Integrand size = 29, antiderivative size = 116 \[ \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc (c+d x)}{d}-\frac {17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {2 a^2 \log (\sin (c+d x))}{d}+\frac {a^2 \log (1+\sin (c+d x))}{8 d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {5 a^3}{4 d (a-a \sin (c+d x))} \]
-a^2*csc(d*x+c)/d-17/8*a^2*ln(1-sin(d*x+c))/d+2*a^2*ln(sin(d*x+c))/d+1/8*a ^2*ln(1+sin(d*x+c))/d+1/4*a^4/d/(a-a*sin(d*x+c))^2+5/4*a^3/d/(a-a*sin(d*x+ c))
Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (-8 \csc (c+d x)-17 \log (1-\sin (c+d x))+16 \log (\sin (c+d x))+\log (1+\sin (c+d x))+\frac {2}{(-1+\sin (c+d x))^2}-\frac {10}{-1+\sin (c+d x)}\right )}{8 d} \]
(a^2*(-8*Csc[c + d*x] - 17*Log[1 - Sin[c + d*x]] + 16*Log[Sin[c + d*x]] + Log[1 + Sin[c + d*x]] + 2/(-1 + Sin[c + d*x])^2 - 10/(-1 + Sin[c + d*x]))) /(8*d)
Time = 0.33 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 99, 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.
| \(\displaystyle \int \csc ^2(c+d x) \sec ^5(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{\sin (c+d x)^2 \cos (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int \frac {\csc ^2(c+d x)}{(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^7 \int \frac {\csc ^2(c+d x)}{a^2 (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^7 \int \left (\frac {\csc ^2(c+d x)}{a^6}+\frac {2 \csc (c+d x)}{a^6}+\frac {17}{8 a^5 (a-a \sin (c+d x))}+\frac {1}{8 a^5 (\sin (c+d x) a+a)}+\frac {5}{4 a^4 (a-a \sin (c+d x))^2}+\frac {1}{2 a^3 (a-a \sin (c+d x))^3}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^7 \left (-\frac {\csc (c+d x)}{a^5}+\frac {2 \log (a \sin (c+d x))}{a^5}-\frac {17 \log (a-a \sin (c+d x))}{8 a^5}+\frac {\log (a \sin (c+d x)+a)}{8 a^5}+\frac {5}{4 a^4 (a-a \sin (c+d x))}+\frac {1}{4 a^3 (a-a \sin (c+d x))^2}\right )}{d}\) |
(a^7*(-(Csc[c + d*x]/a^5) + (2*Log[a*Sin[c + d*x]])/a^5 - (17*Log[a - a*Si n[c + d*x]])/(8*a^5) + Log[a + a*Sin[c + d*x]]/(8*a^5) + 1/(4*a^3*(a - a*S in[c + d*x])^2) + 5/(4*a^4*(a - a*Sin[c + d*x]))))/d
3.9.67.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 0.46 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(154\) |
| default | \(\frac {a^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(154\) |
| risch | \(-\frac {i a^{2} \left (-28 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{5 i \left (d x +c \right )}+28 i {\mathrm e}^{2 i \left (d x +c \right )}-34 \,{\mathrm e}^{3 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} d}-\frac {17 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{4 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{4 d}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(158\) |
| parallelrisch | \(\frac {\left (\frac {7}{2}+17 \left (\frac {3}{4}-\frac {\cos \left (2 d x +2 c \right )}{4}-\sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-\frac {3}{4}+\frac {\cos \left (2 d x +2 c \right )}{4}+\sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 \left (\cos \left (2 d x +2 c \right )-3+4 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {19 \left (\cos \left (d x +c \right )-1\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {7 \cos \left (2 d x +2 c \right )}{2}\right ) a^{2}}{d \left (\cos \left (2 d x +2 c \right )-3+4 \sin \left (d x +c \right )\right )}\) | \(178\) |
| norman | \(\frac {\frac {8 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2}}{2 d}+\frac {4 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {9 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {7 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {7 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {9 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {8 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {17 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}+\frac {a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d}\) | \(320\) |
1/d*(a^2*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c) +tan(d*x+c)))+2*a^2*(1/4/cos(d*x+c)^4+1/2/cos(d*x+c)^2+ln(tan(d*x+c)))+a^2 *(1/4/sin(d*x+c)/cos(d*x+c)^4+5/8/sin(d*x+c)/cos(d*x+c)^2-15/8/sin(d*x+c)+ 15/8*ln(sec(d*x+c)+tan(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.07 \[ \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {18 \, a^{2} \cos \left (d x + c\right )^{2} + 28 \, a^{2} \sin \left (d x + c\right ) - 26 \, a^{2} + 16 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{8 \, {\left (2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{2} - 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
1/8*(18*a^2*cos(d*x + c)^2 + 28*a^2*sin(d*x + c) - 26*a^2 + 16*(2*a^2*cos( d*x + c)^2 - 2*a^2 - (a^2*cos(d*x + c)^2 - 2*a^2)*sin(d*x + c))*log(1/2*si n(d*x + c)) + (2*a^2*cos(d*x + c)^2 - 2*a^2 - (a^2*cos(d*x + c)^2 - 2*a^2) *sin(d*x + c))*log(sin(d*x + c) + 1) - 17*(2*a^2*cos(d*x + c)^2 - 2*a^2 - (a^2*cos(d*x + c)^2 - 2*a^2)*sin(d*x + c))*log(-sin(d*x + c) + 1))/(2*d*co s(d*x + c)^2 - (d*cos(d*x + c)^2 - 2*d)*sin(d*x + c) - 2*d)
Timed out. \[ \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) - \frac {2 \, {\left (9 \, a^{2} \sin \left (d x + c\right )^{2} - 14 \, a^{2} \sin \left (d x + c\right ) + 4 \, a^{2}\right )}}{\sin \left (d x + c\right )^{3} - 2 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )}}{8 \, d} \]
1/8*(a^2*log(sin(d*x + c) + 1) - 17*a^2*log(sin(d*x + c) - 1) + 16*a^2*log (sin(d*x + c)) - 2*(9*a^2*sin(d*x + c)^2 - 14*a^2*sin(d*x + c) + 4*a^2)/(s in(d*x + c)^3 - 2*sin(d*x + c)^2 + sin(d*x + c)))/d
Time = 0.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 34 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 32 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {16 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) + a^{2}\right )}}{\sin \left (d x + c\right )} + \frac {51 \, a^{2} \sin \left (d x + c\right )^{2} - 122 \, a^{2} \sin \left (d x + c\right ) + 75 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{16 \, d} \]
1/16*(2*a^2*log(abs(sin(d*x + c) + 1)) - 34*a^2*log(abs(sin(d*x + c) - 1)) + 32*a^2*log(abs(sin(d*x + c))) - 16*(2*a^2*sin(d*x + c) + a^2)/sin(d*x + c) + (51*a^2*sin(d*x + c)^2 - 122*a^2*sin(d*x + c) + 75*a^2)/(sin(d*x + c ) - 1)^2)/d
Time = 9.86 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95 \[ \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{8\,d}-\frac {17\,a^2\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{8\,d}+\frac {2\,a^2\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\frac {9\,a^2\,{\sin \left (c+d\,x\right )}^2}{4}-\frac {7\,a^2\,\sin \left (c+d\,x\right )}{2}+a^2}{d\,\left ({\sin \left (c+d\,x\right )}^3-2\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )\right )} \]