Integrand size = 26, antiderivative size = 60 \[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}+\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \] Output:
-2*I*a/d/(e*sec(d*x+c))^(1/2)+2*a*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/ cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.66 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.22 \[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=-\frac {4 i a e^{2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )}{3 d \sqrt {1+e^{2 i (c+d x)}} \sqrt {e \sec (c+d x)}} \] Input:
Integrate[(a + I*a*Tan[c + d*x])/Sqrt[e*Sec[c + d*x]],x]
Output:
(((-4*I)/3)*a*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2* I)*(c + d*x))])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[e*Sec[c + d*x]])
Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 3967, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle a \int \frac {1}{\sqrt {e \sec (c+d x)}}dx-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {a \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\) |
Input:
Int[(a + I*a*Tan[c + d*x])/Sqrt[e*Sec[c + d*x]],x]
Output:
((-2*I)*a)/(d*Sqrt[e*Sec[c + d*x]]) + (2*a*EllipticE[(c + d*x)/2, 2])/(d*S qrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (55 ) = 110\).
Time = 4.98 (sec) , antiderivative size = 194, normalized size of antiderivative = 3.23
| method | result | size |
| parts | \(\frac {2 a \left (i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (-\cos \left (d x +c \right )-2-\sec \left (d x +c \right )\right )+i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )+\sin \left (d x +c \right )\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}}-\frac {2 i a}{d \sqrt {e \sec \left (d x +c \right )}}\) | \(194\) |
| risch | \(-\frac {2 i a \sqrt {2}}{d \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) a \sqrt {2}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{d \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(301\) |
| default | \(\frac {a \left (i \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (-4 \cos \left (d x +c \right )-8-4 \sec \left (d x +c \right )\right )+i \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (4 \cos \left (d x +c \right )+8+4 \sec \left (d x +c \right )\right )+i \left (-4 \cos \left (d x +c \right )-4\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+i \ln \left (\frac {4 \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-2 \cos \left (d x +c \right )+2}{\cos \left (d x +c \right )+1}\right )-i \ln \left (\frac {2 \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-\cos \left (d x +c \right )+1}{\cos \left (d x +c \right )+1}\right )+4 \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sin \left (d x +c \right )\right )}{2 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(446\) |
Input:
int((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
2*a/d/(cos(d*x+c)+1)/(e*sec(d*x+c))^(1/2)*(I*(1/(cos(d*x+c)+1))^(1/2)*(cos (d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(-cos (d*x+c)-2-sec(d*x+c))+I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1 ))^(1/2)*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I)*(cos(d*x+c)+2+sec(d*x+c))+ sin(d*x+c))-2*I*a/d/(e*sec(d*x+c))^(1/2)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.45 \[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=\frac {2 i \, \sqrt {2} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}{d \sqrt {e}} \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(1/2),x, algorithm="fricas")
Output:
2*I*sqrt(2)*a*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c)))/(d*sqrt(e))
\[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=i a \left (\int \left (- \frac {i}{\sqrt {e \sec {\left (c + d x \right )}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\sqrt {e \sec {\left (c + d x \right )}}}\, dx\right ) \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))**(1/2),x)
Output:
I*a*(Integral(-I/sqrt(e*sec(c + d*x)), x) + Integral(tan(c + d*x)/sqrt(e*s ec(c + d*x)), x))
\[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\sqrt {e \sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate((I*a*tan(d*x + c) + a)/sqrt(e*sec(d*x + c)), x)
\[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\sqrt {e \sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate((I*a*tan(d*x + c) + a)/sqrt(e*sec(d*x + c)), x)
Timed out. \[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=\int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int((a + a*tan(c + d*x)*1i)/(e/cos(c + d*x))^(1/2),x)
Output:
int((a + a*tan(c + d*x)*1i)/(e/cos(c + d*x))^(1/2), x)
\[ \int \frac {a+i a \tan (c+d x)}{\sqrt {e \sec (c+d x)}} \, dx=\frac {\sqrt {e}\, a \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )}{\sec \left (d x +c \right )}d x \right ) i \right )}{e} \] Input:
int((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(1/2),x)
Output:
(sqrt(e)*a*(int(sqrt(sec(c + d*x))/sec(c + d*x),x) + int((sqrt(sec(c + d*x ))*tan(c + d*x))/sec(c + d*x),x)*i))/e