Integrand size = 26, antiderivative size = 110 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {64 i a^3 \sec ^{13}(c+d x)}{3315 d (a+i a \tan (c+d x))^{13/2}}+\frac {16 i a^2 \sec ^{13}(c+d x)}{255 d (a+i a \tan (c+d x))^{11/2}}+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}} \] Output:
64/3315*I*a^3*sec(d*x+c)^13/d/(a+I*a*tan(d*x+c))^(13/2)+16/255*I*a^2*sec(d *x+c)^13/d/(a+I*a*tan(d*x+c))^(11/2)+2/17*I*a*sec(d*x+c)^13/d/(a+I*a*tan(d *x+c))^(9/2)
Time = 1.90 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {2 \sec ^{12}(c+d x) (68+263 \cos (2 (c+d x))+247 i \sin (2 (c+d x))) (\cos (3 (c+d x))-i \sin (3 (c+d x)))}{3315 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \] Input:
Integrate[Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^(7/2),x]
Output:
(-2*Sec[c + d*x]^12*(68 + 263*Cos[2*(c + d*x)] + (247*I)*Sin[2*(c + d*x)]) *(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]))/(3315*a^3*d*(-I + Tan[c + d*x])^ 3*Sqrt[a + I*a*Tan[c + d*x]])
Time = 0.55 (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.
\(\displaystyle \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (c+d x)^{13}}{(a+i a \tan (c+d x))^{7/2}}dx\) |
\(\Big \downarrow \) 3975 |
\(\displaystyle \frac {8}{17} a \int \frac {\sec ^{13}(c+d x)}{(i \tan (c+d x) a+a)^{9/2}}dx+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{17} a \int \frac {\sec (c+d x)^{13}}{(i \tan (c+d x) a+a)^{9/2}}dx+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}\) |
\(\Big \downarrow \) 3975 |
\(\displaystyle \frac {8}{17} a \left (\frac {4}{15} a \int \frac {\sec ^{13}(c+d x)}{(i \tan (c+d x) a+a)^{11/2}}dx+\frac {2 i a \sec ^{13}(c+d x)}{15 d (a+i a \tan (c+d x))^{11/2}}\right )+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8}{17} a \left (\frac {4}{15} a \int \frac {\sec (c+d x)^{13}}{(i \tan (c+d x) a+a)^{11/2}}dx+\frac {2 i a \sec ^{13}(c+d x)}{15 d (a+i a \tan (c+d x))^{11/2}}\right )+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}\) |
\(\Big \downarrow \) 3974 |
\(\displaystyle \frac {8}{17} a \left (\frac {8 i a^2 \sec ^{13}(c+d x)}{195 d (a+i a \tan (c+d x))^{13/2}}+\frac {2 i a \sec ^{13}(c+d x)}{15 d (a+i a \tan (c+d x))^{11/2}}\right )+\frac {2 i a \sec ^{13}(c+d x)}{17 d (a+i a \tan (c+d x))^{9/2}}\) |
Input:
Int[Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^(7/2),x]
Output:
(((2*I)/17)*a*Sec[c + d*x]^13)/(d*(a + I*a*Tan[c + d*x])^(9/2)) + (8*a*((( (8*I)/195)*a^2*Sec[c + d*x]^13)/(d*(a + I*a*Tan[c + d*x])^(13/2)) + (((2*I )/15)*a*Sec[c + d*x]^13)/(d*(a + I*a*Tan[c + d*x])^(11/2))))/17
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 = 10.07 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {\frac {2 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{8} \left (128 \cos \left (d x +c \right )^{4}+176 \cos \left (d x +c \right )^{2}+195\right )}{3315}+\frac {2 i \left (128 \sec \left (d x +c \right )^{4}+112 \sec \left (d x +c \right )^{6}+91 \sec \left (d x +c \right )^{8}\right )}{3315}}{d \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{3} \left (4 i \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+4 \cos \left (d x +c \right )^{3}-i \sin \left (d x +c \right )-3 \cos \left (d x +c \right )\right )}\) | \(143\) |
Input:
int(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
2/3315/d/(a*(1+I*tan(d*x+c)))^(1/2)/a^3/(4*I*cos(d*x+c)^2*sin(d*x+c)+4*cos (d*x+c)^3-I*sin(d*x+c)-3*cos(d*x+c))*(tan(d*x+c)*sec(d*x+c)^8*(128*cos(d*x +c)^4+176*cos(d*x+c)^2+195)+I*(128*sec(d*x+c)^4+112*sec(d*x+c)^6+91*sec(d* x+c)^8))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (86) = 172\).
Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.57 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {512 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-255 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 68 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 8 i\right )}}{3315 \, {\left (a^{4} d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a^{4} d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a^{4} d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \] Input:
integrate(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")
Output:
-512/3315*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-255*I*e^(4*I*d*x + 4 *I*c) - 68*I*e^(2*I*d*x + 2*I*c) - 8*I)/(a^4*d*e^(16*I*d*x + 16*I*c) + 8*a ^4*d*e^(14*I*d*x + 14*I*c) + 28*a^4*d*e^(12*I*d*x + 12*I*c) + 56*a^4*d*e^( 10*I*d*x + 10*I*c) + 70*a^4*d*e^(8*I*d*x + 8*I*c) + 56*a^4*d*e^(6*I*d*x + 6*I*c) + 28*a^4*d*e^(4*I*d*x + 4*I*c) + 8*a^4*d*e^(2*I*d*x + 2*I*c) + a^4* d)
Timed out. \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**13/(a+I*a*tan(d*x+c))**(7/2),x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (86) = 172\).
Time = 0.50 (sec) , antiderivative size = 902, normalized size of antiderivative = 8.20 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")
Output:
-2/3315*(-331*I*sqrt(a) - 998*sqrt(a)*sin(d*x + c)/(cos(d*x + c) + 1) - 18 38*I*sqrt(a)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 7522*sqrt(a)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 4836*I*sqrt(a)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 27882*sqrt(a)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 8954*I*sqrt(a)* sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 68926*sqrt(a)*sin(d*x + c)^7/(cos(d* x + c) + 1)^7 - 12631*I*sqrt(a)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 1250 52*sqrt(a)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 10540*I*sqrt(a)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 168980*sqrt(a)*sin(d*x + c)^11/(cos(d*x + c ) + 1)^11 - 168980*sqrt(a)*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 + 10540*I *sqrt(a)*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 125052*sqrt(a)*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 + 12631*I*sqrt(a)*sin(d*x + c)^16/(cos(d*x + c ) + 1)^16 - 68926*sqrt(a)*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 + 8954*I*s qrt(a)*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 - 27882*sqrt(a)*sin(d*x + c)^ 19/(cos(d*x + c) + 1)^19 + 4836*I*sqrt(a)*sin(d*x + c)^20/(cos(d*x + c) + 1)^20 - 7522*sqrt(a)*sin(d*x + c)^21/(cos(d*x + c) + 1)^21 + 1838*I*sqrt(a )*sin(d*x + c)^22/(cos(d*x + c) + 1)^22 - 998*sqrt(a)*sin(d*x + c)^23/(cos (d*x + c) + 1)^23 + 331*I*sqrt(a)*sin(d*x + c)^24/(cos(d*x + c) + 1)^24)*( sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2)*(sin(d*x + c)/(cos(d*x + c) + 1 ) - 1)^(7/2)/((a^4 - 12*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 66*a^4*s in(d*x + c)^4/(cos(d*x + c) + 1)^4 - 220*a^4*sin(d*x + c)^6/(cos(d*x + ...
Exception generated. \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^(7/2),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 = 6.70 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {512\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,68{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,255{}\mathrm {i}+8{}\mathrm {i}\right )}{3315\,a^4\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^8} \] Input:
int(1/(cos(c + d*x)^13*(a + a*tan(c + d*x)*1i)^(7/2)),x)
Output:
(512*exp(- c*1i - d*x*1i)*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c* 2i + d*x*2i) + 1))^(1/2)*(exp(c*2i + d*x*2i)*68i + exp(c*4i + d*x*4i)*255i + 8i))/(3315*a^4*d*(exp(c*2i + d*x*2i) + 1)^8)
\[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {too large to display} \] Input:
int(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^(7/2),x)
Output:
(sqrt(a)*( - 36*sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**13*i - 23*int(( - s qrt(tan(c + d*x)*i + 1)*sec(c + d*x)**13*tan(c + d*x)**4)/(tan(c + d*x)**5 *i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*tan(c + d*x)**2*d - 23*int(( - sqrt(tan(c + d*x)*i + 1)* sec(c + d*x)**13*tan(c + d*x)**4)/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*d + 66 *int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**13*tan(c + d*x)**2)/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*tan(c + d*x)**2*d + 66*int(( - sqrt(tan(c + d* x)*i + 1)*sec(c + d*x)**13*tan(c + d*x)**2)/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1) ,x)*d - 20*int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**13*tan(c + d*x)) /(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*tan(c + d*x)**2*d*i - 20*int(( - sqrt(t an(c + d*x)*i + 1)*sec(c + d*x)**13*tan(c + d*x))/(tan(c + d*x)**5*i + 3*t an(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)* i - 1),x)*d*i + int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**13)/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(c + d*x)**3*i + 2*tan(c + d*x)**2 - 3*tan(c + d*x)*i - 1),x)*tan(c + d*x)**2*d + int(( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**13)/(tan(c + d*x)**5*i + 3*tan(c + d*x)**4 - 2*tan(...