\(\int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx\) [673]

Optimal result

Integrand size = 30, antiderivative size = 179 \[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}+\frac {32 i a (e \cos (c+d x))^{7/2} \sec ^4(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {16 i (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d} \]

[Out]

12/35*I*a*(e*cos(d*x+c))^(7/2)*sec(d*x+c)^2/d/(a+I*a*tan(d*x+c))^(1/2)+32/35*I*a*(e*cos(d*x+c))^(7/2)*sec(d*x+
c)^4/d/(a+I*a*tan(d*x+c))^(1/2)-2/7*I*(e*cos(d*x+c))^(7/2)*(a+I*a*tan(d*x+c))^(1/2)/d-16/35*I*(e*cos(d*x+c))^(
7/2)*sec(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3596, 3578, 3583, 3569} \[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {2 i \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{7 d}+\frac {32 i a \sec ^4(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{35 d}+\frac {12 i a \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt {a+i a \tan (c+d x)}} \]

[In]

Int[(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(((12*I)/35)*a*(e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((32*I)/35)*a*(e*Cos[c
+ d*x])^(7/2)*Sec[c + d*x]^4)/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (((2*I)/7)*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a
*Tan[c + d*x]])/d - (((16*I)/35)*(e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/d

Rule 3569

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rule 3578

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*S
ec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(a*f*m)), x] + Dist[a*((m + n)/(m*d^2)), 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] && GtQ[n, 0] && LtQ[m, -
1] && IntegersQ[2*m, 2*n]

Rule 3583

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(b*f*(m + 2*n))), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]

Rule 3596

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*Co
s[e + f*x])^m*(d*Sec[e + f*x])^m, Int[(a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e,
f, m, n}, x] &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \left ((e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx \\ & = -\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (6 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx}{7 e^2} \\ & = \frac {12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (24 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{35 e^2} \\ & = \frac {12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {16 i (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}+\frac {\left (16 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{35 e^4} \\ & = \frac {12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}+\frac {32 i a (e \cos (c+d x))^{7/2} \sec ^4(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {16 i (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.45 \[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {e^3 \sqrt {e \cos (c+d x)} (35 i \cos (c+d x)+i \cos (3 (c+d x))+70 \sin (c+d x)+6 \sin (3 (c+d x))) \sqrt {a+i a \tan (c+d x)}}{70 d} \]

[In]

Integrate[(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(e^3*Sqrt[e*Cos[c + d*x]]*((35*I)*Cos[c + d*x] + I*Cos[3*(c + d*x)] + 70*Sin[c + d*x] + 6*Sin[3*(c + d*x)])*Sq
rt[a + I*a*Tan[c + d*x]])/(70*d)

Maple [A] (verified)

Time = 8.88 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.44

method	result	size
default	\(\frac {2 i \sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, e^{3} \left (-6 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\cos ^{3}\left (d x +c \right )-16 i \sin \left (d x +c \right )+8 \cos \left (d x +c \right )\right )}{35 d}\)	\(78\)

[In]

int((e*cos(d*x+c))^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/35*I/d*(e*cos(d*x+c))^(1/2)*(a*(1+I*tan(d*x+c)))^(1/2)*e^3*(-6*I*cos(d*x+c)^2*sin(d*x+c)+cos(d*x+c)^3-16*I*s
in(d*x+c)+8*cos(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.56 \[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {\sqrt {2} \sqrt {\frac {1}{2}} {\left (-5 i \, e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 35 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 105 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{3}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {5}{2} i \, d x - \frac {5}{2} i \, c\right )}}{140 \, d} \]

[In]

integrate((e*cos(d*x+c))^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/140*sqrt(2)*sqrt(1/2)*(-5*I*e^3*e^(6*I*d*x + 6*I*c) - 35*I*e^3*e^(4*I*d*x + 4*I*c) + 105*I*e^3*e^(2*I*d*x +
2*I*c) + 7*I*e^3)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-5/2*I*d*x - 5/2*I*c)/d

Sympy [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((e*cos(d*x+c))**(7/2)*(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13 \[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {{\left (7 i \, e^{3} \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 i \, e^{3} \cos \left (\frac {7}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) - 35 i \, e^{3} \cos \left (\frac {3}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 105 i \, e^{3} \cos \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 7 \, e^{3} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, e^{3} \sin \left (\frac {7}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 35 \, e^{3} \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 105 \, e^{3} \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )\right )} \sqrt {a} \sqrt {e}}{140 \, d} \]

[In]

integrate((e*cos(d*x+c))^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

1/140*(7*I*e^3*cos(5/2*d*x + 5/2*c) - 5*I*e^3*cos(7/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) - 3
5*I*e^3*cos(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 105*I*e^3*cos(1/5*arctan2(sin(5/2*d*x +
5/2*c), cos(5/2*d*x + 5/2*c))) + 7*e^3*sin(5/2*d*x + 5/2*c) + 5*e^3*sin(7/5*arctan2(sin(5/2*d*x + 5/2*c), cos
(5/2*d*x + 5/2*c))) + 35*e^3*sin(3/5*arctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))) + 105*e^3*sin(1/5*ar
ctan2(sin(5/2*d*x + 5/2*c), cos(5/2*d*x + 5/2*c))))*sqrt(a)*sqrt(e)/d

Giac [F]

\[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} \sqrt {i \, a \tan \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^(7/2)*sqrt(I*a*tan(d*x + c) + a), x)

Mupad [B] (verification not implemented)

Time = 6.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.54 \[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {e^3\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (\sin \left (c+d\,x\right )+\frac {3\,\sin \left (3\,c+3\,d\,x\right )}{35}+\frac {\cos \left (c+d\,x\right )\,1{}\mathrm {i}}{2}+\frac {\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{70}\right )}{d} \]

[In]

int((e*cos(c + d*x))^(7/2)*(a + a*tan(c + d*x)*1i)^(1/2),x)

[Out]

(e^3*(e*cos(c + d*x))^(1/2)*((a*(cos(2*c + 2*d*x) + sin(2*c + 2*d*x)*1i + 1))/(cos(2*c + 2*d*x) + 1))^(1/2)*((
cos(c + d*x)*1i)/2 + sin(c + d*x) + (cos(3*c + 3*d*x)*1i)/70 + (3*sin(3*c + 3*d*x))/35))/d