Integrand size = 21, antiderivative size = 72 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {x}{a^2}+\frac {\text {arctanh}(\sin (c+d x))}{a^2 d}+\frac {\tan (c+d x)}{a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d} \]
-x/a^2+arctanh(sin(d*x+c))/a^2/d+tan(d*x+c)/a^2/d-sec(d*x+c)*tan(d*x+c)/a^ 2/d+1/3*tan(d*x+c)^3/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(767\) vs. \(2(72)=144\).
Time = 7.08 (sec) , antiderivative size = 767, normalized size of antiderivative = 10.65 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 x \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{(a+a \sec (c+d x))^2}-\frac {4 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x)}{d (a+a \sec (c+d x))^2}+\frac {4 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x)}{d (a+a \sec (c+d x))^2}+\frac {2 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{3 d (a+a \sec (c+d x))^2 \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \left (-5 \cos \left (\frac {c}{2}\right )+7 \sin \left (\frac {c}{2}\right )\right )}{3 d (a+a \sec (c+d x))^2 \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {8 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{3 d (a+a \sec (c+d x))^2 \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {2 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{3 d (a+a \sec (c+d x))^2 \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \left (5 \cos \left (\frac {c}{2}\right )+7 \sin \left (\frac {c}{2}\right )\right )}{3 d (a+a \sec (c+d x))^2 \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {8 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \sin \left (\frac {d x}{2}\right )}{3 d (a+a \sec (c+d x))^2 \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]
(-4*x*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2 - (4*Cos [c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x ]^2)/(d*(a + a*Sec[c + d*x])^2) + (4*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d *x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2)/(d*(a + a*Sec[c + d*x])^2) + (2*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d *x])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*(-5*Cos[c/2] + 7*Sin[c/2]))/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[(d*x)/2])/(3*d* (a + a*Sec[c + d*x])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (2*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[(d*x)/2])/(3*d* (a + a*Sec[c + d*x])^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*(5*Cos[c/2] + 7*Sin [c/2]))/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x) /2] + Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*Sin[ (d*x)/2])/(3*d*(a + a*Sec[c + d*x])^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d* x)/2] + Sin[c/2 + (d*x)/2]))
Time = 0.40 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 4376, 3042, 4374, 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.
| \(\displaystyle \int \frac {\tan ^6(c+d x)}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot \left (c+d x+\frac {\pi }{2}\right )^6}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int (a-a \sec (c+d x))^2 \tan ^2(c+d x)dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cot \left (c+d x+\frac {\pi }{2}\right )^2 \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2dx}{a^4}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \frac {\int \left (a^2 \tan ^2(c+d x)+a^2 \sec ^2(c+d x) \tan ^2(c+d x)-2 a^2 \sec (c+d x) \tan ^2(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 \tan ^3(c+d x)}{3 d}+\frac {a^2 \tan (c+d x)}{d}-\frac {a^2 \tan (c+d x) \sec (c+d x)}{d}-a^2 x}{a^4}\) |
(-(a^2*x) + (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d - (a^2*Se c[c + d*x]*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d))/a^4
3.1.80.3.1 Defintions of rubi rules used
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(-\frac {x}{a^{2}}+\frac {2 i \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}+6 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{2} d}\) | \(106\) |
| derivativedivides | \(\frac {-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2} d}\) | \(140\) |
| default | \(\frac {-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2} d}\) | \(140\) |
-x/a^2+2/3*I*(3*exp(5*I*(d*x+c))+6*exp(2*I*(d*x+c))-3*exp(I*(d*x+c))+2)/d/ a^2/(exp(2*I*(d*x+c))+1)^3-1/a^2/d*ln(exp(I*(d*x+c))-I)+1/a^2/d*ln(exp(I*( d*x+c))+I)
Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.35 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {6 \, d x \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{6 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
-1/6*(6*d*x*cos(d*x + c)^3 - 3*cos(d*x + c)^3*log(sin(d*x + c) + 1) + 3*co s(d*x + c)^3*log(-sin(d*x + c) + 1) - 2*(2*cos(d*x + c)^2 - 3*cos(d*x + c) + 1)*sin(d*x + c))/(a^2*d*cos(d*x + c)^3)
\[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\tan ^{6}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (70) = 140\).
Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.72 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (\frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {6 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}}{3 \, d} \]
-1/3*(4*(sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^2 - 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*sin(d *x + c)^4/(cos(d*x + c) + 1)^4 - a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + 6*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - 3*log(sin(d*x + c)/(cos( d*x + c) + 1) + 1)/a^2 + 3*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^2)/d
Time = 2.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} - \frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} + \frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {4 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{2}}}{3 \, d} \]
-1/3*(3*(d*x + c)/a^2 - 3*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^2 + 3*log(a bs(tan(1/2*d*x + 1/2*c) - 1))/a^2 + 4*(3*tan(1/2*d*x + 1/2*c)^5 - tan(1/2* d*x + 1/2*c)^3)/((tan(1/2*d*x + 1/2*c)^2 - 1)^3*a^2))/d
Time = 14.48 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.54 \[ \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {x}{a^2}+\frac {\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2\right )} \]