Integrand size = 28, antiderivative size = 115 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {6 e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )} \] Output:
6*e^4*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^2/d/cos(d*x+c)^(1/2)/(e*sec( d*x+c))^(1/2)-6*e^3*(e*sec(d*x+c))^(1/2)*sin(d*x+c)/a^2/d+4*I*e^2*(e*sec(d *x+c))^(3/2)/d/(a^2+I*a^2*tan(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.44 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {2 i e^3 e^{-i (c+d x)} \left (-1+3 \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (c+d x)}\right )\right ) \sqrt {e \sec (c+d x)}}{a^2 d} \] Input:
Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
((2*I)*e^3*(-1 + 3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1 /2, 3/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(a^2*d*E^(I*(c + d*x )))
Time = 0.53 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3981, 3042, 4255, 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.
\(\displaystyle \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2}dx\) |
\(\Big \downarrow \) 3981 |
\(\displaystyle -\frac {3 e^2 \int (e \sec (c+d x))^{3/2}dx}{a^2}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 e^2 \int \left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx}{a^2}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -\frac {3 e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-e^2 \int \frac {1}{\sqrt {e \sec (c+d x)}}dx\right )}{a^2}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-e^2 \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )}{a^2}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle -\frac {3 e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {e^2 \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )}{a^2}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {e^2 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )}{a^2}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {3 e^2 \left (\frac {2 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{d}-\frac {2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\right )}{a^2}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
Input:
Int[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
(-3*e^2*((-2*e^2*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*S ec[c + d*x]]) + (2*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/d))/a^2 + ((4*I)*e ^2*(e*Sec[c + d*x])^(3/2))/(d*(a^2 + I*a^2*Tan[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 _)])^(n_), x_Symbol] :> Simp[2*d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + 2*n))), x] - Simp[d^2*((m - 2)/(b^2*(m + 2*n))) Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[ {a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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. 218 vs. \(2 (107 ) = 214\).
Time = 4.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.90
method | result | size |
default | \(-\frac {2 \left (\left (1-2 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+i \cos \left (d x +c \right ) \left (-2 \cos \left (d x +c \right )-2\right )+i \left (3 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+3\right ) \operatorname {EllipticE}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+i \left (-3 \cos \left (d x +c \right )^{2}-6 \cos \left (d x +c \right )-3\right ) \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{3}}{a^{2} d \left (\cos \left (d x +c \right )+1\right )}\) | \(219\) |
Input:
int((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-2/a^2/d*((1-2*cos(d*x+c))*sin(d*x+c)+I*cos(d*x+c)*(-2*cos(d*x+c)-2)+I*(3* cos(d*x+c)^2+6*cos(d*x+c)+3)*EllipticE(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(co s(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+I*(-3*cos(d*x+c)^2-6* cos(d*x+c)-3)*EllipticF(I*(cot(d*x+c)-csc(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1 /2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(e*sec(d*x+c))^(1/2)*e^3/(cos(d*x+c )+1)
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (-3 i \, \sqrt {2} e^{\frac {7}{2}} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (-3 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, e^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2} d} \] Input:
integrate((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-2*(-3*I*sqrt(2)*e^(7/2)*e^(I*d*x + I*c)*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, e^(I*d*x + I*c))) + sqrt(2)*(-3*I*e^3*e^(2*I*d*x + 2*I*c) - 2*I*e^3)*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)/(a^2*d)
Timed out. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate((e*sec(d*x+c))**(7/2)/(a+I*a*tan(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate((e*sec(d*x + c))^(7/2)/(I*a*tan(d*x + c) + a)^2, x)
Timed out. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int((e/cos(c + d*x))^(7/2)/(a + a*tan(c + d*x)*1i)^2,x)
Output:
int((e/cos(c + d*x))^(7/2)/(a + a*tan(c + d*x)*1i)^2, x)
\[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x \right ) e^{3}}{a^{2}} \] Input:
int((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x)
Output:
( - sqrt(e)*int((sqrt(sec(c + d*x))*sec(c + d*x)**3)/(tan(c + d*x)**2 - 2* tan(c + d*x)*i - 1),x)*e**3)/a**2