Integrand size = 19, antiderivative size = 102 \[ \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx=-a x-\frac {5 a \text {arctanh}(\sin (c+d x))}{16 d}+\frac {(16 a+5 a \sec (c+d x)) \tan (c+d x)}{16 d}-\frac {(8 a+5 a \sec (c+d x)) \tan ^3(c+d x)}{24 d}+\frac {(6 a+5 a \sec (c+d x)) \tan ^5(c+d x)}{30 d} \]
-a*x-5/16*a*arctanh(sin(d*x+c))/d+1/16*(16*a+5*a*sec(d*x+c))*tan(d*x+c)/d- 1/24*(8*a+5*a*sec(d*x+c))*tan(d*x+c)^3/d+1/30*(6*a+5*a*sec(d*x+c))*tan(d*x +c)^5/d
Time = 0.05 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.75 \[ \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx=-\frac {a \arctan (\tan (c+d x))}{d}-\frac {5 a \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a \tan (c+d x)}{d}-\frac {5 a \sec (c+d x) \tan (c+d x)}{16 d}-\frac {5 a \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {5 a \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a \tan ^3(c+d x)}{3 d}-\frac {5 a \sec ^3(c+d x) \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d}+\frac {a \sec (c+d x) \tan ^5(c+d x)}{d} \]
-((a*ArcTan[Tan[c + d*x]])/d) - (5*a*ArcTanh[Sin[c + d*x]])/(16*d) + (a*Ta n[c + d*x])/d - (5*a*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (5*a*Sec[c + d*x] ^3*Tan[c + d*x])/(24*d) + (5*a*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (a*Tan [c + d*x]^3)/(3*d) - (5*a*Sec[c + d*x]^3*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]*Tan[c + d*x]^5)/d
Time = 0.48 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 4369, 3042, 4369, 27, 3042, 4369, 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.
| \(\displaystyle \int \tan ^6(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )dx\) |
| \(\Big \downarrow \) 4369 |
| \(\displaystyle \frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac {1}{6} \int (5 \sec (c+d x) a+6 a) \tan ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}-\frac {1}{6} \int \cot \left (c+d x+\frac {\pi }{2}\right )^4 \left (5 \csc \left (c+d x+\frac {\pi }{2}\right ) a+6 a\right )dx\) |
| \(\Big \downarrow \) 4369 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int 3 (5 \sec (c+d x) a+8 a) \tan ^2(c+d x)dx-\frac {\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{4 d}\right )+\frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{4} \int (5 \sec (c+d x) a+8 a) \tan ^2(c+d x)dx-\frac {\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{4 d}\right )+\frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{4} \int \cot \left (c+d x+\frac {\pi }{2}\right )^2 \left (5 \csc \left (c+d x+\frac {\pi }{2}\right ) a+8 a\right )dx-\frac {\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{4 d}\right )+\frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}\) |
| \(\Big \downarrow \) 4369 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{4} \left (\frac {\tan (c+d x) (5 a \sec (c+d x)+16 a)}{2 d}-\frac {1}{2} \int (5 \sec (c+d x) a+16 a)dx\right )-\frac {\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{4 d}\right )+\frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \left (-\frac {5 a \text {arctanh}(\sin (c+d x))}{d}-16 a x\right )+\frac {\tan (c+d x) (5 a \sec (c+d x)+16 a)}{2 d}\right )-\frac {\tan ^3(c+d x) (5 a \sec (c+d x)+8 a)}{4 d}\right )+\frac {\tan ^5(c+d x) (5 a \sec (c+d x)+6 a)}{30 d}\) |
((6*a + 5*a*Sec[c + d*x])*Tan[c + d*x]^5)/(30*d) + (-1/4*((8*a + 5*a*Sec[c + d*x])*Tan[c + d*x]^3)/d + (3*((-16*a*x - (5*a*ArcTanh[Sin[c + d*x]])/d) /2 + ((16*a + 5*a*Sec[c + d*x])*Tan[c + d*x])/(2*d)))/4)/6
3.1.11.3.1 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[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-e)*(e*Cot[c + d*x])^(m - 1)*((a*m + b*(m - 1)*Csc [c + d*x])/(d*m*(m - 1))), x] - Simp[e^2/m Int[(e*Cot[c + d*x])^(m - 2)*( a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m , 1]
Time = 1.46 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(\frac {a \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(143\) |
| default | \(\frac {a \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(143\) |
| parts | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(147\) |
