Integrand size = 27, antiderivative size = 65 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \]
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \]
-1/6*(a*Cot[c + d*x]^6)/d - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5 )/(5*d) - (a*Csc[c + d*x]^7)/(7*d)
Time = 0.40 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3313, 3042, 25, 3086, 244, 2009, 3087, 15}
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.
\(\displaystyle \int \cot ^5(c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)}{\sin (c+d x)^8}dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cot ^5(c+d x) \csc ^3(c+d x)dx+a \int \cot ^5(c+d x) \csc ^2(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx+a \int -\sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^5dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {a \int \csc ^2(c+d x) \left (1-\csc ^2(c+d x)\right )^2d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {a \int \left (\csc ^6(c+d x)-2 \csc ^4(c+d x)+\csc ^2(c+d x)\right )d\csc (c+d x)}{d}-a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -a \int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^5dx-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {2}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle -\frac {a \int -\cot ^5(c+d x)d(-\cot (c+d x))}{d}-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {2}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \left (\frac {1}{7} \csc ^7(c+d x)-\frac {2}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)\right )}{d}\) |
-1/6*(a*Cot[c + d*x]^6)/d - (a*(Csc[c + d*x]^3/3 - (2*Csc[c + d*x]^5)/5 + Csc[c + d*x]^7/7))/d
3.6.6.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 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[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[Cos[e + f*x]^ p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[Cos[e + f*x]^p*(d*Sin[e + f*x ])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2 ] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] | | LtQ[p + 1, -n, 2*p + 1])
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(68\) |
default | \(-\frac {a \left (\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(68\) |
parallelrisch | \(-\frac {a \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (7168 \cos \left (2 d x +2 c \right )+385 \sin \left (7 d x +7 c \right )+4025 \sin \left (5 d x +5 c \right )+8925 \sin \left (d x +c \right )+1365 \sin \left (3 d x +3 c \right )+8960 \cos \left (4 d x +4 c \right )+14592\right )}{27525120 d}\) | \(94\) |
risch | \(\frac {2 a \left (140 i {\mathrm e}^{11 i \left (d x +c \right )}+105 \,{\mathrm e}^{12 i \left (d x +c \right )}+112 i {\mathrm e}^{9 i \left (d x +c \right )}-105 \,{\mathrm e}^{10 i \left (d x +c \right )}+456 i {\mathrm e}^{7 i \left (d x +c \right )}+350 \,{\mathrm e}^{8 i \left (d x +c \right )}+112 i {\mathrm e}^{5 i \left (d x +c \right )}-350 \,{\mathrm e}^{6 i \left (d x +c \right )}+140 i {\mathrm e}^{3 i \left (d x +c \right )}+105 \,{\mathrm e}^{4 i \left (d x +c \right )}-105 \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\) | \(147\) |
norman | \(\frac {-\frac {a}{896 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d}+\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 d}+\frac {5 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}-\frac {3 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}-\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {5 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {3 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}+\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {5 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 d}-\frac {a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {a \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{896 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(271\) |
-a/d*(1/7*csc(d*x+c)^7+1/6*csc(d*x+c)^6-2/5*csc(d*x+c)^5-1/2*csc(d*x+c)^4+ 1/3*csc(d*x+c)^3+1/2*csc(d*x+c)^2)
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.63 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {70 \, a \cos \left (d x + c\right )^{4} - 56 \, a \cos \left (d x + c\right )^{2} + 35 \, {\left (3 \, a \cos \left (d x + c\right )^{4} - 3 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 16 \, a}{210 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
1/210*(70*a*cos(d*x + c)^4 - 56*a*cos(d*x + c)^2 + 35*(3*a*cos(d*x + c)^4 - 3*a*cos(d*x + c)^2 + a)*sin(d*x + c) + 16*a)/((d*cos(d*x + c)^6 - 3*d*co s(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {105 \, a \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, a \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, a \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \]
-1/210*(105*a*sin(d*x + c)^5 + 70*a*sin(d*x + c)^4 - 105*a*sin(d*x + c)^3 - 84*a*sin(d*x + c)^2 + 35*a*sin(d*x + c) + 30*a)/(d*sin(d*x + c)^7)
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {105 \, a \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, a \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, a \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \]
-1/210*(105*a*sin(d*x + c)^5 + 70*a*sin(d*x + c)^4 - 105*a*sin(d*x + c)^3 - 84*a*sin(d*x + c)^2 + 35*a*sin(d*x + c) + 30*a)/(d*sin(d*x + c)^7)
Time = 9.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {105\,a\,{\sin \left (c+d\,x\right )}^5+70\,a\,{\sin \left (c+d\,x\right )}^4-105\,a\,{\sin \left (c+d\,x\right )}^3-84\,a\,{\sin \left (c+d\,x\right )}^2+35\,a\,\sin \left (c+d\,x\right )+30\,a}{210\,d\,{\sin \left (c+d\,x\right )}^7} \]