Integrand size = 16, antiderivative size = 283 \[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1+i a) x^3}+\frac {(3-2 i a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (i-a)^2 (i+a) x^2}+\frac {\left (4-9 i a-2 a^2\right ) b^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1+i a) \left (1+a^2\right )^2 x}+\frac {\left (2 a+i \left (1-2 a^2\right )\right ) b^3 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{7/2} (i+a)^{5/2}} \] Output:
-1/3*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(1+I*a)/x^3+1/6*(3-2*I*a)*b*( 1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(I-a)^2/(I+a)/x^2+1/6*(4-9*I*a-2*a^ 2)*b^2*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(1+I*a)/(a^2+1)^2/x+(2*a+I* (-2*a^2+1))*b^3*arctanh((I+a)^(1/2)*(1+I*a+I*b*x)^(1/2)/(I-a)^(1/2)/(1-I*a -I*b*x)^(1/2))/(I-a)^(7/2)/(I+a)^(5/2)
Time = 0.49 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx=\frac {\frac {2 (1+i a) (i+a) (i+a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^3}+\frac {(1-4 i a) b (i+a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^2}-3 i \left (1-2 i a-2 a^2\right ) b^2 \left (\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{(-i+a) x}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{(-1-i a)^{3/2} \sqrt {-1+i a}}\right )}{6 \left (1+a^2\right )^2} \] Input:
Integrate[1/(E^(I*ArcTan[a + b*x])*x^4),x]
Output:
((2*(1 + I*a)*(I + a)*(I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/x^3 + ((1 - (4*I)*a)*b*(I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/x^2 - (3*I)*(1 - (2*I)*a - 2*a^2)*b^2*(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/((-I + a)*x) + (2*b*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/((-1 - I*a)^(3/2)*Sqrt[-1 + I*a])))/(6*(1 + a^2)^2)
Time = 0.65 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {5618, 110, 25, 27, 168, 25, 27, 168, 27, 104, 221}
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.
\(\displaystyle \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 5618 |
\(\displaystyle \int \frac {\sqrt {-i a-i b x+1}}{x^4 \sqrt {i a+i b x+1}}dx\) |
\(\Big \downarrow \) 110 |
\(\displaystyle \frac {\int -\frac {b (2 a+2 b x+3 i)}{x^3 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {b (2 a+2 b x+3 i)}{x^3 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \int \frac {2 a+2 b x+3 i}{x^3 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle -\frac {b \left (-\frac {\int -\frac {b \left (-2 a^2-9 i a-(2 a+3 i) b x+4\right )}{x^2 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{2 \left (a^2+1\right )}-\frac {(2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {b \left (\frac {\int \frac {b \left (-2 a^2-9 i a-(2 a+3 i) b x+4\right )}{x^2 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{2 \left (a^2+1\right )}-\frac {(2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \left (\frac {b \int \frac {-2 a^2-9 i a-(2 a+3 i) b x+4}{x^2 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{2 \left (a^2+1\right )}-\frac {(2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle -\frac {b \left (\frac {b \left (-\frac {\int \frac {3 \left (-2 i a^2+2 a+i\right ) b}{x \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{a^2+1}-\frac {\left (-2 a^2-9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \left (\frac {b \left (-\frac {3 \left (-2 i a^2+2 a+i\right ) b \int \frac {1}{x \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{a^2+1}-\frac {\left (-2 a^2-9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {b \left (\frac {b \left (-\frac {6 \left (-2 i a^2+2 a+i\right ) b \int \frac {1}{-i a+\frac {(1-i a) (i a+i b x+1)}{-i a-i b x+1}-1}d\frac {\sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}}{a^2+1}-\frac {\left (-2 a^2-9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {b \left (\frac {b \left (\frac {6 i \left (-2 i a^2+2 a+i\right ) b \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{\sqrt {-a+i} \sqrt {a+i} \left (a^2+1\right )}-\frac {\left (-2 a^2-9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1+i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1+i a) x^3}\) |
Input:
Int[1/(E^(I*ArcTan[a + b*x])*x^4),x]
Output:
-1/3*(Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/((1 + I*a)*x^3) - (b*(- 1/2*((3*I + 2*a)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/((1 + a^2)*x ^2) + (b*(-(((4 - (9*I)*a - 2*a^2)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I* b*x])/((1 + a^2)*x)) + ((6*I)*(I + 2*a - (2*I)*a^2)*b*ArcTanh[(Sqrt[I + a] *Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(Sqrt[I - a] *Sqrt[I + a]*(1 + a^2))))/(2*(1 + a^2))))/(3*(1 + I*a))
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[(d + e*x)^m*((1 - I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Time = 0.94 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {i \left (2 a^{2} b^{4} x^{4}+9 i a \,b^{4} x^{4}+2 a^{3} b^{3} x^{3}+15 i a^{2} b^{3} x^{3}+3 i a^{3} b^{2} x^{2}-4 b^{4} x^{4}+2 a^{5} b x -3 i a^{4} b x -10 a \,b^{3} x^{3}-3 i b^{3} x^{3}+2 a^{6}-2 a^{2} b^{2} x^{2}+3 i a \,b^{2} x^{2}+4 a^{3} b x -6 i a^{2} b x +6 a^{4}-2 b^{2} x^{2}+2 a b x -3 i b x +6 a^{2}+2\right )}{6 x^{3} \left (i+a \right )^{2} \left (-i+a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {b^{3} \left (2 a^{2}+2 i a -1\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {5}{2}} \left (-i+a \right )}\) | \(281\) |
default | \(\text {Expression too large to display}\) | \(1511\) |
Input:
int(1/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2)/x^4,x,method=_RETURNVERBOSE)
Output:
1/6*I*(-3*I*x*a^4*b+2*a^2*b^4*x^4+3*I*a*b^2*x^2+2*a^3*b^3*x^3+15*I*a^2*b^3 *x^3-4*b^4*x^4+3*I*a^3*b^2*x^2-3*I*b^3*x^3+2*a^5*b*x-10*a*b^3*x^3+9*I*a*b^ 4*x^4+2*a^6-2*a^2*b^2*x^2-3*I*b*x+4*a^3*b*x+6*a^4-2*b^2*x^2-6*I*a^2*b*x+2* a*b*x+6*a^2+2)/x^3/(I+a)^2/(-I+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/2*b^3* (2*I*a+2*a^2-1)/(a^2+1)^(5/2)/(-I+a)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*( b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (194) = 388\).
Time = 0.15 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.44 \[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2)/x^4,x, algorithm="fricas")
Output:
1/6*((2*I*a^2 - 9*a - 4*I)*b^3*x^3 - 3*sqrt((4*a^4 + 8*I*a^3 - 8*a^2 - 4*I *a + 1)*b^6/(a^12 - 2*I*a^11 + 4*a^10 - 10*I*a^9 + 5*a^8 - 20*I*a^7 - 20*I *a^5 - 5*a^4 - 10*I*a^3 - 4*a^2 - 2*I*a - 1))*(a^5 - I*a^4 + 2*a^3 - 2*I*a ^2 + a - I)*x^3*log(-((2*a^2 + 2*I*a - 1)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a^2 + 2*I*a - 1)*b^3 + (a^7 - I*a^6 + 3*a^5 - 3*I*a^4 + 3*a^3 - 3*I*a^2 + a - I)*sqrt((4*a^4 + 8*I*a^3 - 8*a^2 - 4*I*a + 1)*b^6/(a^12 - 2*I*a^11 + 4*a^10 - 10*I*a^9 + 5*a^8 - 20*I*a^7 - 20*I*a^5 - 5*a^4 - 10*I *a^3 - 4*a^2 - 2*I*a - 1)))/((2*a^2 + 2*I*a - 1)*b^3)) + 3*sqrt((4*a^4 + 8 *I*a^3 - 8*a^2 - 4*I*a + 1)*b^6/(a^12 - 2*I*a^11 + 4*a^10 - 10*I*a^9 + 5*a ^8 - 20*I*a^7 - 20*I*a^5 - 5*a^4 - 10*I*a^3 - 4*a^2 - 2*I*a - 1))*(a^5 - I *a^4 + 2*a^3 - 2*I*a^2 + a - I)*x^3*log(-((2*a^2 + 2*I*a - 1)*b^4*x - sqrt (b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a^2 + 2*I*a - 1)*b^3 - (a^7 - I*a^6 + 3*a ^5 - 3*I*a^4 + 3*a^3 - 3*I*a^2 + a - I)*sqrt((4*a^4 + 8*I*a^3 - 8*a^2 - 4* I*a + 1)*b^6/(a^12 - 2*I*a^11 + 4*a^10 - 10*I*a^9 + 5*a^8 - 20*I*a^7 - 20* I*a^5 - 5*a^4 - 10*I*a^3 - 4*a^2 - 2*I*a - 1)))/((2*a^2 + 2*I*a - 1)*b^3)) + ((2*I*a^2 - 9*a - 4*I)*b^2*x^2 + 2*I*a^4 + (-2*I*a^3 + 3*a^2 - 2*I*a + 3)*b*x + 4*I*a^2 + 2*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((a^5 - I*a^4 + 2*a^3 - 2*I*a^2 + a - I)*x^3)