| risch | \(-a x -\frac {i a \left (165 \,{\mathrm e}^{11 i \left (d x +c \right )}-720 \,{\mathrm e}^{10 i \left (d x +c \right )}-25 \,{\mathrm e}^{9 i \left (d x +c \right )}-2160 \,{\mathrm e}^{8 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}-3680 \,{\mathrm e}^{6 i \left (d x +c \right )}-450 \,{\mathrm e}^{5 i \left (d x +c \right )}-3360 \,{\mathrm e}^{4 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}-1488 \,{\mathrm e}^{2 i \left (d x +c \right )}-165 \,{\mathrm e}^{i \left (d x +c \right )}-368\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}\) | \(187\) |
1/d*(a*(1/6*sin(d*x+c)^7/cos(d*x+c)^6-1/24*sin(d*x+c)^7/cos(d*x+c)^4+1/16* sin(d*x+c)^7/cos(d*x+c)^2+1/16*sin(d*x+c)^5+5/48*sin(d*x+c)^3+5/16*sin(d*x +c)-5/16*ln(sec(d*x+c)+tan(d*x+c)))+a*(1/5*tan(d*x+c)^5-1/3*tan(d*x+c)^3+t an(d*x+c)-d*x-c))
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.31 \[ \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx=-\frac {480 \, a d x \cos \left (d x + c\right )^{6} + 75 \, a \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 75 \, a \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (368 \, a \cos \left (d x + c\right )^{5} + 165 \, a \cos \left (d x + c\right )^{4} - 176 \, a \cos \left (d x + c\right )^{3} - 130 \, a \cos \left (d x + c\right )^{2} + 48 \, a \cos \left (d x + c\right ) + 40 \, a\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
-1/480*(480*a*d*x*cos(d*x + c)^6 + 75*a*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 75*a*cos(d*x + c)^6*log(-sin(d*x + c) + 1) - 2*(368*a*cos(d*x + c)^5 + 165*a*cos(d*x + c)^4 - 176*a*cos(d*x + c)^3 - 130*a*cos(d*x + c)^2 + 48* a*cos(d*x + c) + 40*a)*sin(d*x + c))/(d*cos(d*x + c)^6)
\[ \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx=a \left (\int \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )}\, dx\right ) \]
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.31 \[ \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx=\frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a - 5 \, a {\left (\frac {2 \, {\left (33 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
1/480*(32*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d* x + c))*a - 5*a*(2*(33*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 15*sin(d*x + c ))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) + 15*log(sin (d*x + c) + 1) - 15*log(sin(d*x + c) - 1)))/d
Time = 2.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx=-\frac {240 \, {\left (d x + c\right )} a + 75 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 75 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (165 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1095 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3138 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5118 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1945 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
-1/240*(240*(d*x + c)*a + 75*a*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 75*a*l og(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(165*a*tan(1/2*d*x + 1/2*c)^11 - 109 5*a*tan(1/2*d*x + 1/2*c)^9 + 3138*a*tan(1/2*d*x + 1/2*c)^7 - 5118*a*tan(1/ 2*d*x + 1/2*c)^5 + 1945*a*tan(1/2*d*x + 1/2*c)^3 - 315*a*tan(1/2*d*x + 1/2 *c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d
Time = 14.92 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.84 \[ \int (a+a \sec (c+d x)) \tan ^6(c+d x) \, dx=\frac {-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {73\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}-\frac {523\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {853\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}-\frac {389\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {21\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-a\,x-\frac {5\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \]
((21*a*tan(c/2 + (d*x)/2))/8 - (389*a*tan(c/2 + (d*x)/2)^3)/24 + (853*a*ta n(c/2 + (d*x)/2)^5)/20 - (523*a*tan(c/2 + (d*x)/2)^7)/20 + (73*a*tan(c/2 + (d*x)/2)^9)/8 - (11*a*tan(c/2 + (d*x)/2)^11)/8)/(d*(15*tan(c/2 + (d*x)/2) ^4 - 6*tan(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x) /2)^8 - 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) - a*x - (5*a *atanh(tan(c/2 + (d*x)/2)))/(8*d)