\(\int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx\) [679]

Optimal result

Integrand size = 30, antiderivative size = 512 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\frac {3 i \sqrt {a} e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {3 i \sqrt {a} e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {3 i \sqrt {a} e^{5/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {3 i \sqrt {a} e^{5/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}} \]

[Out]

3/8*I*e^(5/2)*arctan(1-2^(1/2)*e^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/a^(1/2)/(e*sec(d*x+c))^(1/2))*a^(1/2)/d/(e*cos
(d*x+c))^(5/2)/(e*sec(d*x+c))^(5/2)*2^(1/2)-3/8*I*e^(5/2)*arctan(1+2^(1/2)*e^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/a^
(1/2)/(e*sec(d*x+c))^(1/2))*a^(1/2)/d/(e*cos(d*x+c))^(5/2)/(e*sec(d*x+c))^(5/2)*2^(1/2)-3/16*I*e^(5/2)*ln(a-2^
(1/2)*a^(1/2)*e^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/(e*sec(d*x+c))^(1/2)+cos(d*x+c)*(a+I*a*tan(d*x+c)))*a^(1/2)/d/(
e*cos(d*x+c))^(5/2)/(e*sec(d*x+c))^(5/2)*2^(1/2)+3/16*I*e^(5/2)*ln(a+2^(1/2)*a^(1/2)*e^(1/2)*(a+I*a*tan(d*x+c)
)^(1/2)/(e*sec(d*x+c))^(1/2)+cos(d*x+c)*(a+I*a*tan(d*x+c)))*a^(1/2)/d/(e*cos(d*x+c))^(5/2)/(e*sec(d*x+c))^(5/2
)*2^(1/2)+1/2*I*a/d/(e*cos(d*x+c))^(5/2)/(a+I*a*tan(d*x+c))^(1/2)-3/4*I*cos(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/
d/(e*cos(d*x+c))^(5/2)

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3596, 3579, 3582, 3576, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\frac {3 i \sqrt {a} e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {3 i \sqrt {a} e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {3 i \sqrt {a} e^{5/2} \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {3 i \sqrt {a} e^{5/2} \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}}+\frac {i a}{2 d \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}} \]

[In]

Int[Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(5/2),x]

[Out]

(((3*I)/4)*Sqrt[a]*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x
]])])/(Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) - (((3*I)/4)*Sqrt[a]*e^(5/2)*ArcTan[1 + (Sqrt[
2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*S
ec[c + d*x])^(5/2)) - (((3*I)/8)*Sqrt[a]*e^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/
Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x]
)^(5/2)) + (((3*I)/8)*Sqrt[a]*e^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[
c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) +
((I/2)*a)/(d*(e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (((3*I)/4)*Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c
+ d*x]])/(d*(e*Cos[c + d*x])^(5/2))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 3576

Int[Sqrt[(d_.)*sec[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-4*b*(d^
2/f), Subst[Int[x^2/(a^2 + d^2*x^4), x], x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]], x] /; FreeQ[{a, b,
d, e, f}, x] && EqQ[a^2 + b^2, 0]

Rule 3579

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 - 1)/(f*(m + n - 1))), x] + Dist[a*((m + 2*n - 2)/(m + n - 1)), 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] &&
GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

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 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}& = \frac {\int (e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)} \, dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}+\frac {(3 a) \int \frac {(e \sec (c+d x))^{5/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{4 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}}+\frac {\left (3 e^2\right ) \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx}{8 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}}-\frac {\left (3 i a e^4\right ) \text {Subst}\left (\int \frac {x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}}+\frac {\left (3 i a e^3\right ) \text {Subst}\left (\int \frac {a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{4 d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (3 i a e^3\right ) \text {Subst}\left (\int \frac {a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{4 d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}}-\frac {\left (3 i a e^2\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (3 i a e^2\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (3 i \sqrt {a} e^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}+2 x}{-\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {\left (3 i \sqrt {a} e^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}-2 x}{-\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = -\frac {3 i \sqrt {a} e^{5/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {3 i \sqrt {a} e^{5/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}}-\frac {\left (3 i \sqrt {a} e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {\left (3 i \sqrt {a} e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {3 i \sqrt {a} e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {3 i \sqrt {a} e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{4 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}-\frac {3 i \sqrt {a} e^{5/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {3 i \sqrt {a} e^{5/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{8 \sqrt {2} d (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}+\frac {i a}{2 d (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}-\frac {3 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.71 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\frac {\sqrt {\cos (c+d x)} \left (\frac {3 i e^{-\frac {1}{2} i (2 c+5 d x)} \left (-e^{-2 i c}\right )^{3/4} \left (1+e^{2 i (c+d x)}\right )^2 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (\arctan \left (\frac {e^{\frac {i d x}{2}}}{\sqrt [4]{-e^{-2 i c}}}\right )-\text {arctanh}\left (\frac {e^{\frac {i d x}{2}}}{\sqrt [4]{-e^{-2 i c}}}\right )\right )}{4 \sqrt {2}}-3 i \cos ^{\frac {3}{2}}(c+d x)+2 \sqrt {\cos (c+d x)} (i \cos (c+d x)+\sin (c+d x))\right ) \sqrt {a+i a \tan (c+d x)}}{4 d (e \cos (c+d x))^{5/2}} \]

