Integrand size = 21, antiderivative size = 237 \[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {x}{a^3}-\frac {125 \text {arctanh}(\sin (c+d x))}{128 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {115 \sec (c+d x) \tan (c+d x)}{128 a^3 d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {5 \sec (c+d x) \tan ^3(c+d x)}{8 a^3 d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\sec (c+d x) \tan ^5(c+d x)}{2 a^3 d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a^3 d}-\frac {3 \tan ^7(c+d x)}{7 a^3 d} \] Output:
x/a^3-125/128*arctanh(sin(d*x+c))/a^3/d-tan(d*x+c)/a^3/d+115/128*sec(d*x+c )*tan(d*x+c)/a^3/d+5/64*sec(d*x+c)^3*tan(d*x+c)/a^3/d+1/3*tan(d*x+c)^3/a^3 /d-5/8*sec(d*x+c)*tan(d*x+c)^3/a^3/d-5/48*sec(d*x+c)^3*tan(d*x+c)^3/a^3/d- 1/5*tan(d*x+c)^5/a^3/d+1/2*sec(d*x+c)*tan(d*x+c)^5/a^3/d+1/8*sec(d*x+c)^3* tan(d*x+c)^5/a^3/d-3/7*tan(d*x+c)^7/a^3/d
Time = 3.14 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.53 \[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (1680000 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec (c) \sec ^8(c+d x) (470400 d x \cos (c)+376320 d x \cos (c+2 d x)+376320 d x \cos (3 c+2 d x)+188160 d x \cos (3 c+4 d x)+188160 d x \cos (5 c+4 d x)+53760 d x \cos (5 c+6 d x)+53760 d x \cos (7 c+6 d x)+6720 d x \cos (7 c+8 d x)+6720 d x \cos (9 c+8 d x)+519680 \sin (c)+133175 \sin (d x)+133175 \sin (2 c+d x)-544768 \sin (c+2 d x)+286720 \sin (3 c+2 d x)+63595 \sin (2 c+3 d x)+63595 \sin (4 c+3 d x)-254464 \sin (3 c+4 d x)+161280 \sin (5 c+4 d x)+65135 \sin (4 c+5 d x)+65135 \sin (6 c+5 d x)-118784 \sin (5 c+6 d x)+27195 \sin (6 c+7 d x)+27195 \sin (8 c+7 d x)-14848 \sin (7 c+8 d x))\right )}{215040 a^3 d (1+\sec (c+d x))^3} \] Input:
Integrate[Tan[c + d*x]^12/(a + a*Sec[c + d*x])^3,x]
Output:
(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(1680000*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*Sec[c + d *x]^8*(470400*d*x*Cos[c] + 376320*d*x*Cos[c + 2*d*x] + 376320*d*x*Cos[3*c + 2*d*x] + 188160*d*x*Cos[3*c + 4*d*x] + 188160*d*x*Cos[5*c + 4*d*x] + 537 60*d*x*Cos[5*c + 6*d*x] + 53760*d*x*Cos[7*c + 6*d*x] + 6720*d*x*Cos[7*c + 8*d*x] + 6720*d*x*Cos[9*c + 8*d*x] + 519680*Sin[c] + 133175*Sin[d*x] + 133 175*Sin[2*c + d*x] - 544768*Sin[c + 2*d*x] + 286720*Sin[3*c + 2*d*x] + 635 95*Sin[2*c + 3*d*x] + 63595*Sin[4*c + 3*d*x] - 254464*Sin[3*c + 4*d*x] + 1 61280*Sin[5*c + 4*d*x] + 65135*Sin[4*c + 5*d*x] + 65135*Sin[6*c + 5*d*x] - 118784*Sin[5*c + 6*d*x] + 27195*Sin[6*c + 7*d*x] + 27195*Sin[8*c + 7*d*x] - 14848*Sin[7*c + 8*d*x])))/(215040*a^3*d*(1 + Sec[c + d*x])^3)
Time = 0.68 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4376, 25, 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.
| \(\displaystyle \int \frac {\tan ^{12}(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot \left (c+d x+\frac {\pi }{2}\right )^{12}}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int -(a-a \sec (c+d x))^3 \tan ^6(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (a-a \sec (c+d x))^3 \tan ^6(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3dx}{a^6}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle -\frac {\int \left (a^3 \tan ^6(c+d x)-a^3 \sec ^3(c+d x) \tan ^6(c+d x)+3 a^3 \sec ^2(c+d x) \tan ^6(c+d x)-3 a^3 \sec (c+d x) \tan ^6(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {125 a^3 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan (c+d x)}{d}-\frac {a^3 \tan ^5(c+d x) \sec ^3(c+d x)}{8 d}+\frac {5 a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{48 d}-\frac {5 a^3 \tan (c+d x) \sec ^3(c+d x)}{64 d}-\frac {a^3 \tan ^5(c+d x) \sec (c+d x)}{2 d}+\frac {5 a^3 \tan ^3(c+d x) \sec (c+d x)}{8 d}-\frac {115 a^3 \tan (c+d x) \sec (c+d x)}{128 d}-a^3 x}{a^6}\) |
Input:
Int[Tan[c + d*x]^12/(a + a*Sec[c + d*x])^3,x]
Output:
-((-(a^3*x) + (125*a^3*ArcTanh[Sin[c + d*x]])/(128*d) + (a^3*Tan[c + d*x]) /d - (115*a^3*Sec[c + d*x]*Tan[c + d*x])/(128*d) - (5*a^3*Sec[c + d*x]^3*T an[c + d*x])/(64*d) - (a^3*Tan[c + d*x]^3)/(3*d) + (5*a^3*Sec[c + d*x]*Tan [c + d*x]^3)/(8*d) + (5*a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*d) + (a^3*T an[c + d*x]^5)/(5*d) - (a^3*Sec[c + d*x]*Tan[c + d*x]^5)/(2*d) - (a^3*Sec[ c + d*x]^3*Tan[c + d*x]^5)/(8*d) + (3*a^3*Tan[c + d*x]^7)/(7*d))/a^6)
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 = 4.96 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(\frac {x}{a^{3}}-\frac {i \left (27195 \,{\mathrm e}^{15 i \left (d x +c \right )}+65135 \,{\mathrm e}^{13 i \left (d x +c \right )}+161280 \,{\mathrm e}^{12 i \left (d x +c \right )}+63595 \,{\mathrm e}^{11 i \left (d x +c \right )}+286720 \,{\mathrm e}^{10 i \left (d x +c \right )}+133175 \,{\mathrm e}^{9 i \left (d x +c \right )}+519680 \,{\mathrm e}^{8 i \left (d x +c \right )}-133175 \,{\mathrm e}^{7 i \left (d x +c \right )}+544768 \,{\mathrm e}^{6 i \left (d x +c \right )}-63595 \,{\mathrm e}^{5 i \left (d x +c \right )}+254464 \,{\mathrm e}^{4 i \left (d x +c \right )}-65135 \,{\mathrm e}^{3 i \left (d x +c \right )}+118784 \,{\mathrm e}^{2 i \left (d x +c \right )}-27195 \,{\mathrm e}^{i \left (d x +c \right )}+14848\right )}{6720 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}+\frac {125 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d \,a^{3}}-\frac {125 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 d \,a^{3}}\) | \(227\) |
| derivativedivides | \(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {13}{14 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {65}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {143}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {79}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {49}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {29}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {253}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {125 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128}+\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{8}}+\frac {13}{14 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}+\frac {65}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {143}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {79}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {49}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {29}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {253}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {125 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128}}{d \,a^{3}}\) | \(290\) |
| default | \(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {13}{14 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {65}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {143}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {79}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {49}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {29}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {253}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {125 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128}+\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{8}}+\frac {13}{14 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}+\frac {65}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {143}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {79}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {49}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {29}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {253}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {125 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128}}{d \,a^{3}}\) | \(290\) |
