^3*(a + b*Sin[c + d*x])))/(b^2*d)
3.13.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 0.84 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(\frac {-\frac {-a^{4}+2 a^{2} b^{2}-b^{4}}{a^{3} b^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {\left (a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{4} b^{2}}-\frac {1}{2 a^{2} \sin \left (d x +c \right )^{2}}+\frac {\left (-2 a^{2}+3 b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \sin \left (d x +c \right )}}{d}\) | \(129\) |
| default | \(\frac {-\frac {-a^{4}+2 a^{2} b^{2}-b^{4}}{a^{3} b^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {\left (a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{4} b^{2}}-\frac {1}{2 a^{2} \sin \left (d x +c \right )^{2}}+\frac {\left (-2 a^{2}+3 b^{2}\right ) \ln \left (\sin \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \sin \left (d x +c \right )}}{d}\) | \(129\) |
| parallelrisch | \(\frac {\left (a -b \right ) \left (a +b \right ) \left (a^{2}+3 b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-a^{4} \left (a +b \sin \left (d x +c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b^{2} \left (a^{2}-\frac {3 b^{2}}{2}\right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {b^{2} a^{2} \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+22\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 a \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} \left (3+\cos \left (2 d x +2 c \right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}+\frac {11 a^{2} b^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+a^{4}+3 b^{4}\right ) a}{a^{4} b^{2} d \left (a +b \sin \left (d x +c \right )\right )}\) | \(237\) |
| norman | \(\frac {-\frac {1}{8 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {3 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2} d}+\frac {3 b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d}-\frac {\left (4 a^{4}-15 a^{2} b^{2}+12 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4} b d}-\frac {\left (8 a^{4}-33 a^{2} b^{2}+24 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4} b d}-\frac {\left (8 a^{4}-33 a^{2} b^{2}+24 b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4} b d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}+\frac {\left (a^{4}+2 a^{2} b^{2}-3 b^{4}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{2} a^{4} d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2} d}-\frac {\left (2 a^{2}-3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4} d}\) | \(391\) |
| risch | \(-\frac {i x}{b^{2}}-\frac {2 i c}{b^{2} d}+\frac {6 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-4 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}-6 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-4 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+12 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{i \left (d x +c \right )}-4 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} b^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{3} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{2} d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{2} d}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{4} d}\) | \(394\) |
1/d*(-(-a^4+2*a^2*b^2-b^4)/a^3/b^2/(a+b*sin(d*x+c))+(a^4+2*a^2*b^2-3*b^4)/ a^4/b^2*ln(a+b*sin(d*x+c))-1/2/a^2/sin(d*x+c)^2+(-2*a^2+3*b^2)/a^4*ln(sin( d*x+c))+2/a^3*b/sin(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (129) = 258\).
