Integrand size = 21, antiderivative size = 138 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^4}{4 b^5 d}-\frac {4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^5}{5 b^5 d}+\frac {\left (3 a^2+b^2\right ) (a+b \tan (c+d x))^6}{3 b^5 d}-\frac {4 a (a+b \tan (c+d x))^7}{7 b^5 d}+\frac {(a+b \tan (c+d x))^8}{8 b^5 d} \]
1/4*(a^2+b^2)^2*(a+b*tan(d*x+c))^4/b^5/d-4/5*a*(a^2+b^2)*(a+b*tan(d*x+c))^ 5/b^5/d+1/3*(3*a^2+b^2)*(a+b*tan(d*x+c))^6/b^5/d-4/7*a*(a+b*tan(d*x+c))^7/ b^5/d+1/8*(a+b*tan(d*x+c))^8/b^5/d
Time = 0.67 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.83 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\frac {1}{4} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^4-\frac {4}{5} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^5+\frac {1}{3} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^6-\frac {4}{7} a (a+b \tan (c+d x))^7+\frac {1}{8} (a+b \tan (c+d x))^8}{b^5 d} \]
(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^4)/4 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^5)/5 + ((3*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/3 - (4*a*(a + b*Tan[ c + d*x])^7)/7 + (a + b*Tan[c + d*x])^8/8)/(b^5*d)
Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3987, 27, 476, 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.
| \(\displaystyle \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^6 (a+b \tan (c+d x))^3dx\) |
| \(\Big \downarrow \) 3987 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )^2}{b^4}d(b \tan (c+d x))}{b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )^2d(b \tan (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left ((a+b \tan (c+d x))^7-4 a (a+b \tan (c+d x))^6+2 \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^5-4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^4+\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^3\right )d(b \tan (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^6-\frac {4}{5} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^5+\frac {1}{4} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^4+\frac {1}{8} (a+b \tan (c+d x))^8-\frac {4}{7} a (a+b \tan (c+d x))^7}{b^5 d}\) |
(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^4)/4 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^5)/5 + ((3*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/3 - (4*a*(a + b*Tan[ c + d*x])^7)/7 + (a + b*Tan[c + d*x])^8/8)/(b^5*d)
3.6.32.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(b*f) Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]
Time = 69.61 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{2 \cos \left (d x +c \right )^{6}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(173\) |
| default | \(\frac {-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{2 \cos \left (d x +c \right )^{6}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(173\) |
| risch | \(-\frac {16 \left (-56 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+24 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-210 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+70 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-322 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-7 i a^{3}-420 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-70 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+3 i a \,b^{2}+105 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-210 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+70 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-196 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-42 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+210 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+84 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-245 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-70 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\) | \(275\) |
1/d*(-a^3*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+1/2*a^2*b/ cos(d*x+c)^6+3*a*b^2*(1/7*sin(d*x+c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos( d*x+c)^5+8/105*sin(d*x+c)^3/cos(d*x+c)^3)+b^3*(1/8*sin(d*x+c)^4/cos(d*x+c) ^8+1/12*sin(d*x+c)^4/cos(d*x+c)^6+1/24*sin(d*x+c)^4/cos(d*x+c)^4))
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {105 \, b^{3} + 140 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (8 \, {\left (7 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (7 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 45 \, a b^{2} \cos \left (d x + c\right ) + 3 \, {\left (7 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{8}} \]
1/840*(105*b^3 + 140*(3*a^2*b - b^3)*cos(d*x + c)^2 + 8*(8*(7*a^3 - 3*a*b^ 2)*cos(d*x + c)^7 + 4*(7*a^3 - 3*a*b^2)*cos(d*x + c)^5 + 45*a*b^2*cos(d*x + c) + 3*(7*a^3 - 3*a*b^2)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^8 )
\[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \sec ^{6}{\left (c + d x \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {105 \, b^{3} \tan \left (d x + c\right )^{8} + 360 \, a b^{2} \tan \left (d x + c\right )^{7} + 140 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \tan \left (d x + c\right )^{6} + 168 \, {\left (a^{3} + 6 \, a b^{2}\right )} \tan \left (d x + c\right )^{5} + 1260 \, a^{2} b \tan \left (d x + c\right )^{2} + 210 \, {\left (6 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{4} + 840 \, a^{3} \tan \left (d x + c\right ) + 280 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{3}}{840 \, d} \]
1/840*(105*b^3*tan(d*x + c)^8 + 360*a*b^2*tan(d*x + c)^7 + 140*(3*a^2*b + 2*b^3)*tan(d*x + c)^6 + 168*(a^3 + 6*a*b^2)*tan(d*x + c)^5 + 1260*a^2*b*ta n(d*x + c)^2 + 210*(6*a^2*b + b^3)*tan(d*x + c)^4 + 840*a^3*tan(d*x + c) + 280*(2*a^3 + 3*a*b^2)*tan(d*x + c)^3)/d
Time = 0.79 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.20 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {105 \, b^{3} \tan \left (d x + c\right )^{8} + 360 \, a b^{2} \tan \left (d x + c\right )^{7} + 420 \, a^{2} b \tan \left (d x + c\right )^{6} + 280 \, b^{3} \tan \left (d x + c\right )^{6} + 168 \, a^{3} \tan \left (d x + c\right )^{5} + 1008 \, a b^{2} \tan \left (d x + c\right )^{5} + 1260 \, a^{2} b \tan \left (d x + c\right )^{4} + 210 \, b^{3} \tan \left (d x + c\right )^{4} + 560 \, a^{3} \tan \left (d x + c\right )^{3} + 840 \, a b^{2} \tan \left (d x + c\right )^{3} + 1260 \, a^{2} b \tan \left (d x + c\right )^{2} + 840 \, a^{3} \tan \left (d x + c\right )}{840 \, d} \]
1/840*(105*b^3*tan(d*x + c)^8 + 360*a*b^2*tan(d*x + c)^7 + 420*a^2*b*tan(d *x + c)^6 + 280*b^3*tan(d*x + c)^6 + 168*a^3*tan(d*x + c)^5 + 1008*a*b^2*t an(d*x + c)^5 + 1260*a^2*b*tan(d*x + c)^4 + 210*b^3*tan(d*x + c)^4 + 560*a ^3*tan(d*x + c)^3 + 840*a*b^2*tan(d*x + c)^3 + 1260*a^2*b*tan(d*x + c)^2 + 840*a^3*tan(d*x + c))/d
Time = 4.86 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {2\,a^3}{3}+a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {a^3}{5}+\frac {6\,a\,b^2}{5}\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (\frac {a^2\,b}{2}+\frac {b^3}{3}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {3\,a^2\,b}{2}+\frac {b^3}{4}\right )+a^3\,\mathrm {tan}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^8}{8}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}}{d} \]