Input:
int(tan(d*x+c)^12/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
x/a^3-1/6720*I*(27195*exp(15*I*(d*x+c))+65135*exp(13*I*(d*x+c))+161280*exp (12*I*(d*x+c))+63595*exp(11*I*(d*x+c))+286720*exp(10*I*(d*x+c))+133175*exp (9*I*(d*x+c))+519680*exp(8*I*(d*x+c))-133175*exp(7*I*(d*x+c))+544768*exp(6 *I*(d*x+c))-63595*exp(5*I*(d*x+c))+254464*exp(4*I*(d*x+c))-65135*exp(3*I*( d*x+c))+118784*exp(2*I*(d*x+c))-27195*exp(I*(d*x+c))+14848)/d/a^3/(exp(2*I *(d*x+c))+1)^8+125/128/d/a^3*ln(exp(I*(d*x+c))-I)-125/128/d/a^3*ln(exp(I*( d*x+c))+I)
Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.62 \[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {26880 \, d x \cos \left (d x + c\right )^{8} - 13125 \, \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) + 13125 \, \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (14848 \, \cos \left (d x + c\right )^{7} - 27195 \, \cos \left (d x + c\right )^{6} + 7424 \, \cos \left (d x + c\right )^{5} + 17710 \, \cos \left (d x + c\right )^{4} - 14592 \, \cos \left (d x + c\right )^{3} - 1960 \, \cos \left (d x + c\right )^{2} + 5760 \, \cos \left (d x + c\right ) - 1680\right )} \sin \left (d x + c\right )}{26880 \, a^{3} d \cos \left (d x + c\right )^{8}} \] Input:
integrate(tan(d*x+c)^12/(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/26880*(26880*d*x*cos(d*x + c)^8 - 13125*cos(d*x + c)^8*log(sin(d*x + c) + 1) + 13125*cos(d*x + c)^8*log(-sin(d*x + c) + 1) - 2*(14848*cos(d*x + c) ^7 - 27195*cos(d*x + c)^6 + 7424*cos(d*x + c)^5 + 17710*cos(d*x + c)^4 - 1 4592*cos(d*x + c)^3 - 1960*cos(d*x + c)^2 + 5760*cos(d*x + c) - 1680)*sin( d*x + c))/(a^3*d*cos(d*x + c)^8)
\[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\tan ^{12}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(tan(d*x+c)**12/(a+a*sec(d*x+c))**3,x)
Output:
Integral(tan(c + d*x)**12/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3
Time = 0.13 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.81 \[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {11375 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {79723 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {269879 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {550089 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {749973 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {212625 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {26565 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}\right )}}{a^{3} - \frac {8 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {56 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {56 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {8 \, a^{3} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{3} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {26880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {13125 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} - \frac {13125 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{13440 \, d} \] Input:
integrate(tan(d*x+c)^12/(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
-1/13440*(2*(315*sin(d*x + c)/(cos(d*x + c) + 1) - 11375*sin(d*x + c)^3/(c os(d*x + c) + 1)^3 + 79723*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 269879*si n(d*x + c)^7/(cos(d*x + c) + 1)^7 + 550089*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 749973*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 212625*sin(d*x + c)^ 13/(cos(d*x + c) + 1)^13 - 26565*sin(d*x + c)^15/(cos(d*x + c) + 1)^15)/(a ^3 - 8*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 28*a^3*sin(d*x + c)^4/(co s(d*x + c) + 1)^4 - 56*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 70*a^3*si n(d*x + c)^8/(cos(d*x + c) + 1)^8 - 56*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 28*a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 8*a^3*sin(d*x + c) ^14/(cos(d*x + c) + 1)^14 + a^3*sin(d*x + c)^16/(cos(d*x + c) + 1)^16) - 2 6880*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 + 13125*log(sin(d*x + c)/ (cos(d*x + c) + 1) + 1)/a^3 - 13125*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^3)/d
Time = 0.59 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {13440 \, {\left (d x + c\right )}}{a^{3}} - \frac {13125 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac {13125 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (26565 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 212625 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 749973 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 550089 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 269879 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 79723 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 11375 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{8} a^{3}}}{13440 \, d} \] Input:
