Integrand size = 26, antiderivative size = 88 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {8 i (a+i a \tan (c+d x))^{5/2}}{5 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{9/2}}{9 a^5 d} \]
-8/5*I*(a+I*a*tan(d*x+c))^(5/2)/a^3/d+8/7*I*(a+I*a*tan(d*x+c))^(7/2)/a^4/d -2/9*I*(a+I*a*tan(d*x+c))^(9/2)/a^5/d
Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)} \left (107 i+110 \tan (c+d x)-35 i \tan ^2(c+d x)\right )}{315 a d} \]
(2*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]]*(107*I + 110*Tan[c + d *x] - (35*I)*Tan[c + d*x]^2))/(315*a*d)
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3042, 3968, 53, 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.
\(\displaystyle \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (c+d x)^6}{\sqrt {a+i a \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i \int (a-i a \tan (c+d x))^2 (i \tan (c+d x) a+a)^{3/2}d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i \int \left ((i \tan (c+d x) a+a)^{7/2}-4 a (i \tan (c+d x) a+a)^{5/2}+4 a^2 (i \tan (c+d x) a+a)^{3/2}\right )d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (\frac {8}{5} a^2 (a+i a \tan (c+d x))^{5/2}+\frac {2}{9} (a+i a \tan (c+d x))^{9/2}-\frac {8}{7} a (a+i a \tan (c+d x))^{7/2}\right )}{a^5 d}\) |
((-I)*((8*a^2*(a + I*a*Tan[c + d*x])^(5/2))/5 - (8*a*(a + I*a*Tan[c + d*x] )^(7/2))/7 + (2*(a + I*a*Tan[c + d*x])^(9/2))/9))/(a^5*d)
3.4.34.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Time = 1.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}\right )}{d \,a^{5}}\) | \(63\) |
default | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}\right )}{d \,a^{5}}\) | \(63\) |
2*I/d/a^5*(-1/9*(a+I*a*tan(d*x+c))^(9/2)+4/7*a*(a+I*a*tan(d*x+c))^(7/2)-4/ 5*a^2*(a+I*a*tan(d*x+c))^(5/2))
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {32 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (8 i \, e^{\left (9 i \, d x + 9 i \, c\right )} + 36 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 63 i \, e^{\left (5 i \, d x + 5 i \, c\right )}\right )}}{315 \, {\left (a d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
-32/315*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(8*I*e^(9*I*d*x + 9*I*c) + 36*I*e^(7*I*d*x + 7*I*c) + 63*I*e^(5*I*d*x + 5*I*c))/(a*d*e^(8*I*d*x + 8*I*c) + 4*a*d*e^(6*I*d*x + 6*I*c) + 6*a*d*e^(4*I*d*x + 4*I*c) + 4*a*d*e^( 2*I*d*x + 2*I*c) + a*d)
\[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (64) = 128\).
Time = 0.24 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.92 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 i \, {\left (315 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} - \frac {42 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac {35 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{4}}{a^{4}}\right )}}{315 \, a d} \]
-2/315*I*(315*sqrt(I*a*tan(d*x + c) + a) - 42*(3*(I*a*tan(d*x + c) + a)^(5 /2) - 10*(I*a*tan(d*x + c) + a)^(3/2)*a + 15*sqrt(I*a*tan(d*x + c) + a)*a^ 2)/a^2 + (35*(I*a*tan(d*x + c) + a)^(9/2) - 180*(I*a*tan(d*x + c) + a)^(7/ 2)*a + 378*(I*a*tan(d*x + c) + a)^(5/2)*a^2 - 420*(I*a*tan(d*x + c) + a)^( 3/2)*a^3 + 315*sqrt(I*a*tan(d*x + c) + a)*a^4)/a^4)/(a*d)
\[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{6}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
Time = 7.16 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.48 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,256{}\mathrm {i}}{315\,a\,d}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{315\,a\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{105\,a\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,320{}\mathrm {i}}{63\,a\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{9\,a\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4} \]
((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)* 320i)/(63*a*d*(exp(c*2i + d*x*2i) + 1)^3) - ((a - (a*(exp(c*2i + d*x*2i)*1 i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*128i)/(315*a*d*(exp(c*2i + d*x *2i) + 1)) - ((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*32i)/(105*a*d*(exp(c*2i + d*x*2i) + 1)^2) - ((a - (a*(exp(c*2 i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*256i)/(315*a*d) - ((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2) *32i)/(9*a*d*(exp(c*2i + d*x*2i) + 1)^4)