\(\int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx\) [220]

Optimal result

Integrand size = 28, antiderivative size = 156 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=-\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{77 d e^6}-\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}} \]

[Out]

-2/77*a^4*sin(d*x+c)/d/e^5/(e*sec(d*x+c))^(1/2)-2/77*a^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip
ticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*(e*sec(d*x+c))^(1/2)/d/e^6-4/11*I*a*(a+I*a*tan(d*x+c))^3/d/(
e*sec(d*x+c))^(11/2)+4/77*I*(a^4+I*a^4*tan(d*x+c))/d/e^2/(e*sec(d*x+c))^(7/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3577, 3854, 3856, 2720} \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=-\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{77 d e^6}-\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}} \]

[In]

Int[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(11/2),x]

[Out]

(-2*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(77*d*e^6) - (2*a^4*Sin[c + d*x])/(
77*d*e^5*Sqrt[e*Sec[c + d*x]]) - (((4*I)/11)*a*(a + I*a*Tan[c + d*x])^3)/(d*(e*Sec[c + d*x])^(11/2)) + (((4*I)
/77)*(a^4 + I*a^4*Tan[c + d*x]))/(d*e^2*(e*Sec[c + d*x])^(7/2))

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3577

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[2*b*(d
*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n - 1)/(f*m)), x] - Dist[b^2*((m + 2*n - 2)/(d^2*m)), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]

Rule 3854

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
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}-\frac {a^2 \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{7/2}} \, dx}{11 e^2} \\ & = -\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {\left (3 a^4\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{77 e^4} \\ & = -\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {a^4 \int \sqrt {e \sec (c+d x)} \, dx}{77 e^6} \\ & = -\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {\left (a^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 e^6} \\ & = -\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{77 d e^6}-\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.50 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=-\frac {a^4 \sqrt {e \sec (c+d x)} \left (37 i \cos (c+d x)+11 i \cos (3 (c+d x))-3 \sin (c+d x)+4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (3 (c+d x))-i \sin (3 (c+d x)))-3 \sin (3 (c+d x))\right ) (\cos (3 c+7 d x)+i \sin (3 c+7 d x))}{154 d e^6 (\cos (d x)+i \sin (d x))^4} \]

[In]

Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(11/2),x]

[Out]

-1/154*(a^4*Sqrt[e*Sec[c + d*x]]*((37*I)*Cos[c + d*x] + (11*I)*Cos[3*(c + d*x)] - 3*Sin[c + d*x] + 4*Sqrt[Cos[
c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)]) - 3*Sin[3*(c + d*x)])*(Cos[3*c + 7
*d*x] + I*Sin[3*c + 7*d*x]))/(d*e^6*(Cos[d*x] + I*Sin[d*x])^4)

Maple [A] (verified)

Time = 22.54 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.29

method	result	size
default	\(\frac {2 a^{4} \left (-56 i \left (\cos ^{5}\left (d x +c \right )\right )+56 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+44 i \left (\cos ^{3}\left (d x +c \right )\right )+i \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-\sin \left (d x +c \right )\right )}{77 e^{5} d \sqrt {e \sec \left (d x +c \right )}}\)	\(201\)
risch	\(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (7 \,{\mathrm e}^{4 i \left (d x +c \right )}+13 \,{\mathrm e}^{2 i \left (d x +c \right )}+4\right ) a^{4} \sqrt {2}}{154 d \,e^{5} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, F\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) a^{4} \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{77 d \sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}\, e^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\)	\(260\)
parts	\(-\frac {2 a^{4} \left (-7 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+15 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}}+15 i \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, 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}}-9 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-15 \sin \left (d x +c \right )\right )}{77 d \sqrt {e \sec \left (d x +c \right )}\, e^{5}}-\frac {2 a^{4} \left (-7 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+4 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}}+4 i \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, 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}}+13 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4 \sin \left (d x +c \right )\right )}{77 d \sqrt {e \sec \left (d x +c \right )}\, e^{5}}-\frac {8 i a^{4} \left (7 \left (\cos ^{5}\left (d x +c \right )\right )-11 \left (\cos ^{3}\left (d x +c \right )\right )\right )}{77 d \sqrt {e \sec \left (d x +c \right )}\, e^{5}}-\frac {8 i a^{4}}{11 d \left (e \sec \left (d x +c \right )\right )^{\frac {11}{2}}}+\frac {4 a^{4} \left (21 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+10 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}}+10 i \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, 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}}-6 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-10 \sin \left (d x +c \right )\right )}{77 d \sqrt {e \sec \left (d x +c \right )}\, e^{5}}\)	\(598\)

[In]

int((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(11/2),x,method=_RETURNVERBOSE)

[Out]

2/77*a^4/e^5/d/(e*sec(d*x+c))^(1/2)*(-56*I*cos(d*x+c)^5+56*sin(d*x+c)*cos(d*x+c)^4+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)+44*I*cos(d*x+c)^3+I*sec(d*x+c)*(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)-16*cos(d*x+c)^2*si
n(d*x+c)-sin(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 = 110, normalized size of antiderivative = 0.71 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=\frac {4 i \, \sqrt {2} a^{4} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {2} {\left (-7 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 20 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 17 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{154 \, d e^{6}} \]

[In]

integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(11/2),x, algorithm="fricas")

[Out]

1/154*(4*I*sqrt(2)*a^4*sqrt(e)*weierstrassPInverse(-4, 0, e^(I*d*x + I*c)) + sqrt(2)*(-7*I*a^4*e^(6*I*d*x + 6*
I*c) - 20*I*a^4*e^(4*I*d*x + 4*I*c) - 17*I*a^4*e^(2*I*d*x + 2*I*c) - 4*I*a^4)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1)
)*e^(1/2*I*d*x + 1/2*I*c))/(d*e^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(d*x+c))**4/(e*sec(d*x+c))**(11/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(11/2),x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)^4/(e*sec(d*x + c))^(11/2), x)

Giac [F]

\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(11/2),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^4/(e*sec(d*x + c))^(11/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]

[In]

int((a + a*tan(c + d*x)*1i)^4/(e/cos(c + d*x))^(11/2),x)

[Out]

int((a + a*tan(c + d*x)*1i)^4/(e/cos(c + d*x))^(11/2), x)