Integrand size = 16, antiderivative size = 170 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx=-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac {5 i a \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}+\frac {11 a^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 x}+\frac {3}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {3}{8} i a^3 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
-1/3*(1-I*a*x)^(1/4)*(1+I*a*x)^(3/4)/x^3+5/12*I*a*(1-I*a*x)^(1/4)*(1+I*a*x )^(3/4)/x^2+11/24*a^2*(1-I*a*x)^(1/4)*(1+I*a*x)^(3/4)/x+3/8*I*a^3*arctan(( 1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))-3/8*I*a^3*arctanh((1+I*a*x)^(1/4)/(1-I*a*x )^(1/4))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx=\frac {\sqrt [4]{1-i a x} \left (-8+2 i a x+a^2 x^2+11 i a^3 x^3-18 i a^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {i+a x}{i-a x}\right )\right )}{24 x^3 \sqrt [4]{1+i a x}} \]
((1 - I*a*x)^(1/4)*(-8 + (2*I)*a*x + a^2*x^2 + (11*I)*a^3*x^3 - (18*I)*a^3 *x^3*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(24*x^3*(1 + I* a*x)^(1/4))
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5585, 110, 27, 168, 27, 168, 27, 104, 25, 827, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {\sqrt [4]{1-i a x}}{x^4 \sqrt [4]{1+i a x}}dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {1}{3} \int -\frac {a (4 a x+5 i)}{2 x^3 (1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} a \int \frac {4 a x+5 i}{x^3 (1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {1}{6} a \left (-\frac {1}{2} \int -\frac {a (11-10 i a x)}{2 x^2 (1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \int \frac {11-10 i a x}{x^2 (1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \left (-\int \frac {9 i a}{2 x (1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}\right )-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \left (-\frac {9}{2} i a \int \frac {1}{x (1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}\right )-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \left (-18 i a \int -\frac {\sqrt {i a x+1}}{\sqrt {1-i a x} \left (1-\frac {i a x+1}{1-i a x}\right )}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}\right )-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \left (18 i a \int \frac {\sqrt {i a x+1}}{\sqrt {1-i a x} \left (1-\frac {i a x+1}{1-i a x}\right )}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}\right )-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \left (-18 i a \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}\right )-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}\right )-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \left (-18 i a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}\right )-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}\right )-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {1}{6} a \left (\frac {1}{4} a \left (-18 i a \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\right )-\frac {11 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}\right )-\frac {5 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{2 x^2}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}\) |
-1/3*((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/x^3 - (a*((((-5*I)/2)*(1 - I*a* x)^(1/4)*(1 + I*a*x)^(3/4))/x^2 + (a*((-11*(1 - I*a*x)^(1/4)*(1 + I*a*x)^( 3/4))/x - (18*I)*a*(ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]/2 - ArcTan h[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]/2)))/4))/6
3.1.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[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] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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] && !Intege rQ[(I*n - 1)/2]
\[\int \frac {1}{\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}\, x^{4}}d x\]
Time = 0.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx=\frac {-9 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 9 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 9 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 9 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) + 2 \, {\left (11 \, a^{2} x^{2} + 10 i \, a x - 8\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{3}} \]
1/48*(-9*I*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + 1) - 9*a^3*x^ 3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + I) + 9*a^3*x^3*log(sqrt(I*sqrt (a^2*x^2 + 1)/(a*x + I)) - I) + 9*I*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/( a*x + I)) - 1) + 2*(11*a^2*x^2 + 10*I*a*x - 8)*sqrt(a^2*x^2 + 1)*sqrt(I*sq rt(a^2*x^2 + 1)/(a*x + I)))/x^3
\[ \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx=\int \frac {1}{x^{4} \sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}\, dx \]
\[ \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx=\int { \frac {1}{x^{4} \sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}} \,d x } \]
Exception generated. \[ \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:The choice was done assuming 0=[0,0 ]Warning, replacing 0 by -46, a substitution variable should perhaps be pu rged.Warn
Timed out. \[ \int \frac {e^{-\frac {1}{2} i \arctan (a x)}}{x^4} \, dx=\int \frac {1}{x^4\,\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}} \,d x \]