\(\int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx\) [245]

Optimal result

Integrand size = 28, antiderivative size = 178 \[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {22 e^8 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^7 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^3 d}+\frac {22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^3 d}-\frac {4 i e^2 (e \sec (c+d x))^{11/2}}{3 a d (a+i a \tan (c+d x))^2} \]

[Out]

-22/21*I*e^4*(e*sec(d*x+c))^(7/2)/a^3/d+22/15*e^5*(e*sec(d*x+c))^(5/2)*sin(d*x+c)/a^3/d-22/5*e^8*(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)+22/5*e^7*sin(d*x+c)*(e*sec(d*x+c))^(1/2)/a^3/d-4/3*I*e^2*(e*sec(d*x+c))^(11/2)/a/d/(a+I*a*tan(d*x+c))^2

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3581, 3582, 3853, 3856, 2719} \[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {22 e^8 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 e^7 \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 a^3 d}+\frac {22 e^5 \sin (c+d x) (e \sec (c+d x))^{5/2}}{15 a^3 d}-\frac {22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}-\frac {4 i e^2 (e \sec (c+d x))^{11/2}}{3 a d (a+i a \tan (c+d x))^2} \]

[In]

Int[(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(-22*e^8*EllipticE[(c + d*x)/2, 2])/(5*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((22*I)/21)*e^4*(e*Se
c[c + d*x])^(7/2))/(a^3*d) + (22*e^7*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*d) + (22*e^5*(e*Sec[c + d*x])^(
5/2)*Sin[c + d*x])/(15*a^3*d) - (((4*I)/3)*e^2*(e*Sec[c + d*x])^(11/2))/(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))^{11/2}}{3 a d (a+i a \tan (c+d x))^2}+\frac {\left (11 e^2\right ) \int \frac {(e \sec (c+d x))^{11/2}}{a+i a \tan (c+d x)} \, dx}{3 a^2} \\ & = -\frac {22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}-\frac {4 i e^2 (e \sec (c+d x))^{11/2}}{3 a d (a+i a \tan (c+d x))^2}+\frac {\left (11 e^4\right ) \int (e \sec (c+d x))^{7/2} \, dx}{3 a^3} \\ & = -\frac {22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^3 d}-\frac {4 i e^2 (e \sec (c+d x))^{11/2}}{3 a d (a+i a \tan (c+d x))^2}+\frac {\left (11 e^6\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 a^3} \\ & = -\frac {22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^7 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^3 d}+\frac {22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^3 d}-\frac {4 i e^2 (e \sec (c+d x))^{11/2}}{3 a d (a+i a \tan (c+d x))^2}-\frac {\left (11 e^8\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a^3} \\ & = -\frac {22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^7 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^3 d}+\frac {22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^3 d}-\frac {4 i e^2 (e \sec (c+d x))^{11/2}}{3 a d (a+i a \tan (c+d x))^2}-\frac {\left (11 e^8\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = -\frac {22 e^8 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {22 i e^4 (e \sec (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^7 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^3 d}+\frac {22 e^5 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^3 d}-\frac {4 i e^2 (e \sec (c+d x))^{11/2}}{3 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 = 2.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.72 \[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {e^6 (e \sec (c+d x))^{3/2} \left (-556-868 \cos (2 (c+d x))+77 e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+203 i \sec (c+d x) \sin (3 (c+d x))+143 i \tan (c+d x)\right ) (-i+\tan (c+d x))}{210 a^3 d} \]

[In]

Integrate[(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^3,x]

[Out]

-1/210*(e^6*(e*Sec[c + d*x])^(3/2)*(-556 - 868*Cos[2*(c + d*x)] + (77*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeo
metric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + (203*I)*Sec[c + d*x]*Sin[3*(c + d*x)] +
(143*I)*Tan[c + d*x])*(-I + Tan[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. 453 vs. \(2 (179 ) = 358\).

Time = 9.28 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.55

method	result	size
default	\(-\frac {2 \sqrt {e \sec \left (d x +c \right )}\, e^{7} \left (231 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )-231 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )+462 i \cos \left (d x +c \right ) E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-462 i \cos \left (d x +c \right ) F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+231 i E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-231 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+140 i-231 \sin \left (d x +c \right )+140 i \sec \left (d x +c \right )+63 \tan \left (d x +c \right )-15 i \left (\sec ^{2}\left (d x +c \right )\right )+63 \sec \left (d x +c \right ) \tan \left (d x +c \right )-15 i \left (\sec ^{3}\left (d x +c \right )\right )\right )}{105 a^{3} d \left (\cos \left (d x +c \right )+1\right )}\)	\(454\)

[In]

int((e*sec(d*x+c))^(15/2)/(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-2/105/a^3/d*(e*sec(d*x+c))^(1/2)*e^7/(cos(d*x+c)+1)*(231*I*cos(d*x+c)^2*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I
)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)-231*I*cos(d*x+c)^2*EllipticF(I*(csc(d*x+c)-cot(d*
x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)+462*I*cos(d*x+c)*EllipticE(I*(csc(d*x+c)-c
ot(d*x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)-462*I*cos(d*x+c)*EllipticF(I*(csc(d*x
+c)-cot(d*x+c)),I)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)+231*I*(cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)-231*I*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2
)*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(1/(cos(d*x+c)+1))^(1/2)+140*I-231*sin(d*x+c)+140*I*sec(d*x+c)+63*tan
(d*x+c)-15*I*sec(d*x+c)^2+63*sec(d*x+c)*tan(d*x+c)-15*I*sec(d*x+c)^3)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.20 \[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (231 i \, e^{7} e^{\left (7 i \, d x + 7 i \, c\right )} + 847 i \, e^{7} e^{\left (5 i \, d x + 5 i \, c\right )} + 1133 i \, e^{7} e^{\left (3 i \, d x + 3 i \, c\right )} + 637 i \, e^{7} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 231 \, \sqrt {2} {\left (i \, e^{7} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, e^{7} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, e^{7} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{7}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{105 \, {\left (a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]

[In]

integrate((e*sec(d*x+c))^(15/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-2/105*(sqrt(2)*(231*I*e^7*e^(7*I*d*x + 7*I*c) + 847*I*e^7*e^(5*I*d*x + 5*I*c) + 1133*I*e^7*e^(3*I*d*x + 3*I*c
) + 637*I*e^7*e^(I*d*x + I*c))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 231*sqrt(2)*(I*e^7*
e^(6*I*d*x + 6*I*c) + 3*I*e^7*e^(4*I*d*x + 4*I*c) + 3*I*e^7*e^(2*I*d*x + 2*I*c) + I*e^7)*sqrt(e)*weierstrassZe
ta(-4, 0, weierstrassPInverse(-4, 0, e^(I*d*x + I*c))))/(a^3*d*e^(6*I*d*x + 6*I*c) + 3*a^3*d*e^(4*I*d*x + 4*I*
c) + 3*a^3*d*e^(2*I*d*x + 2*I*c) + a^3*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate((e*sec(d*x+c))**(15/2)/(a+I*a*tan(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((e*sec(d*x+c))^(15/2)/(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]

\[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {15}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((e*sec(d*x+c))^(15/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((e*sec(d*x + c))^(15/2)/(I*a*tan(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \sec (c+d x))^{15/2}}{(a+i a \tan (c+d x))^3} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{15/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]

[In]

int((e/cos(c + d*x))^(15/2)/(a + a*tan(c + d*x)*1i)^3,x)

[Out]

int((e/cos(c + d*x))^(15/2)/(a + a*tan(c + d*x)*1i)^3, x)