Integrand size = 28, antiderivative size = 145 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\frac {30 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{77 a d e^4}+\frac {18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac {30 \sin (c+d x)}{77 a d e^3 \sqrt {e \sec (c+d x)}}+\frac {2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \] Output:
30/77*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*(e*sec(d*x+c ))^(1/2)/a/d/e^4+18/77*sin(d*x+c)/a/d/e/(e*sec(d*x+c))^(5/2)+30/77*sin(d*x +c)/a/d/e^3/(e*sec(d*x+c))^(1/2)+2/11*I/d/(e*sec(d*x+c))^(7/2)/(a+I*a*tan( d*x+c))
Time = 1.74 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=-\frac {(e \sec (c+d x))^{3/2} \left (-148 \cos (c+d x)+34 \cos (3 (c+d x))+2 \cos (5 (c+d x))+240 i \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (c+d x)+i \sin (c+d x))+78 i \sin (c+d x)+87 i \sin (3 (c+d x))+9 i \sin (5 (c+d x))\right )}{616 a d e^5 (-i+\tan (c+d x))} \] Input:
Integrate[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])),x]
Output:
-1/616*((e*Sec[c + d*x])^(3/2)*(-148*Cos[c + d*x] + 34*Cos[3*(c + d*x)] + 2*Cos[5*(c + d*x)] + (240*I)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]* (Cos[c + d*x] + I*Sin[c + d*x]) + (78*I)*Sin[c + d*x] + (87*I)*Sin[3*(c + d*x)] + (9*I)*Sin[5*(c + d*x)]))/(a*d*e^5*(-I + Tan[c + d*x]))
Time = 0.65 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 3983, 3042, 4256, 3042, 4256, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \frac {1}{(a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}dx\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \int \frac {1}{(e \sec (c+d x))^{7/2}}dx}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {9 \left (\frac {5 \int \frac {1}{(e \sec (c+d x))^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {5 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {9 \left (\frac {5 \left (\frac {\int \sqrt {e \sec (c+d x)}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {5 \left (\frac {\int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {9 \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {9 \left (\frac {5 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a}+\frac {2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}}\) |
Input:
Int[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])),x]
Output:
(9*((2*Sin[c + d*x])/(7*d*e*(e*Sec[c + d*x])^(5/2)) + (5*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*d*e^2) + (2*Sin[c + d*x])/(3*d*e*Sqrt[e*Sec[c + d*x]])))/(7*e^2)))/(11*a) + ((2*I)/11)/(d*( e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x]))
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[a*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ (b*f*(m + 2*n))), x] + Simp[Simplify[m + n]/(a*(m + 2*n)) Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x ] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2* n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Time = 10.99 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\frac {2 \sin \left (d x +c \right ) \left (7 \cos \left (d x +c \right )^{4}+9 \cos \left (d x +c \right )^{2}+15\right )}{77}+\frac {2 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (15+15 \sec \left (d x +c \right )\right )}{77}+\frac {2 i \cos \left (d x +c \right )^{5}}{11}}{a d \sqrt {e \sec \left (d x +c \right )}\, e^{3}}\) | \(128\) |
Input:
int(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/a/d*(2/77*sin(d*x+c)*(7*cos(d*x+c)^4+9*cos(d*x+c)^2+15)+2/77*I*(1/(cos(d *x+c)+1))^(1/2)*EllipticF(I*(cot(d*x+c)-csc(d*x+c)),I)*(cos(d*x+c)/(cos(d* x+c)+1))^(1/2)*(15+15*sec(d*x+c))+2/11*I*cos(d*x+c)^5)/(e*sec(d*x+c))^(1/2 )/e^3
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-11 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 121 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 70 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 226 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 53 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - 480 i \, \sqrt {2} \sqrt {e} e^{\left (6 i \, d x + 6 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{1232 \, a d e^{4}} \] Input:
integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Output:
1/1232*(sqrt(2)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(-11*I*e^(10*I*d*x + 10* I*c) - 121*I*e^(8*I*d*x + 8*I*c) + 70*I*e^(6*I*d*x + 6*I*c) + 226*I*e^(4*I *d*x + 4*I*c) + 53*I*e^(2*I*d*x + 2*I*c) + 7*I)*e^(1/2*I*d*x + 1/2*I*c) - 480*I*sqrt(2)*sqrt(e)*e^(6*I*d*x + 6*I*c)*weierstrassPInverse(-4, 0, e^(I* d*x + I*c)))*e^(-6*I*d*x - 6*I*c)/(a*d*e^4)
\[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=- \frac {i \int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{2}} \tan {\left (c + d x \right )} - i \left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx}{a} \] Input:
integrate(1/(e*sec(d*x+c))**(7/2)/(a+I*a*tan(d*x+c)),x)
Output:
-I*Integral(1/((e*sec(c + d*x))**(7/2)*tan(c + d*x) - I*(e*sec(c + d*x))** (7/2)), x)/a
Exception generated. \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\int { \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}} \,d x } \] Input:
integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x, algorithm="giac")
Output:
integrate(1/((e*sec(d*x + c))^(7/2)*(I*a*tan(d*x + c) + a)), x)
Timed out. \[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\int \frac {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:
int(1/((e/cos(c + d*x))^(7/2)*(a + a*tan(c + d*x)*1i)),x)
Output:
int(1/((e/cos(c + d*x))^(7/2)*(a + a*tan(c + d*x)*1i)), x)
\[ \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4} \tan \left (d x +c \right ) i +\sec \left (d x +c \right )^{4}}d x \right )}{a \,e^{4}} \] Input:
int(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x)
Output:
(sqrt(e)*int(sqrt(sec(c + d*x))/(sec(c + d*x)**4*tan(c + d*x)*i + sec(c + d*x)**4),x))/(a*e**4)