[In]

Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(5/2),x]

[Out]

(Sqrt[Cos[c + d*x]]*((((3*I)/4)*(-E^((-2*I)*c))^(3/4)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[(1 + E^((2*I)*(c + d*x)
))/E^(I*(c + d*x))]*(ArcTan[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)] - ArcTanh[E^((I/2)*d*x)/(-E^((-2*I)*c))^(1/4)
]))/(Sqrt[2]*E^((I/2)*(2*c + 5*d*x))) - (3*I)*Cos[c + d*x]^(3/2) + 2*Sqrt[Cos[c + d*x]]*(I*Cos[c + d*x] + Sin[
c + d*x]))*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*(e*Cos[c + d*x])^(5/2))

Maple [A] (verified)

Time = 10.12 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (3 i \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )-i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+2 i \tan \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+3 \,\operatorname {arctanh}\left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+3 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+3 \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+2 i \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+\sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+2 \tan \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-2 \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\right )}{d \left (i \cos \left (d x +c \right )+i-\sin \left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, e^{2} \sqrt {e \cos \left (d x +c \right )}}\)	\(359\)

[In]

int((a+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

(1/8+1/8*I)/d*(a*(1+I*tan(d*x+c)))^(1/2)/(I*cos(d*x+c)+I-sin(d*x+c))/(1/(cos(d*x+c)+1))^(1/2)/e^2/(e*cos(d*x+c
))^(1/2)*(3*I*cos(d*x+c)*arctanh(1/2*(cos(d*x+c)+sin(d*x+c)+1)/(cos(d*x+c)+1)/(1/(cos(d*x+c)+1))^(1/2))-3*I*co
s(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)+3*I*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-I*(1/(cos(d*x+c)+1))^(1/2)+2*I*tan(d
*x+c)*(1/(cos(d*x+c)+1))^(1/2)+3*arctanh(1/2*(-cos(d*x+c)+sin(d*x+c)-1)/(cos(d*x+c)+1)/(1/(cos(d*x+c)+1))^(1/2
))*cos(d*x+c)+3*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+3*sin(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)+2*I*sec(d*x+c)*(1/(c
os(d*x+c)+1))^(1/2)+(1/(cos(d*x+c)+1))^(1/2)+2*tan(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)-2*sec(d*x+c)*(1/(cos(d*x+c)
+1))^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\frac {\sqrt {2} \sqrt {\frac {1}{2}} \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}} {\left (-3 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - {\left (d e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3}\right )} \sqrt {\frac {9 i \, a}{16 \, d^{2} e^{5}}} \log \left (\frac {4}{3} i \, d e^{3} \sqrt {\frac {9 i \, a}{16 \, d^{2} e^{5}}} + \sqrt {2} \sqrt {\frac {1}{2}} \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 {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + {\left (d e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3}\right )} \sqrt {\frac {9 i \, a}{16 \, d^{2} e^{5}}} \log \left (-\frac {4}{3} i \, d e^{3} \sqrt {\frac {9 i \, a}{16 \, d^{2} e^{5}}} + \sqrt {2} \sqrt {\frac {1}{2}} \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 {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) - {\left (d e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3}\right )} \sqrt {-\frac {9 i \, a}{16 \, d^{2} e^{5}}} \log \left (\frac {4}{3} i \, d e^{3} \sqrt {-\frac {9 i \, a}{16 \, d^{2} e^{5}}} + \sqrt {2} \sqrt {\frac {1}{2}} \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 {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + {\left (d e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3}\right )} \sqrt {-\frac {9 i \, a}{16 \, d^{2} e^{5}}} \log \left (-\frac {4}{3} i \, d e^{3} \sqrt {-\frac {9 i \, a}{16 \, d^{2} e^{5}}} + \sqrt {2} \sqrt {\frac {1}{2}} \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 {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )}{2 \, {\left (d e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3}\right )}} \]

[In]

integrate((a+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-3*I*e^(4*I*d*x + 4*
I*c) + I*e^(2*I*d*x + 2*I*c))*e^(-1/2*I*d*x - 1/2*I*c) - (d*e^3*e^(4*I*d*x + 4*I*c) + 2*d*e^3*e^(2*I*d*x + 2*I
*c) + d*e^3)*sqrt(9/16*I*a/(d^2*e^5))*log(4/3*I*d*e^3*sqrt(9/16*I*a/(d^2*e^5)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2
*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c)) + (d*e^3*e^(4*I*d*x + 4*I*c)
+ 2*d*e^3*e^(2*I*d*x + 2*I*c) + d*e^3)*sqrt(9/16*I*a/(d^2*e^5))*log(-4/3*I*d*e^3*sqrt(9/16*I*a/(d^2*e^5)) + sq
rt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c)) -
(d*e^3*e^(4*I*d*x + 4*I*c) + 2*d*e^3*e^(2*I*d*x + 2*I*c) + d*e^3)*sqrt(-9/16*I*a/(d^2*e^5))*log(4/3*I*d*e^3*sq
rt(-9/16*I*a/(d^2*e^5)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*
e^(-1/2*I*d*x - 1/2*I*c)) + (d*e^3*e^(4*I*d*x + 4*I*c) + 2*d*e^3*e^(2*I*d*x + 2*I*c) + d*e^3)*sqrt(-9/16*I*a/(
d^2*e^5))*log(-4/3*I*d*e^3*sqrt(-9/16*I*a/(d^2*e^5)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(
a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c)))/(d*e^3*e^(4*I*d*x + 4*I*c) + 2*d*e^3*e^(2*I*d*x + 2*I*
c) + d*e^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(d*x+c))**(1/2)/(e*cos(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2249 vs. \(2 (384) = 768\).