\[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx=- i \int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a x^{4} + b x^{5} - i x^{4}}\, dx \] Input:
integrate(1/(1+I*(b*x+a))*(1+(b*x+a)**2)**(1/2)/x**4,x)
Output:
-I*Integral(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(a*x**4 + b*x**5 - I*x**4 ), x)
\[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx=\int { \frac {\sqrt {{\left (b x + a\right )}^{2} + 1}}{{\left (i \, b x + i \, a + 1\right )} x^{4}} \,d x } \] Input:
integrate(1/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2)/x^4,x, algorithm="maxima")
Output:
integrate(sqrt((b*x + a)^2 + 1)/((I*b*x + I*a + 1)*x^4), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (194) = 388\).
Time = 0.17 (sec) , antiderivative size = 884, normalized size of antiderivative = 3.12 \[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2)/x^4,x, algorithm="giac")
Output:
1/2*(2*a^2*b^3 + 2*I*a*b^3 - b^3)*log(abs(2*x*abs(b) - 2*sqrt((b*x + a)^2 + 1) - 2*sqrt(a^2 + 1))/abs(2*x*abs(b) - 2*sqrt((b*x + a)^2 + 1) + 2*sqrt( a^2 + 1)))/((a^5 - I*a^4 + 2*a^3 - 2*I*a^2 + a - I)*sqrt(a^2 + 1)) + 1/3*( -8*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^5*b^3 + 24*(-I*x*abs(b) + I*sq rt((b*x + a)^2 + 1))*a^7*b^3 - 24*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a ^6*b^2*abs(b) - 8*I*a^8*b^2*abs(b) + 6*(x*abs(b) - sqrt((b*x + a)^2 + 1))^ 5*a^2*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^4*b^3 + 18*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^6*b^3 - 12*(x*abs(b) - sqrt((b*x + a)^2 + 1))^ 2*a^5*b^2*abs(b) + 12*a^7*b^2*abs(b) + 6*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^5*a*b^3 - 32*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^3*b^3 + 54*(-I*x *abs(b) + I*sqrt((b*x + a)^2 + 1))*a^5*b^3 - 60*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^4*b^2*abs(b) - 20*I*a^6*b^2*abs(b) - 3*(x*abs(b) - sqrt((b* x + a)^2 + 1))^5*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^2*b^3 + 3 9*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^4*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^3*b^2*abs(b) + 36*a^5*b^2*abs(b) - 24*I*(x*abs(b) - sqrt((b *x + a)^2 + 1))^3*a*b^3 + 36*(-I*x*abs(b) + I*sqrt((b*x + a)^2 + 1))*a^3*b ^3 - 48*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^2*b^2*abs(b) - 12*I*a^4*b ^2*abs(b) + 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^2*b^3 - 12*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a*b^2*abs(b) + 36*a^3*b^2*abs(b) + 6*(-I*x*abs(b ) + I*sqrt((b*x + a)^2 + 1))*a*b^3 - 12*I*(x*abs(b) - sqrt((b*x + a)^2 ...
Timed out. \[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^2+1}}{x^4\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )} \,d x \] Input:
int(((a + b*x)^2 + 1)^(1/2)/(x^4*(a*1i + b*x*1i + 1)),x)
Output:
int(((a + b*x)^2 + 1)^(1/2)/(x^4*(a*1i + b*x*1i + 1)), x)
\[ \int \frac {e^{-i \arctan (a+b x)}}{x^4} \, dx=\int \frac {\sqrt {1+\left (b x +a \right )^{2}}}{\left (1+i \left (b x +a \right )\right ) x^{4}}d x \] Input:
int(1/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2)/x^4,x)
Output:
int(1/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2)/x^4,x)