integrate(tan(d*x+c)^12/(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
1/13440*(13440*(d*x + c)/a^3 - 13125*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^ 3 + 13125*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^3 + 2*(26565*tan(1/2*d*x + 1/2*c)^15 - 212625*tan(1/2*d*x + 1/2*c)^13 + 749973*tan(1/2*d*x + 1/2*c)^1 1 - 550089*tan(1/2*d*x + 1/2*c)^9 + 269879*tan(1/2*d*x + 1/2*c)^7 - 79723* tan(1/2*d*x + 1/2*c)^5 + 11375*tan(1/2*d*x + 1/2*c)^3 - 315*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^8*a^3))/d
Time = 13.84 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.12 \[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {x}{a^3}-\frac {125\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a^3\,d}-\frac {-\frac {253\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {2025\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}-\frac {35713\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{320}+\frac {183363\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2240}-\frac {269879\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6720}+\frac {11389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960}-\frac {325\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )} \] Input:
int(tan(c + d*x)^12/(a + a/cos(c + d*x))^3,x)
Output:
x/a^3 - (125*atanh(tan(c/2 + (d*x)/2)))/(64*a^3*d) - ((3*tan(c/2 + (d*x)/2 ))/64 - (325*tan(c/2 + (d*x)/2)^3)/192 + (11389*tan(c/2 + (d*x)/2)^5)/960 - (269879*tan(c/2 + (d*x)/2)^7)/6720 + (183363*tan(c/2 + (d*x)/2)^9)/2240 - (35713*tan(c/2 + (d*x)/2)^11)/320 + (2025*tan(c/2 + (d*x)/2)^13)/64 - (2 53*tan(c/2 + (d*x)/2)^15)/64)/(d*(28*a^3*tan(c/2 + (d*x)/2)^4 - 8*a^3*tan( c/2 + (d*x)/2)^2 - 56*a^3*tan(c/2 + (d*x)/2)^6 + 70*a^3*tan(c/2 + (d*x)/2) ^8 - 56*a^3*tan(c/2 + (d*x)/2)^10 + 28*a^3*tan(c/2 + (d*x)/2)^12 - 8*a^3*t an(c/2 + (d*x)/2)^14 + a^3*tan(c/2 + (d*x)/2)^16 + a^3))
Time = 0.16 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.72 \[ \int \frac {\tan ^{12}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {14848 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-51968 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+44800 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-13440 \cos \left (d x +c \right ) \sin \left (d x +c \right )+13125 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{8}-52500 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{6}+78750 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}-52500 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}+13125 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-13125 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{8}+52500 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6}-78750 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}+52500 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}-13125 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+13440 \sin \left (d x +c \right )^{8} d x -27195 \sin \left (d x +c \right )^{7}-53760 \sin \left (d x +c \right )^{6} d x +63875 \sin \left (d x +c \right )^{5}+80640 \sin \left (d x +c \right )^{4} d x -48125 \sin \left (d x +c \right )^{3}-53760 \sin \left (d x +c \right )^{2} d x +13125 \sin \left (d x +c \right )+13440 d x}{13440 a^{3} d \left (\sin \left (d x +c \right )^{8}-4 \sin \left (d x +c \right )^{6}+6 \sin \left (d x +c \right )^{4}-4 \sin \left (d x +c \right )^{2}+1\right )} \] Input:
int(tan(d*x+c)^12/(a+a*sec(d*x+c))^3,x)
Output:
(14848*cos(c + d*x)*sin(c + d*x)**7 - 51968*cos(c + d*x)*sin(c + d*x)**5 + 44800*cos(c + d*x)*sin(c + d*x)**3 - 13440*cos(c + d*x)*sin(c + d*x) + 13 125*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8 - 52500*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6 + 78750*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 - 52500*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2 + 13125*log(tan((c + d*x)/ 2) - 1) - 13125*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8 + 52500*log(tan( (c + d*x)/2) + 1)*sin(c + d*x)**6 - 78750*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4 + 52500*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 - 13125*log(ta n((c + d*x)/2) + 1) + 13440*sin(c + d*x)**8*d*x - 27195*sin(c + d*x)**7 - 53760*sin(c + d*x)**6*d*x + 63875*sin(c + d*x)**5 + 80640*sin(c + d*x)**4* d*x - 48125*sin(c + d*x)**3 - 53760*sin(c + d*x)**2*d*x + 13125*sin(c + d* x) + 13440*d*x)/(13440*a**3*d*(sin(c + d*x)**8 - 4*sin(c + d*x)**6 + 6*sin (c + d*x)**4 - 4*sin(c + d*x)**2 + 1))