\(\int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx\) [247]

Optimal result

Integrand size = 28, antiderivative size = 141 \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2} \]

[Out]

14/3*I*e^4*(e*sec(d*x+c))^(3/2)/a^3/d+14*e^6*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2
*d*x+1/2*c),2^(1/2))/a^3/d/cos(d*x+c)^(1/2)/(e*sec(d*x+c))^(1/2)-14*e^5*sin(d*x+c)*(e*sec(d*x+c))^(1/2)/a^3/d+
4*I*e^2*(e*sec(d*x+c))^(7/2)/a/d/(a+I*a*tan(d*x+c))^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3581, 3582, 3853, 3856, 2719} \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {14 e^5 \sin (c+d x) \sqrt {e \sec (c+d x)}}{a^3 d}+\frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2} \]

[In]

Int[(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(14*e^6*EllipticE[(c + d*x)/2, 2])/(a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((14*I)/3)*e^4*(e*Sec[c
+ d*x])^(3/2))/(a^3*d) - (14*e^5*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(a^3*d) + ((4*I)*e^2*(e*Sec[c + d*x])^(7/2
))/(a*d*(a + I*a*Tan[c + d*x])^2)

Rule 2719

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]

Rule 3581

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] - Dist[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] || (Integers
Q[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]

Rule 3582

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] + Dist[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] &&  !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 3853

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] + Dist[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]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}-\frac {\left (7 e^2\right ) \int \frac {(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx}{a^2} \\ & = \frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}-\frac {\left (7 e^4\right ) \int (e \sec (c+d x))^{3/2} \, dx}{a^3} \\ & = \frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}+\frac {\left (7 e^6\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{a^3} \\ & = \frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}+\frac {\left (7 e^6\right ) \int \sqrt {\cos (c+d x)} \, dx}{a^3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.85 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.66 \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\frac {i e^4 (e \sec (c+d x))^{3/2} \left (35+33 \cos (2 (c+d x))-7 \left (1+e^{2 i (c+d x)}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+9 i \sin (2 (c+d x))\right )}{3 a^3 d} \]

[In]

Integrate[(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^3,x]

[Out]

((I/3)*e^4*(e*Sec[c + d*x])^(3/2)*(35 + 33*Cos[2*(c + d*x)] - 7*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric
2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (9*I)*Sin[2*(c + d*x)]))/(a^3*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (152 ) = 304\).

Time = 10.35 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.15

method	result	size
default	\(\frac {2 \sqrt {e \sec \left (d x +c \right )}\, e^{5} \left (21 i \left (\cos ^{2}\left (d x +c \right )\right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-21 i \left (\cos ^{2}\left (d x +c \right )\right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+42 i \cos \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-42 i \cos \left (d x +c \right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+21 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-21 i E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+12 i \left (\cos ^{2}\left (d x +c \right )\right )+12 i \cos \left (d x +c \right )+12 \sin \left (d x +c \right ) \cos \left (d x +c \right )+i-9 \sin \left (d x +c \right )+i \sec \left (d x +c \right )\right )}{3 a^{3} d \left (\cos \left (d x +c \right )+1\right )}\)	\(444\)

[In]

int((e*sec(d*x+c))^(11/2)/(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

2/3/a^3/d*(e*sec(d*x+c))^(1/2)*e^5/(cos(d*x+c)+1)*(21*I*cos(d*x+c)^2*(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)-21*I*cos(d*x+c)^2*(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)+42*I*cos(d*x+c)*(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)-42*I*cos(d*x+c)*(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)+21*I*(1/(cos(d*x+c)+1))^(1/2)*Ellip
ticF(I*(-csc(d*x+c)+cot(d*x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-21*I*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(I
*(-csc(d*x+c)+cot(d*x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+12*I*cos(d*x+c)^2+12*I*cos(d*x+c)+12*sin(d*x+c)
*cos(d*x+c)+I-9*sin(d*x+c)+I*sec(d*x+c))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06 \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (-21 i \, e^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 35 i \, e^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - 12 i \, e^{5}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 21 \, \sqrt {2} {\left (-i \, e^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - i \, e^{5} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{3 \, {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )}} \]

[In]

integrate((e*sec(d*x+c))^(11/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-2/3*(sqrt(2)*(-21*I*e^5*e^(4*I*d*x + 4*I*c) - 35*I*e^5*e^(2*I*d*x + 2*I*c) - 12*I*e^5)*sqrt(e/(e^(2*I*d*x + 2
*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 21*sqrt(2)*(-I*e^5*e^(3*I*d*x + 3*I*c) - I*e^5*e^(I*d*x + I*c))*sqrt(e)*
weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))/(a^3*d*e^(3*I*d*x + 3*I*c) + a^3*d*e^(I*d
*x + I*c))

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate((e*sec(d*x+c))**(11/2)/(a+I*a*tan(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((e*sec(d*x+c))^(11/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((e*sec(d*x+c))^(11/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((e*sec(d*x + c))^(11/2)/(I*a*tan(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{11/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]

[In]

int((e/cos(c + d*x))^(11/2)/(a + a*tan(c + d*x)*1i)^3,x)

[Out]

int((e/cos(c + d*x))^(11/2)/(a + a*tan(c + d*x)*1i)^3, x)