Integrand size = 26, antiderivative size = 88 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d} \]
-8/13*I*(a+I*a*tan(d*x+c))^(13/2)/a^3/d+8/15*I*(a+I*a*tan(d*x+c))^(15/2)/a ^4/d-2/17*I*(a+I*a*tan(d*x+c))^(17/2)/a^5/d
Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 a^3 (-i+\tan (c+d x))^6 \sqrt {a+i a \tan (c+d x)} \left (331 i+494 \tan (c+d x)-195 i \tan ^2(c+d x)\right )}{3315 d} \]
(2*a^3*(-I + Tan[c + d*x])^6*Sqrt[a + I*a*Tan[c + d*x]]*(331*I + 494*Tan[c + d*x] - (195*I)*Tan[c + d*x]^2))/(3315*d)
Time = 0.26 (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.
\(\displaystyle \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (c+d x)^6 (a+i a \tan (c+d x))^{7/2}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i \int (a-i a \tan (c+d x))^2 (i \tan (c+d x) a+a)^{11/2}d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i \int \left ((i \tan (c+d x) a+a)^{15/2}-4 a (i \tan (c+d x) a+a)^{13/2}+4 a^2 (i \tan (c+d x) a+a)^{11/2}\right )d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (\frac {8}{13} a^2 (a+i a \tan (c+d x))^{13/2}+\frac {2}{17} (a+i a \tan (c+d x))^{17/2}-\frac {8}{15} a (a+i a \tan (c+d x))^{15/2}\right )}{a^5 d}\) |
((-I)*((8*a^2*(a + I*a*Tan[c + d*x])^(13/2))/13 - (8*a*(a + I*a*Tan[c + d* x])^(15/2))/15 + (2*(a + I*a*Tan[c + d*x])^(17/2))/17))/(a^5*d)
3.4.20.3.1 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 = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72
\[\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {17}{2}}}{17}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {15}{2}}}{15}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}\right )}{d \,a^{5}}\]
2*I/d/a^5*(-1/17*(a+I*a*tan(d*x+c))^(17/2)+4/15*a*(a+I*a*tan(d*x+c))^(15/2 )-4/13*a^2*(a+I*a*tan(d*x+c))^(13/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (64) = 128\).
Time = 0.27 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.86 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {512 \, \sqrt {2} {\left (8 i \, a^{3} e^{\left (17 i \, d x + 17 i \, c\right )} + 68 i \, a^{3} e^{\left (15 i \, d x + 15 i \, c\right )} + 255 i \, a^{3} e^{\left (13 i \, d x + 13 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{3315 \, {\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-512/3315*sqrt(2)*(8*I*a^3*e^(17*I*d*x + 17*I*c) + 68*I*a^3*e^(15*I*d*x + 15*I*c) + 255*I*a^3*e^(13*I*d*x + 13*I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1 ))/(d*e^(16*I*d*x + 16*I*c) + 8*d*e^(14*I*d*x + 14*I*c) + 28*d*e^(12*I*d*x + 12*I*c) + 56*d*e^(10*I*d*x + 10*I*c) + 70*d*e^(8*I*d*x + 8*I*c) + 56*d* e^(6*I*d*x + 6*I*c) + 28*d*e^(4*I*d*x + 4*I*c) + 8*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {2 i \, {\left (195 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {17}{2}} - 884 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a + 1020 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a^{2}\right )}}{3315 \, a^{5} d} \]
-2/3315*I*(195*(I*a*tan(d*x + c) + a)^(17/2) - 884*(I*a*tan(d*x + c) + a)^ (15/2)*a + 1020*(I*a*tan(d*x + c) + a)^(13/2)*a^2)/(a^5*d)
\[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{6} \,d x } \]
Time = 17.40 (sec) , antiderivative size = 562, normalized size of antiderivative = 6.39 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {a^3\,\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}}\,4096{}\mathrm {i}}{3315\,d}-\frac {a^3\,\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}}\,2048{}\mathrm {i}}{3315\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a^3\,\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}}\,512{}\mathrm {i}}{1105\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {a^3\,\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}}\,56320{}\mathrm {i}}{663\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {a^3\,\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}}\,205312{}\mathrm {i}}{663\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {a^3\,\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}}\,540672{}\mathrm {i}}{1105\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {a^3\,\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}}\,1341952{}\mathrm {i}}{3315\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6}+\frac {a^3\,\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}}\,44032{}\mathrm {i}}{255\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^7}-\frac {a^3\,\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}}\,512{}\mathrm {i}}{17\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^8} \]
(a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1 /2)*56320i)/(663*d*(exp(c*2i + d*x*2i) + 1)^3) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*2048i)/(3315*d*(exp(c *2i + d*x*2i) + 1)) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c *2i + d*x*2i) + 1))^(1/2)*512i)/(1105*d*(exp(c*2i + d*x*2i) + 1)^2) - (a^3 *(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)* 4096i)/(3315*d) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*205312i)/(663*d*(exp(c*2i + d*x*2i) + 1)^4) + (a^3*( a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*54 0672i)/(1105*d*(exp(c*2i + d*x*2i) + 1)^5) - (a^3*(a - (a*(exp(c*2i + d*x* 2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*1341952i)/(3315*d*(exp(c* 2i + d*x*2i) + 1)^6) + (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp( c*2i + d*x*2i) + 1))^(1/2)*44032i)/(255*d*(exp(c*2i + d*x*2i) + 1)^7) - (a ^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2 )*512i)/(17*d*(exp(c*2i + d*x*2i) + 1)^8)