Integrand size = 26, antiderivative size = 88 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d} \] Output:
-8/9*I*(a+I*a*tan(d*x+c))^(9/2)/a^3/d+8/11*I*(a+I*a*tan(d*x+c))^(11/2)/a^4 /d-2/13*I*(a+I*a*tan(d*x+c))^(13/2)/a^5/d
Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {2 i a (-i+\tan (c+d x))^4 \sqrt {a+i a \tan (c+d x)} \left (-203+270 i \tan (c+d x)+99 \tan ^2(c+d x)\right )}{1287 d} \] Input:
Integrate[Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
(((2*I)/1287)*a*(-I + Tan[c + d*x])^4*Sqrt[a + I*a*Tan[c + d*x]]*(-203 + ( 270*I)*Tan[c + d*x] + 99*Tan[c + d*x]^2))/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.
\(\displaystyle \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (c+d x)^6 (a+i a \tan (c+d x))^{3/2}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i \int (a-i a \tan (c+d x))^2 (i \tan (c+d x) a+a)^{7/2}d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i \int \left ((i \tan (c+d x) a+a)^{11/2}-4 a (i \tan (c+d x) a+a)^{9/2}+4 a^2 (i \tan (c+d x) a+a)^{7/2}\right )d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (\frac {8}{9} a^2 (a+i a \tan (c+d x))^{9/2}+\frac {2}{13} (a+i a \tan (c+d x))^{13/2}-\frac {8}{11} a (a+i a \tan (c+d x))^{11/2}\right )}{a^5 d}\) |
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])^(9/2))/9 - (8*a*(a + I*a*Tan[c + d*x] )^(11/2))/11 + (2*(a + I*a*Tan[c + d*x])^(13/2))/13))/(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 = 0.55 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}\right )}{d \,a^{5}}\) | \(63\) |
default | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}\right )}{d \,a^{5}}\) | \(63\) |
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*(-1/13*(a+I*a*tan(d*x+c))^(13/2)+4/11*a*(a+I*a*tan(d*x+c))^(11/2 )-4/9*a^2*(a+I*a*tan(d*x+c))^(9/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (64) = 128\).
Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.52 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {128 \, \sqrt {2} {\left (8 i \, a e^{\left (13 i \, d x + 13 i \, c\right )} + 52 i \, a e^{\left (11 i \, d x + 11 i \, c\right )} + 143 i \, a e^{\left (9 i \, d x + 9 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{1287 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:
integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")
Output:
-128/1287*sqrt(2)*(8*I*a*e^(13*I*d*x + 13*I*c) + 52*I*a*e^(11*I*d*x + 11*I *c) + 143*I*a*e^(9*I*d*x + 9*I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))/(d*e^ (12*I*d*x + 12*I*c) + 6*d*e^(10*I*d*x + 10*I*c) + 15*d*e^(8*I*d*x + 8*I*c) + 20*d*e^(6*I*d*x + 6*I*c) + 15*d*e^(4*I*d*x + 4*I*c) + 6*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**6*(a+I*a*tan(d*x+c))**(3/2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {2 i \, {\left (99 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 468 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 572 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2}\right )}}{1287 \, a^{5} d} \] Input:
integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")
Output:
-2/1287*I*(99*(I*a*tan(d*x + c) + a)^(13/2) - 468*(I*a*tan(d*x + c) + a)^( 11/2)*a + 572*(I*a*tan(d*x + c) + a)^(9/2)*a^2)/(a^5*d)
Exception generated. \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \] 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 = 4.51 (sec) , antiderivative size = 420, normalized size of antiderivative = 4.77 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {a\,\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}}\,1024{}\mathrm {i}}{1287\,d}-\frac {a\,\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}}{1287\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a\,\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}}{429\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {a\,\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}}\,27136{}\mathrm {i}}{1287\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {a\,\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}}\,58624{}\mathrm {i}}{1287\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {a\,\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}}\,5120{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {a\,\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}}{13\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6} \] Input:
int((a + a*tan(c + d*x)*1i)^(3/2)/cos(c + d*x)^6,x)
Output:
(a*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2 )*27136i)/(1287*d*(exp(c*2i + d*x*2i) + 1)^3) - (a*(a - (a*(exp(c*2i + d*x *2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*512i)/(1287*d*(exp(c*2i + d*x*2i) + 1)) - (a*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*128i)/(429*d*(exp(c*2i + d*x*2i) + 1)^2) - (a*(a - (a* (exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*1024i)/(1 287*d) - (a*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*58624i)/(1287*d*(exp(c*2i + d*x*2i) + 1)^4) + (a*(a - (a*(exp(c *2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*5120i)/(143*d*( exp(c*2i + d*x*2i) + 1)^5) - (a*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/( exp(c*2i + d*x*2i) + 1))^(1/2)*128i)/(13*d*(exp(c*2i + d*x*2i) + 1)^6)
\[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {2 \sqrt {a}\, a i \left (-\sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{6}+7 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{6} \tan \left (d x +c \right )d x \right ) d \right )}{d} \] Input:
int(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^(3/2),x)
Output:
(2*sqrt(a)*a*i*( - sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**6 + 7*int(sqrt(t an(c + d*x)*i + 1)*sec(c + d*x)**6*tan(c + d*x),x)*d))/d