Integrand size = 21, antiderivative size = 129 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^2 \cot (c+d x)}{d}-\frac {5 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d} \]
2*a^2*arctanh(sin(d*x+c))/d-4*a^2*cot(d*x+c)/d-5/3*a^2*cot(d*x+c)^3/d-2/5* a^2*cot(d*x+c)^5/d-2*a^2*csc(d*x+c)/d-2/3*a^2*csc(d*x+c)^3/d-2/5*a^2*csc(d *x+c)^5/d+a^2*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(317\) vs. \(2(129)=258\).
Time = 1.42 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.46 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^2 \cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x))^2 \left (-3840 \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+3840 \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc (2 c) \csc ^4\left (\frac {1}{2} (c+d x)\right ) \csc (c+d x) (320 \sin (2 c)-596 \sin (d x)+864 \sin (2 d x)+216 \sin (c-d x)-416 \sin (c+d x)+624 \sin (2 (c+d x))-416 \sin (3 (c+d x))+104 \sin (4 (c+d x))-596 \sin (2 c+d x)-680 \sin (3 c+d x)+894 \sin (c+2 d x)+224 \sin (2 (c+2 d x))+894 \sin (3 c+2 d x)+480 \sin (4 c+2 d x)-776 \sin (c+3 d x)-596 \sin (2 c+3 d x)-596 \sin (4 c+3 d x)-120 \sin (5 c+3 d x)+149 \sin (3 c+4 d x)+149 \sin (5 c+4 d x))\right )}{7680 d} \]
(a^2*Cos[c + d*x]*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(-3840*Cos[c + d *x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3840*Cos[c + d*x]*Log[Cos[( c + d*x)/2] + Sin[(c + d*x)/2]] + Csc[2*c]*Csc[(c + d*x)/2]^4*Csc[c + d*x] *(320*Sin[2*c] - 596*Sin[d*x] + 864*Sin[2*d*x] + 216*Sin[c - d*x] - 416*Si n[c + d*x] + 624*Sin[2*(c + d*x)] - 416*Sin[3*(c + d*x)] + 104*Sin[4*(c + d*x)] - 596*Sin[2*c + d*x] - 680*Sin[3*c + d*x] + 894*Sin[c + 2*d*x] + 224 *Sin[2*(c + 2*d*x)] + 894*Sin[3*c + 2*d*x] + 480*Sin[4*c + 2*d*x] - 776*Si n[c + 3*d*x] - 596*Sin[2*c + 3*d*x] - 596*Sin[4*c + 3*d*x] - 120*Sin[5*c + 3*d*x] + 149*Sin[3*c + 4*d*x] + 149*Sin[5*c + 4*d*x])))/(7680*d)
Time = 0.50 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 4360, 3042, 3352, 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.
| \(\displaystyle \int \csc ^6(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^6(c+d x) \sec ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\sin \left (c+d x-\frac {\pi }{2}\right )^2 \cos \left (c+d x-\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \csc ^6(c+d x)+a^2 \csc ^6(c+d x) \sec ^2(c+d x)+2 a^2 \csc ^6(c+d x) \sec (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 \tan (c+d x)}{d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {5 a^2 \cot ^3(c+d x)}{3 d}-\frac {4 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc (c+d x)}{d}\) |
(2*a^2*ArcTanh[Sin[c + d*x]])/d - (4*a^2*Cot[c + d*x])/d - (5*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x])/d - (2 *a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) + (a^2*Tan[c + d *x])/d
3.1.35.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 1.33 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19
| method | result | size |
| norman | \(\frac {\frac {a^{2}}{40 d}+\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 d}+\frac {31 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{12 d}-\frac {5 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}+\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(154\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+2 a^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )}{d}\) | \(155\) |
| default | \(\frac {a^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+2 a^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )}{d}\) | \(155\) |
| parallelrisch | \(\frac {a^{2} \left (3 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+240 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-240 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+32 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-240 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+240 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-600 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+310 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-120 d}\) | \(165\) |
| risch | \(-\frac {4 i a^{2} \left (15 \,{\mathrm e}^{7 i \left (d x +c \right )}-60 \,{\mathrm e}^{6 i \left (d x +c \right )}+85 \,{\mathrm e}^{5 i \left (d x +c \right )}-40 \,{\mathrm e}^{4 i \left (d x +c \right )}-27 \,{\mathrm e}^{3 i \left (d x +c \right )}+108 \,{\mathrm e}^{2 i \left (d x +c \right )}-97 \,{\mathrm e}^{i \left (d x +c \right )}+28\right )}{15 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(171\) |
(1/40/d*a^2+4/15*a^2/d*tan(1/2*d*x+1/2*c)^2+31/12*a^2/d*tan(1/2*d*x+1/2*c) ^4-5*a^2/d*tan(1/2*d*x+1/2*c)^6+1/8/d*a^2*tan(1/2*d*x+1/2*c)^8)/tan(1/2*d* x+1/2*c)^5/(-1+tan(1/2*d*x+1/2*c)^2)-2*a^2/d*ln(tan(1/2*d*x+1/2*c)-1)+2*a^ 2/d*ln(tan(1/2*d*x+1/2*c)+1)
Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.60 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {56 \, a^{2} \cos \left (d x + c\right )^{4} - 82 \, a^{2} \cos \left (d x + c\right )^{3} - 32 \, a^{2} \cos \left (d x + c\right )^{2} + 76 \, a^{2} \cos \left (d x + c\right ) - 15 \, {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 15 \, {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 15 \, a^{2}}{15 \, {\left (d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
-1/15*(56*a^2*cos(d*x + c)^4 - 82*a^2*cos(d*x + c)^3 - 32*a^2*cos(d*x + c) ^2 + 76*a^2*cos(d*x + c) - 15*(a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(sin(d*x + c) + 1)*sin(d*x + c) + 15*(a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(-sin(d*x + c) + 1)*sin (d*x + c) - 15*a^2)/((d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c))*sin(d*x + c))
Timed out. \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3 \, a^{2} {\left (\frac {15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{5}} - 5 \, \tan \left (d x + c\right )\right )} + \frac {{\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{2}}{\tan \left (d x + c\right )^{5}}}{15 \, d} \]
-1/15*(a^2*(2*(15*sin(d*x + c)^4 + 5*sin(d*x + c)^2 + 3)/sin(d*x + c)^5 - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) + 3*a^2*((15*tan(d*x + c)^4 + 5*tan(d*x + c)^2 + 1)/tan(d*x + c)^5 - 5*tan(d*x + c)) + (15*tan( d*x + c)^4 + 10*tan(d*x + c)^2 + 3)*a^2/tan(d*x + c)^5)/d
Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {240 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 240 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {345 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{120 \, d} \]
1/120*(240*a^2*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 240*a^2*log(abs(tan(1/ 2*d*x + 1/2*c) - 1)) + 15*a^2*tan(1/2*d*x + 1/2*c) - 240*a^2*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - (345*a^2*tan(1/2*d*x + 1/2*c)^4 + 3 5*a^2*tan(1/2*d*x + 1/2*c)^2 + 3*a^2)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 13.73 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.96 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {-39\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {62\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {32\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {a^2}{5}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]