Integrand size = 30, antiderivative size = 125 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {16 i a^2 \sqrt {e \sec (c+d x)}}{21 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {8 i a \sqrt {a+i a \tan (c+d x)}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}} \] Output:
16/21*I*a^2*(e*sec(d*x+c))^(1/2)/d/e^4/(a+I*a*tan(d*x+c))^(1/2)-8/21*I*a*( a+I*a*tan(d*x+c))^(1/2)/d/e^2/(e*sec(d*x+c))^(3/2)-2/7*I*(a+I*a*tan(d*x+c) )^(3/2)/d/(e*sec(d*x+c))^(7/2)
Time = 1.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.78 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {a (\cos (d x)-i \sin (d x)) (-7 i+9 i \cos (2 (c+d x))+12 \sin (2 (c+d x))) (\cos (c+2 d x)+i \sin (c+2 d x)) \sqrt {a+i a \tan (c+d x)}}{21 d e^3 \sqrt {e \sec (c+d x)}} \] Input:
Integrate[(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(7/2),x]
Output:
(a*(Cos[d*x] - I*Sin[d*x])*(-7*I + (9*I)*Cos[2*(c + d*x)] + 12*Sin[2*(c + d*x)])*(Cos[c + 2*d*x] + I*Sin[c + 2*d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(21 *d*e^3*Sqrt[e*Sec[c + d*x]])
Time = 0.59 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3978, 3042, 3978, 3042, 3969}
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.
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}}dx\) |
\(\Big \downarrow \) 3978 |
\(\displaystyle \frac {4 a \int \frac {\sqrt {i \tan (c+d x) a+a}}{(e \sec (c+d x))^{3/2}}dx}{7 e^2}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \int \frac {\sqrt {i \tan (c+d x) a+a}}{(e \sec (c+d x))^{3/2}}dx}{7 e^2}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3978 |
\(\displaystyle \frac {4 a \left (\frac {2 a \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}dx}{3 e^2}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}\right )}{7 e^2}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \left (\frac {2 a \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}dx}{3 e^2}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}\right )}{7 e^2}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3969 |
\(\displaystyle \frac {4 a \left (\frac {4 i a \sqrt {e \sec (c+d x)}}{3 d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}\right )}{7 e^2}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}\) |
Input:
Int[(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(7/2),x]
Output:
(((-2*I)/7)*(a + I*a*Tan[c + d*x])^(3/2))/(d*(e*Sec[c + d*x])^(7/2)) + (4* a*((((4*I)/3)*a*Sqrt[e*Sec[c + d*x]])/(d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (((2*I)/3)*Sqrt[a + I*a*Tan[c + d*x]])/(d*(e*Sec[c + d*x])^(3/2))))/(7*e^ 2)
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/ (a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ [Simplify[m + n], 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/( a*f*m)), x] + Simp[a*((m + n)/(m*d^2)) Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b ^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
Time = 6.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.64
method | result | size |
default | \(-\frac {2 a \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (3 i \cos \left (d x +c \right )^{3}-3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}-4 i \cos \left (d x +c \right )-8 \sin \left (d x +c \right )\right )}{21 d \sqrt {e \sec \left (d x +c \right )}\, e^{3}}\) | \(80\) |
risch | \(-\frac {i a \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}-7 \cos \left (d x +c \right )+35 i \sin \left (d x +c \right )\right )}{42 e^{3} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(92\) |
Input:
int((a+I*a*tan(d*x+c))^(3/2)/(e*sec(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/21/d*a*(a*(1+I*tan(d*x+c)))^(1/2)*(3*I*cos(d*x+c)^3-3*sin(d*x+c)*cos(d* x+c)^2-4*I*cos(d*x+c)-8*sin(d*x+c))/(e*sec(d*x+c))^(1/2)/e^3
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {{\left (-3 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 17 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 21 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{42 \, d e^{4}} \] Input:
integrate((a+I*a*tan(d*x+c))^(3/2)/(e*sec(d*x+c))^(7/2),x, algorithm="fric as")
Output:
1/42*(-3*I*a*e^(6*I*d*x + 6*I*c) - 17*I*a*e^(4*I*d*x + 4*I*c) + 7*I*a*e^(2 *I*d*x + 2*I*c) + 21*I*a)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I *d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c)/(d*e^4)
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*tan(d*x+c))**(3/2)/(e*sec(d*x+c))**(7/2),x)
Output:
Timed out
Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {{\left (-3 i \, a \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) - 14 i \, a \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 21 i \, a \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 14 \, a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 21 \, a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{42 \, d e^{\frac {7}{2}}} \] Input:
integrate((a+I*a*tan(d*x+c))^(3/2)/(e*sec(d*x+c))^(7/2),x, algorithm="maxi ma")
Output:
1/42*(-3*I*a*cos(7/2*d*x + 7/2*c) - 14*I*a*cos(3/2*d*x + 3/2*c) + 21*I*a*c os(1/2*d*x + 1/2*c) + 3*a*sin(7/2*d*x + 7/2*c) + 14*a*sin(3/2*d*x + 3/2*c) + 21*a*sin(1/2*d*x + 1/2*c))*sqrt(a)/(d*e^(7/2))
Exception generated. \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+I*a*tan(d*x+c))^(3/2)/(e*sec(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 = 2.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {a\,\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,4{}\mathrm {i}-\cos \left (4\,c+4\,d\,x\right )\,3{}\mathrm {i}+38\,\sin \left (2\,c+2\,d\,x\right )+3\,\sin \left (4\,c+4\,d\,x\right )+7{}\mathrm {i}\right )}{84\,d\,e^4} \] Input:
int((a + a*tan(c + d*x)*1i)^(3/2)/(e/cos(c + d*x))^(7/2),x)
Output:
(a*(e/cos(c + d*x))^(1/2)*((a*(cos(2*c + 2*d*x) + sin(2*c + 2*d*x)*1i + 1) )/(cos(2*c + 2*d*x) + 1))^(1/2)*(cos(2*c + 2*d*x)*4i - cos(4*c + 4*d*x)*3i + 38*sin(2*c + 2*d*x) + 3*sin(4*c + 4*d*x) + 7i))/(84*d*e^4)
\[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, \sqrt {a}\, a \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )}{\sec \left (d x +c \right )^{4}}d x \right ) i +\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}}{\sec \left (d x +c \right )^{4}}d x \right )}{e^{4}} \] Input:
int((a+I*a*tan(d*x+c))^(3/2)/(e*sec(d*x+c))^(7/2),x)
Output:
(sqrt(e)*sqrt(a)*a*(int((sqrt(sec(c + d*x))*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x))/sec(c + d*x)**4,x)*i + int((sqrt(sec(c + d*x))*sqrt(tan(c + d*x)* i + 1))/sec(c + d*x)**4,x)))/e**4