Time = 0.39 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.56 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {3 \, a^{2} b^{3} \sin \left (d x + c\right ) + 2 \, a^{5} - 5 \, a^{3} b^{2} + 6 \, a b^{4} - 2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} - 3 \, b^{5} - {\left (a^{4} b + 2 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (2 \, a^{3} b^{2} - 3 \, a b^{4} - {\left (2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, a^{2} b^{3} - 3 \, b^{5} - {\left (2 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right )}{2 \, {\left (a^{5} b^{2} d \cos \left (d x + c\right )^{2} - a^{5} b^{2} d + {\left (a^{4} b^{3} d \cos \left (d x + c\right )^{2} - a^{4} b^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
-1/2*(3*a^2*b^3*sin(d*x + c) + 2*a^5 - 5*a^3*b^2 + 6*a*b^4 - 2*(a^5 - 2*a^ 3*b^2 + 3*a*b^4)*cos(d*x + c)^2 + 2*(a^5 + 2*a^3*b^2 - 3*a*b^4 - (a^5 + 2* a^3*b^2 - 3*a*b^4)*cos(d*x + c)^2 + (a^4*b + 2*a^2*b^3 - 3*b^5 - (a^4*b + 2*a^2*b^3 - 3*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) - 2*(2*a^3*b^2 - 3*a*b^4 - (2*a^3*b^2 - 3*a*b^4)*cos(d*x + c)^2 + (2*a^2*b^ 3 - 3*b^5 - (2*a^2*b^3 - 3*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*sin (d*x + c)))/(a^5*b^2*d*cos(d*x + c)^2 - a^5*b^2*d + (a^4*b^3*d*cos(d*x + c )^2 - a^4*b^3*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {3 \, a b^{3} \sin \left (d x + c\right ) - a^{2} b^{2} + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )^{2}}{a^{3} b^{3} \sin \left (d x + c\right )^{3} + a^{4} b^{2} \sin \left (d x + c\right )^{2}} - \frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{4}} + \frac {2 \, {\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} b^{2}}}{2 \, d} \]
1/2*((3*a*b^3*sin(d*x + c) - a^2*b^2 + 2*(a^4 - 2*a^2*b^2 + 3*b^4)*sin(d*x + c)^2)/(a^3*b^3*sin(d*x + c)^3 + a^4*b^2*sin(d*x + c)^2) - 2*(2*a^2 - 3* b^2)*log(sin(d*x + c))/a^4 + 2*(a^4 + 2*a^2*b^2 - 3*b^4)*log(b*sin(d*x + c ) + a)/(a^4*b^2))/d
Time = 0.35 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {2 \, {\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{2}} + \frac {2 \, {\left (a^{4} \sin \left (d x + c\right ) + 2 \, a^{2} b^{2} \sin \left (d x + c\right ) - 3 \, b^{4} \sin \left (d x + c\right ) + 4 \, a^{3} b - 4 \, a b^{3}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{4} b} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{2} - 9 \, b^{2} \sin \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) - a^{2}}{a^{4} \sin \left (d x + c\right )^{2}}}{2 \, d} \]
-1/2*(2*(2*a^2 - 3*b^2)*log(abs(sin(d*x + c)))/a^4 - 2*(a^4 + 2*a^2*b^2 - 3*b^4)*log(abs(b*sin(d*x + c) + a))/(a^4*b^2) + 2*(a^4*sin(d*x + c) + 2*a^ 2*b^2*sin(d*x + c) - 3*b^4*sin(d*x + c) + 4*a^3*b - 4*a*b^3)/((b*sin(d*x + c) + a)*a^4*b) - (6*a^2*sin(d*x + c)^2 - 9*b^2*sin(d*x + c)^2 + 4*a*b*sin (d*x + c) - a^2)/(a^4*sin(d*x + c)^2))/d
Time = 13.80 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.14 \[ \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{2}-8\,b^2\right )+\frac {a^2}{2}-3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^4-5\,a^2\,b^2+2\,b^4\right )}{a\,b}}{d\,\left (4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{b^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-3\,b^2\right )}{a^4\,d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4+2\,a^2\,b^2-3\,b^4\right )}{a^4\,b^2\,d} \]
(b*tan(c/2 + (d*x)/2))/(a^3*d) - (tan(c/2 + (d*x)/2)^2*(a^2/2 - 8*b^2) + a ^2/2 - 3*a*b*tan(c/2 + (d*x)/2) + (4*tan(c/2 + (d*x)/2)^3*(2*a^4 + 2*b^4 - 5*a^2*b^2))/(a*b))/(d*(4*a^4*tan(c/2 + (d*x)/2)^2 + 4*a^4*tan(c/2 + (d*x) /2)^4 + 8*a^3*b*tan(c/2 + (d*x)/2)^3)) - log(tan(c/2 + (d*x)/2)^2 + 1)/(b^ 2*d) - tan(c/2 + (d*x)/2)^2/(8*a^2*d) - (log(tan(c/2 + (d*x)/2))*(2*a^2 - 3*b^2))/(a^4*d) + (log(a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c/2 + (d*x)/2)^2 )*(a^4 - 3*b^4 + 2*a^2*b^2))/(a^4*b^2*d)