Time = 0.57 (sec) , antiderivative size = 2249, normalized size of antiderivative = 4.39 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-32*(6*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + I*sqrt(2)*sin(4*d*x + 4*c) + 2*I*sqrt(2)*sin(2
*d*x + 2*c) + sqrt(2))*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, sqrt(2)*sin(1
/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 6*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c
) + I*sqrt(2)*sin(4*d*x + 4*c) + 2*I*sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) + 1, -sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 6*(s
qrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + I*sqrt(2)*sin(4*d*x + 4*c) + 2*I*sqrt(2)*sin(2*d*x + 2*
c) + sqrt(2))*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1, sqrt(2)*sin(1/4*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 6*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + I*sqr
t(2)*sin(4*d*x + 4*c) + 2*I*sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*
c), cos(2*d*x + 2*c))) - 1, -sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 6*(-I*sqrt(2)
*cos(4*d*x + 4*c) - 2*I*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c) - I*s
qrt(2))*arctan2(sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c
), cos(2*d*x + 2*c))), sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*
x + 2*c), cos(2*d*x + 2*c))) + 1) + 6*(I*sqrt(2)*cos(4*d*x + 4*c) + 2*I*sqrt(2)*cos(2*d*x + 2*c) - sqrt(2)*sin
(4*d*x + 4*c) - 2*sqrt(2)*sin(2*d*x + 2*c) + I*sqrt(2))*arctan2(-sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))), -sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 3*(sqrt(2)*cos(4*d*x + 4
*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + I*sqrt(2)*sin(4*d*x + 4*c) + 2*I*sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*log(2*
sqrt(2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c
))) + 2*(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1)*cos(1/2*arctan2(sin(2*d*x + 2*c), c
os(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*cos(1/4*arctan2(sin(2*d*x + 2*c
), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 3*(sqrt(
2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + I*sqrt(2)*sin(4*d*x + 4*c) + 2*I*sqrt(2)*sin(2*d*x + 2*c) +
sqrt(2))*log(-2*sqrt(2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))*sin(1/4*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c))) - 2*(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1)*cos(1/2*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*cos(1/4*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))
) + 1) + 3*(I*sqrt(2)*cos(4*d*x + 4*c) + 2*I*sqrt(2)*cos(2*d*x + 2*c) - sqrt(2)*sin(4*d*x + 4*c) - 2*sqrt(2)*s
in(2*d*x + 2*c) + I*sqrt(2))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(
sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sq
rt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 3*(-I*sqrt(2)*cos(4*d*x + 4*c) - 2*I*sqrt(2)
*cos(2*d*x + 2*c) + sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c) - I*sqrt(2))*log(2*cos(1/4*arctan2(s
in(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*c
os(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) + 2) + 3*(I*sqrt(2)*cos(4*d*x + 4*c) + 2*I*sqrt(2)*cos(2*d*x + 2*c) - sqrt(2)*sin(4*d*x + 4*c) - 2*sqrt
(2)*sin(2*d*x + 2*c) + I*sqrt(2))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) +
2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 3*(-I*sqrt(2)*cos(4*d*x + 4*c) - 2*I*sq
rt(2)*cos(2*d*x + 2*c) + sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c) - I*sqrt(2))*log(2*cos(1/4*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt
(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d
*x + 2*c))) + 2) + 48*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 16*cos(3/4*arctan2(sin(2*d*x + 2*
c), cos(2*d*x + 2*c))) + 48*I*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 16*I*sin(3/4*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c))))*sqrt(a)*sqrt(e)/((-1024*I*e^3*cos(4*d*x + 4*c) - 2048*I*e^3*cos(2*d*x + 2*c)
+ 1024*e^3*sin(4*d*x + 4*c) + 2048*e^3*sin(2*d*x + 2*c) - 1024*I*e^3)*d)

Giac [F]

\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(I*a*tan(d*x + c) + a)/(e*cos(d*x + c))^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((a + a*tan(c + d*x)*1i)^(1/2)/(e*cos(c + d*x))^(5/2),x)

[Out]

int((a + a*tan(c + d*x)*1i)^(1/2)/(e*cos(c + d*x))^(5/2), x)