\(\int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx\) [114]

Optimal result

Integrand size = 16, antiderivative size = 233 \[ \int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]

[Out]

2467/192*a^4*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)-1/4*(1-I*a*x)^(1/4)/x^4/(1+I*a*x)^(1/4)+17/24*I*a*(1-I*a*x)^(1/4)
/x^3/(1+I*a*x)^(1/4)+113/96*a^2*(1-I*a*x)^(1/4)/x^2/(1+I*a*x)^(1/4)-521/192*I*a^3*(1-I*a*x)^(1/4)/x/(1+I*a*x)^
(1/4)+475/64*a^4*arctan((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))-475/64*a^4*arctanh((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5170, 100, 156, 160, 12, 95, 304, 209, 212} \[ \int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}} \]

[In]

Int[1/(E^(((5*I)/2)*ArcTan[a*x])*x^5),x]

[Out]

(2467*a^4*(1 - I*a*x)^(1/4))/(192*(1 + I*a*x)^(1/4)) - (1 - I*a*x)^(1/4)/(4*x^4*(1 + I*a*x)^(1/4)) + (((17*I)/
24)*a*(1 - I*a*x)^(1/4))/(x^3*(1 + I*a*x)^(1/4)) + (113*a^2*(1 - I*a*x)^(1/4))/(96*x^2*(1 + I*a*x)^(1/4)) - ((
(521*I)/192)*a^3*(1 - I*a*x)^(1/4))/(x*(1 + I*a*x)^(1/4)) + (475*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4
)])/64 - (475*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)])/64

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 209

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

Rule 212

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

Rule 304

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

Rule 5170

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a*x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))
), x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[(I*n - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^{5/4}}{x^5 (1+i a x)^{5/4}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}-\frac {1}{4} \int \frac {\frac {17 i a}{2}+8 a^2 x}{x^4 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {1}{12} \int \frac {-\frac {113 a^2}{4}+\frac {51}{2} i a^3 x}{x^3 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {1}{24} \int \frac {-\frac {521 i a^3}{8}-\frac {113 a^4 x}{2}}{x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {1}{24} \int \frac {\frac {1425 a^4}{16}-\frac {521}{8} i a^5 x}{x (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx \\ & = \frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}-\frac {i \int \frac {1425 i a^5}{32 x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{12 a} \\ & = \frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {1}{128} \left (475 a^4\right ) \int \frac {1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx \\ & = \frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {1}{32} \left (475 a^4\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = \frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}-\frac {1}{64} \left (475 a^4\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{64} \left (475 a^4\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = \frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {475}{64} a^4 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.42 \[ \int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\frac {\sqrt [4]{1-i a x} \left (-48+136 i a x+226 a^2 x^2-521 i a^3 x^3+2467 a^4 x^4-2850 a^4 x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {i+a x}{i-a x}\right )\right )}{192 x^4 \sqrt [4]{1+i a x}} \]

[In]

Integrate[1/(E^(((5*I)/2)*ArcTan[a*x])*x^5),x]

[Out]

((1 - I*a*x)^(1/4)*(-48 + (136*I)*a*x + 226*a^2*x^2 - (521*I)*a^3*x^3 + 2467*a^4*x^4 - 2850*a^4*x^4*Hypergeome
tric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(192*x^4*(1 + I*a*x)^(1/4))

Maple [F]

\[\int \frac {1}{{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}} x^{5}}d x\]

[In]

int(1/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x)

[Out]

int(1/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=-\frac {2 \, {\left (2467 i \, a^{4} x^{4} + 521 \, a^{3} x^{3} + 226 i \, a^{2} x^{2} - 136 \, a x - 48 i\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1425 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 1425 \, {\left (-i \, a^{5} x^{5} - a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 1425 \, {\left (i \, a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 1425 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right )}{384 \, {\left (a x^{5} - i \, x^{4}\right )}} \]

[In]

integrate(1/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm="fricas")

[Out]

-1/384*(2*(2467*I*a^4*x^4 + 521*a^3*x^3 + 226*I*a^2*x^2 - 136*a*x - 48*I)*sqrt(a^2*x^2 + 1)*sqrt(I*sqrt(a^2*x^
2 + 1)/(a*x + I)) + 1425*(a^5*x^5 - I*a^4*x^4)*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + 1) + 1425*(-I*a^5*x^5
- a^4*x^4)*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + I) + 1425*(I*a^5*x^5 + a^4*x^4)*log(sqrt(I*sqrt(a^2*x^2
+ 1)/(a*x + I)) - I) - 1425*(a^5*x^5 - I*a^4*x^4)*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) - 1))/(a*x^5 - I*x^4
)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\text {Timed out} \]

[In]

integrate(1/((1+I*a*x)/(a**2*x**2+1)**(1/2))**(5/2)/x**5,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\int { \frac {1}{x^{5} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm="maxima")

[Out]

integrate(1/(x^5*((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(5/2)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {5}{2} i \arctan (a x)}}{x^5} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0]Warning, replacing 0 by 81, a substitution variable should perhaps be p
urged.Warni

Mupad [F(-1)]

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

[In]

int(1/(x^5*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(5/2)),x)

[Out]

int(1/(x^5*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(5/2)), x)