Integrand size = 26, antiderivative size = 88 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {8 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d} \] Output:
-8/3*I*(a+I*a*tan(d*x+c))^(3/2)/a^3/d+8/5*I*(a+I*a*tan(d*x+c))^(5/2)/a^4/d -2/7*I*(a+I*a*tan(d*x+c))^(7/2)/a^5/d
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.67 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {2 (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \left (-71+54 i \tan (c+d x)+15 \tan ^2(c+d x)\right )}{105 a^2 d} \] Input:
Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
(-2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]]*(-71 + (54*I)*Tan[c + d *x] + 15*Tan[c + d*x]^2))/(105*a^2*d)
Time = 0.28 (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.
\(\displaystyle \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (c+d x)^6}{(a+i a \tan (c+d x))^{3/2}}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i \int (a-i a \tan (c+d x))^2 \sqrt {i \tan (c+d x) a+a}d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i \int \left ((i \tan (c+d x) a+a)^{5/2}-4 a (i \tan (c+d x) a+a)^{3/2}+4 a^2 \sqrt {i \tan (c+d x) a+a}\right )d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (\frac {8}{3} a^2 (a+i a \tan (c+d x))^{3/2}+\frac {2}{7} (a+i a \tan (c+d x))^{7/2}-\frac {8}{5} a (a+i a \tan (c+d x))^{5/2}\right )}{a^5 d}\) |
Input:
Int[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
((-I)*((8*a^2*(a + I*a*Tan[c + d*x])^(3/2))/3 - (8*a*(a + I*a*Tan[c + d*x] )^(5/2))/5 + (2*(a + I*a*Tan[c + d*x])^(7/2))/7))/(a^5*d)
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 = 1.00 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,a^{5}}\) | \(63\) |
default | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,a^{5}}\) | \(63\) |
Input:
int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
2*I/d/a^5*(-1/7*(a+I*a*tan(d*x+c))^(7/2)+4/5*a*(a+I*a*tan(d*x+c))^(5/2)-4/ 3*a^2*(a+I*a*tan(d*x+c))^(3/2))
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.23 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {16 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (8 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 28 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 35 i \, e^{\left (3 i \, d x + 3 i \, c\right )}\right )}}{105 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")
Output:
-16/105*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(8*I*e^(7*I*d*x + 7*I*c) + 28*I*e^(5*I*d*x + 5*I*c) + 35*I*e^(3*I*d*x + 3*I*c))/(a^2*d*e^(6*I*d*x + 6*I*c) + 3*a^2*d*e^(4*I*d*x + 4*I*c) + 3*a^2*d*e^(2*I*d*x + 2*I*c) + a^2 *d)
\[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(sec(d*x+c)**6/(a+I*a*tan(d*x+c))**(3/2),x)
Output:
Integral(sec(c + d*x)**6/(I*a*(tan(c + d*x) - I))**(3/2), x)
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {2 i \, {\left (15 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 84 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 140 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2}\right )}}{105 \, a^{5} d} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")
Output:
-2/105*I*(15*(I*a*tan(d*x + c) + a)^(7/2) - 84*(I*a*tan(d*x + c) + a)^(5/2 )*a + 140*(I*a*tan(d*x + c) + a)^(3/2)*a^2)/(a^5*d)
Exception generated. \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(3/2),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 = 3.94 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.75 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{105\,a^2\,d}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{105\,a^2\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,16{}\mathrm {i}}{35\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,16{}\mathrm {i}}{7\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3} \] Input:
int(1/(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i)^(3/2)),x)
Output:
((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)* 16i)/(7*a^2*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)*64i)/(105*a^2*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)*16i)/(35*a^2*d*(exp(c*2i + d*x*2i) + 1)^2) - ((a - (a*(exp(c *2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*128i)/(105*a^2* d)
\[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (-2 \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{6} i -9 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{6} \tan \left (d x +c \right )^{2}}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) \tan \left (d x +c \right )^{2} d -9 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{6} \tan \left (d x +c \right )^{2}}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) d +7 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{6} \tan \left (d x +c \right )}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) \tan \left (d x +c \right )^{2} d i +7 \left (\int \frac {\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{6} \tan \left (d x +c \right )}{\tan \left (d x +c \right )^{3} i +\tan \left (d x +c \right )^{2}+\tan \left (d x +c \right ) i +1}d x \right ) d i \right )}{a^{2} d \left (\tan \left (d x +c \right )^{2}+1\right )} \] Input:
int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
(sqrt(a)*( - 2*sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**6*i - 9*int((sqrt(ta n(c + d*x)*i + 1)*sec(c + d*x)**6*tan(c + d*x)**2)/(tan(c + d*x)**3*i + ta n(c + d*x)**2 + tan(c + d*x)*i + 1),x)*tan(c + d*x)**2*d - 9*int((sqrt(tan (c + d*x)*i + 1)*sec(c + d*x)**6*tan(c + d*x)**2)/(tan(c + d*x)**3*i + tan (c + d*x)**2 + tan(c + d*x)*i + 1),x)*d + 7*int((sqrt(tan(c + d*x)*i + 1)* sec(c + d*x)**6*tan(c + d*x))/(tan(c + d*x)**3*i + tan(c + d*x)**2 + tan(c + d*x)*i + 1),x)*tan(c + d*x)**2*d*i + 7*int((sqrt(tan(c + d*x)*i + 1)*se c(c + d*x)**6*tan(c + d*x))/(tan(c + d*x)**3*i + tan(c + d*x)**2 + tan(c + d*x)*i + 1),x)*d*i))/(a**2*d*(tan(c + d*x)**2 + 1))