Integrand size = 21, antiderivative size = 65 \[ \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {(4 a+5 b) \cot (c+d x)}{5 d}-\frac {(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d} \]
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.46 \[ \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {8 a \cot (c+d x)}{15 d}-\frac {2 b \cot (c+d x)}{3 d}-\frac {4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d} \]
(-8*a*Cot[c + d*x])/(15*d) - (2*b*Cot[c + d*x])/(3*d) - (4*a*Cot[c + d*x]* Csc[c + d*x]^2)/(15*d) - (b*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d)
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3491, 3042, 4254, 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.
\(\displaystyle \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \sin (c+d x)^2}{\sin (c+d x)^6}dx\) |
\(\Big \downarrow \) 3491 |
\(\displaystyle \frac {1}{5} (4 a+5 b) \int \csc ^4(c+d x)dx-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} (4 a+5 b) \int \csc (c+d x)^4dx-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {(4 a+5 b) \int \left (\cot ^2(c+d x)+1\right )d\cot (c+d x)}{5 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(4 a+5 b) \left (\frac {1}{3} \cot ^3(c+d x)+\cot (c+d x)\right )}{5 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d}\) |
-1/5*((4*a + 5*b)*(Cot[c + d*x] + Cot[c + d*x]^3/3))/d - (a*Cot[c + d*x]*C sc[c + d*x]^4)/(5*d)
3.1.73.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x _)]^2), x_Symbol] :> Simp[A*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Simp[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)) Int[(b*Sin[e + f* x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 1.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{15}\right ) \cot \left (d x +c \right )+b \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(56\) |
default | \(\frac {a \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{15}\right ) \cot \left (d x +c \right )+b \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(56\) |
risch | \(\frac {4 i \left (15 b \,{\mathrm e}^{6 i \left (d x +c \right )}-40 a \,{\mathrm e}^{4 i \left (d x +c \right )}-35 b \,{\mathrm e}^{4 i \left (d x +c \right )}+20 a \,{\mathrm e}^{2 i \left (d x +c \right )}+25 b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 a -5 b \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}\) | \(87\) |
parallelrisch | \(\frac {-3 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +\left (-25 a -20 b \right ) \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-150 a -180 b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +\left (\frac {25 a}{3}+\frac {20 b}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 a +60 b \right )}{480 d}\) | \(107\) |
norman | \(\frac {-\frac {a}{160 d}+\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {5 \left (7 a +8 b \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {5 \left (7 a +8 b \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {\left (31 a +20 b \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {\left (31 a +20 b \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}-\frac {\left (203 a +220 b \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {\left (203 a +220 b \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(190\) |
1/d*(a*(-8/15-1/5*csc(d*x+c)^4-4/15*csc(d*x+c)^2)*cot(d*x+c)+b*(-2/3-1/3*c sc(d*x+c)^2)*cot(d*x+c))
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.25 \[ \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (a + b\right )} \cos \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
-1/15*(2*(4*a + 5*b)*cos(d*x + c)^5 - 5*(4*a + 5*b)*cos(d*x + c)^3 + 15*(a + b)*cos(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))
\[ \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=\int \left (a + b \sin ^{2}{\left (c + d x \right )}\right ) \csc ^{6}{\left (c + d x \right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {15 \, {\left (a + b\right )} \tan \left (d x + c\right )^{4} + 5 \, {\left (2 \, a + b\right )} \tan \left (d x + c\right )^{2} + 3 \, a}{15 \, d \tan \left (d x + c\right )^{5}} \]
Time = 0.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {15 \, a \tan \left (d x + c\right )^{4} + 15 \, b \tan \left (d x + c\right )^{4} + 10 \, a \tan \left (d x + c\right )^{2} + 5 \, b \tan \left (d x + c\right )^{2} + 3 \, a}{15 \, d \tan \left (d x + c\right )^{5}} \]
-1/15*(15*a*tan(d*x + c)^4 + 15*b*tan(d*x + c)^4 + 10*a*tan(d*x + c)^2 + 5 *b*tan(d*x + c)^2 + 3*a)/(d*tan(d*x + c)^5)
Time = 14.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \csc ^6(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {a\,{\mathrm {cot}\left (c+d\,x\right )}^5}{5\,d}-\frac {\mathrm {cot}\left (c+d\,x\right )\,\left (a+b\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {2\,a}{3}+\frac {b}{3}\right )}{d} \]