Integrand size = 26, antiderivative size = 96 \[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}+\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}} \] Output:
-2/5*I*a/d/(e*sec(d*x+c))^(5/2)+6/5*a*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2) )/d/e^2/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)+2/5*a*sin(d*x+c)/d/e/(e*sec( d*x+c))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=-\frac {a \left (2+2 \cos (2 (c+d x))-2 \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )-3 i \sin (2 (c+d x))\right ) (-i+\tan (c+d x))}{5 d e^2 \sqrt {e \sec (c+d x)}} \] Input:
Integrate[(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(5/2),x]
Output:
-1/5*(a*(2 + 2*Cos[2*(c + d*x)] - 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeo metric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - (3*I)*Sin[2*(c + d*x)])*( -I + Tan[c + d*x]))/(d*e^2*Sqrt[e*Sec[c + d*x]])
Time = 0.44 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3967, 3042, 4256, 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.
| \(\displaystyle \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle a \int \frac {1}{(e \sec (c+d x))^{5/2}}dx-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle a \left (\frac {3 \int \frac {1}{\sqrt {e \sec (c+d x)}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}\right )-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}\right )-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle a \left (\frac {3 \int \sqrt {\cos (c+d x)}dx}{5 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}\right )-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}\right )-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle a \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}\right )-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}\) |
Input:
Int[(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(5/2),x]
Output:
(((-2*I)/5)*a)/(d*(e*Sec[c + d*x])^(5/2)) + a*((6*EllipticE[(c + d*x)/2, 2 ])/(5*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*Sin[c + d*x])/(5 *d*e*(e*Sec[c + d*x])^(3/2)))
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[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (83 ) = 166\).
Time = 10.00 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.19
| method | result | size |
| parts | \(\frac {2 a \left (\sin \left (d x +c \right ) \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )+3\right )-3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{2}}-\frac {2 i a}{5 d \left (e \sec \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(210\) |
| default | \(\frac {2 a \left (\sin \left (d x +c \right ) \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )+3\right )+i \left (-\cos \left (d x +c \right )^{3}-\cos \left (d x +c \right )^{2}\right )+3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )-3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{2}}\) | \(216\) |
| risch | \(-\frac {i \left ({\mathrm e}^{2 i \left (d x +c \right )}+7\right ) a \sqrt {2}}{10 d \,e^{2} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {3 i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) a \sqrt {2}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{5 d \,e^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(320\) |
Input:
int((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
2/5*a/d/(cos(d*x+c)+1)/(e*sec(d*x+c))^(1/2)/e^2*(sin(d*x+c)*(cos(d*x+c)^2+ cos(d*x+c)+3)-3*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/ 2)*(cos(d*x+c)+2+sec(d*x+c))*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)+3*I*(1 /(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+2+sec (d*x+c))*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I))-2/5*I*a/d/(e*sec(d*x+c))^ (5/2)
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.14 \[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=\frac {{\left (12 i \, \sqrt {2} a \sqrt {e} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (-i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i \, a\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{10 \, d e^{3}} \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(5/2),x, algorithm="fricas")
Output:
1/10*(12*I*sqrt(2)*a*sqrt(e)*e^(I*d*x + I*c)*weierstrassZeta(-4, 0, weiers trassPInverse(-4, 0, e^(I*d*x + I*c))) + sqrt(2)*(-I*a*e^(4*I*d*x + 4*I*c) + 4*I*a*e^(2*I*d*x + 2*I*c) + 5*I*a)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^ (1/2*I*d*x + 1/2*I*c))*e^(-I*d*x - I*c)/(d*e^3)
\[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=i a \left (\int \left (- \frac {i}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))**(5/2),x)
Output:
I*a*(Integral(-I/(e*sec(c + d*x))**(5/2), x) + Integral(tan(c + d*x)/(e*se c(c + d*x))**(5/2), x))
\[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((I*a*tan(d*x + c) + a)/(e*sec(d*x + c))^(5/2), x)
\[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=\int { \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(5/2),x, algorithm="giac")
Output:
integrate((I*a*tan(d*x + c) + a)/(e*sec(d*x + c))^(5/2), x)
Timed out. \[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=\int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:
int((a + a*tan(c + d*x)*1i)/(e/cos(c + d*x))^(5/2),x)
Output:
int((a + a*tan(c + d*x)*1i)/(e/cos(c + d*x))^(5/2), x)
\[ \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, a \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \tan \left (d x +c \right )}{\sec \left (d x +c \right )^{3}}d x \right ) i \right )}{e^{3}} \] Input:
int((a+I*a*tan(d*x+c))/(e*sec(d*x+c))^(5/2),x)
Output:
(sqrt(e)*a*(int(sqrt(sec(c + d*x))/sec(c + d*x)**3,x) + int((sqrt(sec(c + d*x))*tan(c + d*x))/sec(c + d*x)**3,x)*i))/e**3