Integrand size = 26, antiderivative size = 110 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {64 i a^3 \sec ^7(c+d x)}{693 d (a+i a \tan (c+d x))^{7/2}}+\frac {16 i a^2 \sec ^7(c+d x)}{99 d (a+i a \tan (c+d x))^{5/2}}+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}} \] Output:
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 = 0.76 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 \sec ^6(c+d x) (44+107 \cos (2 (c+d x))+91 i \sin (2 (c+d x))) (i \cos (3 (c+d x))+\sin (3 (c+d x)))}{693 d \sqrt {a+i a \tan (c+d x)}} \] Input:
Integrate[Sec[c + d*x]^7/Sqrt[a + I*a*Tan[c + d*x]],x]
Output:
(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 = 0.54 (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 |
\(\displaystyle \frac {8}{11} a \left (\frac {8 i a^2 \sec ^7(c+d x)}{63 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^{5/2}}\right )+\frac {2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\) |
Input:
Int[Sec[c + d*x]^7/Sqrt[a + I*a*Tan[c + d*x]],x]
Output:
(((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
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 = 4.91 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\frac {2 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{5} \left (128 \cos \left (d x +c \right )^{4}+80 \cos \left (d x +c \right )^{2}+63\right )}{693}+\frac {2 i \left (128 \sec \left (d x +c \right )+16 \sec \left (d x +c \right )^{3}+7 \sec \left (d x +c \right )^{5}\right )}{693}}{d \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}}\) | \(91\) |
Input:
int(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/d*(2/693*tan(d*x+c)*sec(d*x+c)^5*(128*cos(d*x+c)^4+80*cos(d*x+c)^2+63)+2 /693*I*(128*sec(d*x+c)+16*sec(d*x+c)^3+7*sec(d*x+c)^5))/(a*(1+I*tan(d*x+c) ))^(1/2)
Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {64 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-99 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 44 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 8 i\right )}}{693 \, {\left (a d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \] Input:
integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-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)
\[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sec ^{7}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \] Input:
integrate(sec(d*x+c)**7/(a+I*a*tan(d*x+c))**(1/2),x)
Output:
Integral(sec(c + d*x)**7/sqrt(I*a*(tan(c + d*x) - I)), x)
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 = 0.22 (sec) , antiderivative size = 474, normalized size of antiderivative = 4.31 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
Output:
-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))
Exception generated. \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Time = 3.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95 \[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {64\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,44{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,99{}\mathrm {i}+8{}\mathrm {i}\right )}{693\,a\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \] Input:
int(1/(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^(1/2)),x)
Output:
(64*exp(- c*1i - d*x*1i)*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2 i + d*x*2i) + 1))^(1/2)*(exp(c*2i + d*x*2i)*44i + exp(c*4i + d*x*4i)*99i + 8i))/(693*a*d*(exp(c*2i + d*x*2i) + 1)^5)
\[ \int \frac {\sec ^7(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 \sqrt {a}\, i \left (-\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{7}+5 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{7} \tan \left (d x +c \right )}{\tan \left (d x +c \right )^{2}+1}d x \right ) \tan \left (d x +c \right )^{2} d +5 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{7} \tan \left (d x +c \right )}{\tan \left (d x +c \right )^{2}+1}d x \right ) d \right )}{a d \left (\tan \left (d x +c \right )^{2}+1\right )} \] Input:
int(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^(1/2),x)
Output:
(2*sqrt(a)*i*( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**7 + 5*int((sqrt(ta n(c + d*x)*i + 1)*sec(c + d*x)**7*tan(c + d*x))/(tan(c + d*x)**2 + 1),x)*t an(c + d*x)**2*d + 5*int((sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**7*tan(c + d*x))/(tan(c + d*x)**2 + 1),x)*d))/(a*d*(tan(c + d*x)**2 + 1))