Integrand size = 28, antiderivative size = 70 \[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}+\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{a d} \] Output:
-2*I*e^2*(e*sec(d*x+c))^(1/2)/a/d+2*e^2*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
Time = 1.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.70 \[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=\frac {2 e^2 \left (-i+\sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )\right ) \sqrt {e \sec (c+d x)}}{a d} \] Input:
Integrate[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x]),x]
Output:
(2*e^2*(-I + Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[e*Sec[c + d*x]])/(a*d)
Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3042, 3982, 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.
\(\displaystyle \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)}dx\) |
\(\Big \downarrow \) 3982 |
\(\displaystyle \frac {e^2 \int \sqrt {e \sec (c+d x)}dx}{a}-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {e^2 \int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a}-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{a d}-\frac {2 i e^2 \sqrt {e \sec (c+d x)}}{a d}\) |
Input:
Int[(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x]),x]
Output:
((-2*I)*e^2*Sqrt[e*Sec[c + d*x]])/(a*d) + (2*e^2*Sqrt[Cos[c + d*x]]*Ellipt icF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(a*d)
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[d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Simp[d^2*((m - 2)/(a*(m + n - 1))) 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] && LtQ[n, 0] && GtQ[m, 1] && !IL tQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 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 = 2.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {i e^{2} \left (-2+2 \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {e \sec \left (d x +c \right )}}{a d}\) | \(87\) |
Input:
int((e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
I*e^2/a/d*(-2+2*EllipticF(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^ (1/2)*(cos(d*x+c)+1)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(e*sec(d*x+c))^(1/ 2)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90 \[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=-\frac {2 \, {\left (i \, \sqrt {2} e^{2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + i \, \sqrt {2} e^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{a d} \] Input:
integrate((e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Output:
-2*(I*sqrt(2)*e^2*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c ) + I*sqrt(2)*e^(5/2)*weierstrassPInverse(-4, 0, e^(I*d*x + I*c)))/(a*d)
\[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \] Input:
integrate((e*sec(d*x+c))**(5/2)/(a+I*a*tan(d*x+c)),x)
Output:
-I*Integral((e*sec(c + d*x))**(5/2)/(tan(c + d*x) - I), x)/a
Exception generated. \[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}}}{i \, a \tan \left (d x + c\right ) + a} \,d x } \] Input:
integrate((e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="giac")
Output:
integrate((e*sec(d*x + c))^(5/2)/(I*a*tan(d*x + c) + a), x)
Timed out. \[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \] Input:
int((e/cos(c + d*x))^(5/2)/(a + a*tan(c + d*x)*1i),x)
Output:
int((e/cos(c + d*x))^(5/2)/(a + a*tan(c + d*x)*1i), x)
\[ \int \frac {(e \sec (c+d x))^{5/2}}{a+i a \tan (c+d x)} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\tan \left (d x +c \right ) i +1}d x \right ) e^{2}}{a} \] Input:
int((e*sec(d*x+c))^(5/2)/(a+I*a*tan(d*x+c)),x)
Output:
(sqrt(e)*int((sqrt(sec(c + d*x))*sec(c + d*x)**2)/(tan(c + d*x)*i + 1),x)* e**2)/a