Integrand size = 16, antiderivative size = 203 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=-\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac {55}{8} i a^3 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {55}{8} i a^3 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
-287/24*I*a^3*(1+I*a*x)^(1/4)/(1-I*a*x)^(1/4)-1/3*(1+I*a*x)^(1/4)/x^3/(1-I *a*x)^(1/4)-13/12*I*a*(1+I*a*x)^(1/4)/x^2/(1-I*a*x)^(1/4)+61/24*a^2*(1+I*a *x)^(1/4)/x/(1-I*a*x)^(1/4)+55/8*I*a^3*arctan((1+I*a*x)^(1/4)/(1-I*a*x)^(1 /4))+55/8*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 = 106, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\frac {-8-34 i a x+87 a^2 x^2-226 i a^3 x^3+287 a^4 x^4+110 a^3 x^3 (i+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )}{24 x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \]
(-8 - (34*I)*a*x + 87*a^2*x^2 - (226*I)*a^3*x^3 + 287*a^4*x^4 + 110*a^3*x^ 3*(I + a*x)*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)])/(24*x^3*( 1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))
Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5585, 109, 27, 168, 27, 168, 27, 172, 27, 104, 756, 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.
| \(\displaystyle \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {(1+i a x)^{5/4}}{x^4 (1-i a x)^{5/4}}dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {1}{3} \int -\frac {a (13 i-12 a x)}{2 x^3 (1-i a x)^{5/4} (i a x+1)^{3/4}}dx-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} a \int \frac {13 i-12 a x}{x^3 (1-i a x)^{5/4} (i a x+1)^{3/4}}dx-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{2} \int \frac {a (52 i a x+61)}{2 x^2 (1-i a x)^{5/4} (i a x+1)^{3/4}}dx-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \int \frac {52 i a x+61}{x^2 (1-i a x)^{5/4} (i a x+1)^{3/4}}dx-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (-\int -\frac {a (165 i-122 a x)}{2 x (1-i a x)^{5/4} (i a x+1)^{3/4}}dx-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {1}{2} a \int \frac {165 i-122 a x}{x (1-i a x)^{5/4} (i a x+1)^{3/4}}dx-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {1}{2} a \left (\frac {2 i \int \frac {165 a}{2 x \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx}{a}+\frac {574 i \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {1}{2} a \left (165 i \int \frac {1}{x \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx+\frac {574 i \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {1}{2} a \left (660 i \int \frac {1}{\frac {i a x+1}{1-i a x}-1}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {574 i \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {1}{2} a \left (660 i \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}\right )+\frac {574 i \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {1}{2} a \left (660 i \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\right )+\frac {574 i \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {1}{2} a \left (660 i \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\right )+\frac {574 i \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {61 \sqrt [4]{1+i a x}}{x \sqrt [4]{1-i a x}}\right )-\frac {13 i \sqrt [4]{1+i a x}}{2 x^2 \sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}\) |
-1/3*(1 + I*a*x)^(1/4)/(x^3*(1 - I*a*x)^(1/4)) + (a*((((-13*I)/2)*(1 + I*a *x)^(1/4))/(x^2*(1 - I*a*x)^(1/4)) - (a*((-61*(1 + I*a*x)^(1/4))/(x*(1 - I *a*x)^(1/4)) + (a*(((574*I)*(1 + I*a*x)^(1/4))/(1 - I*a*x)^(1/4) + (660*I) *(-1/2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - ArcTanh[(1 + I*a*x)^( 1/4)/(1 - I*a*x)^(1/4)]/2)))/2))/4))/6
3.1.86.3.1 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[(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] + Simp[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])
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + Simp[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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) 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]
\[\int \frac {{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}}{x^{4}}d x\]
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\frac {165 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 165 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 165 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 165 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (287 i \, a^{3} x^{3} - 61 \, a^{2} x^{2} + 26 i \, a x + 8\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{3}} \]
1/48*(165*I*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + 1) - 165*a^3 *x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + I) + 165*a^3*x^3*log(sqrt(I *sqrt(a^2*x^2 + 1)/(a*x + I)) - I) - 165*I*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) - 1) - 2*(287*I*a^3*x^3 - 61*a^2*x^2 + 26*I*a*x + 8)*sqrt (I*sqrt(a^2*x^2 + 1)/(a*x + I)))/x^3
Timed out. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\int { \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}{x^{4}} \,d x } \]
Exception generated. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{\frac {5}{2} i \arctan (a x)}}{x^4} \, dx=\int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}}{x^4} \,d x \]