Integrand size = 26, antiderivative size = 110 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {64 i a^3 \sec ^7(c+d x)}{693 d (a+i a \tan (c+d x))^{7/2}}+\frac {16 i a^2 \sec ^7(c+d x)}{99 d (a+i a \tan (c+d x))^{5/2}}+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}} \]
64/693*I*a^3*sec(d*x+c)^7/d/(a+I*a*tan(d*x+c))^(7/2)+16/99*I*a^2*sec(d*x+c )^7/d/(a+I*a*tan(d*x+c))^(5/2)+2/11*I*a*sec(d*x+c)^7/d/(a+I*a*tan(d*x+c))^ (3/2)
Time = 0.86 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 \sec ^6(c+d x) (44+107 \cos (2 (c+d x))+91 i \sin (2 (c+d x))) (i \cos (3 (c+d x))+\sin (3 (c+d x)))}{693 d \sqrt {a+i a \tan (c+d x)}} \]
(2*Sec[c + d*x]^6*(44 + 107*Cos[2*(c + d*x)] + (91*I)*Sin[2*(c + d*x)])*(I *Cos[3*(c + d*x)] + Sin[3*(c + d*x)]))/(693*d*Sqrt[a + I*a*Tan[c + d*x]])
Time = 0.53 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 3975, 3042, 3975, 3042, 3974}
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.
\(\displaystyle \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (c+d x)^7}{\sqrt {a+i a \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 3975 |
\(\displaystyle \frac {8}{11} a \int \frac {\sec ^7(c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{11} a \int \frac {\sec (c+d x)^7}{(i \tan (c+d x) a+a)^{3/2}}dx+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3975 |
\(\displaystyle \frac {8}{11} a \left (\frac {4}{9} a \int \frac {\sec ^7(c+d x)}{(i \tan (c+d x) a+a)^{5/2}}dx+\frac {2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{11} a \left (\frac {4}{9} a \int \frac {\sec (c+d x)^7}{(i \tan (c+d x) a+a)^{5/2}}dx+\frac {2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3974 |
\(\displaystyle \frac {8}{11} a \left (\frac {8 i a^2 \sec ^7(c+d x)}{63 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\) |
(((2*I)/11)*a*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (8*a*(((( 8*I)/63)*a^2*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((2*I)/9) *a*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(5/2))))/11
3.4.41.3.1 Defintions of rubi rules used
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[2*b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^ (n - 1)/(f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m/2 + n - 1], 0]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] + Simp[a*((m + 2*n - 2)/(m + n - 1)) Int[(d*Se c[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m/2 + n - 1], 0] && !Inte gerQ[n]
Time = 6.94 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\frac {256 i \sec \left (d x +c \right )}{693}+\frac {256 \sec \left (d x +c \right ) \tan \left (d x +c \right )}{693}+\frac {32 i \left (\sec ^{3}\left (d x +c \right )\right )}{693}+\frac {160 \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{693}+\frac {2 i \left (\sec ^{5}\left (d x +c \right )\right )}{99}+\frac {2 \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{11}}{d \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}}\) | \(99\) |
2/693/d/(a*(1+I*tan(d*x+c)))^(1/2)*(128*I*sec(d*x+c)+128*sec(d*x+c)*tan(d* x+c)+16*I*sec(d*x+c)^3+80*tan(d*x+c)*sec(d*x+c)^3+7*I*sec(d*x+c)^5+63*tan( d*x+c)*sec(d*x+c)^5)
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {64 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-99 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 44 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 8 i\right )}}{693 \, {\left (a d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
-64/693*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-99*I*e^(4*I*d*x + 4*I* c) - 44*I*e^(2*I*d*x + 2*I*c) - 8*I)/(a*d*e^(10*I*d*x + 10*I*c) + 5*a*d*e^ (8*I*d*x + 8*I*c) + 10*a*d*e^(6*I*d*x + 6*I*c) + 10*a*d*e^(4*I*d*x + 4*I*c ) + 5*a*d*e^(2*I*d*x + 2*I*c) + a*d)
\[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sec ^{7}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (86) = 172\).
Time = 0.38 (sec) , antiderivative size = 474, normalized size of antiderivative = 4.31 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 \, {\left (-151 i \, \sqrt {a} - \frac {542 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {484 i \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {22 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {627 i \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1452 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1452 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {627 i \, \sqrt {a} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {22 \, \sqrt {a} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {484 i \, \sqrt {a} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {542 \, \sqrt {a} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {151 i \, \sqrt {a} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1}}{693 \, {\left (a - \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d \sqrt {-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}} \]
-2/693*(-151*I*sqrt(a) - 542*sqrt(a)*sin(d*x + c)/(cos(d*x + c) + 1) + 484 *I*sqrt(a)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 22*sqrt(a)*sin(d*x + c)^3 /(cos(d*x + c) + 1)^3 - 627*I*sqrt(a)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 1452*sqrt(a)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1452*sqrt(a)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 627*I*sqrt(a)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 22*sqrt(a)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 484*I*sqrt(a)*sin (d*x + c)^10/(cos(d*x + c) + 1)^10 - 542*sqrt(a)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 151*I*sqrt(a)*sin(d*x + c)^12/(cos(d*x + c) + 1)^12)*sqrt(s in(d*x + c)/(cos(d*x + c) + 1) + 1)*sqrt(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/((a - 6*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 - 20*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a*sin(d *x + c)^8/(cos(d*x + c) + 1)^8 - 6*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12)*d*sqrt(-2*I*sin(d*x + c)/(cos( d*x + c) + 1) + sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1))
\[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{7}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
Time = 6.63 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {64\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,44{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,99{}\mathrm {i}+8{}\mathrm {i}\right )}{693\,a\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \]