3.188 \(\int \frac{e^{3 i \tan ^{-1}(a+b x)}}{x^4} \, dx\)

Optimal. Leaf size=338 \[ \frac{\left (-2 a^2+51 i a+52\right ) b^3 \sqrt{i a+i b x+1}}{6 (-a+i) (a+i)^4 \sqrt{-i a-i b x+1}}-\frac{\left (-6 i a^2-18 a+11 i\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right )}{(-a+i)^{3/2} (a+i)^{9/2}}+\frac{(19+16 i a) b^2 \sqrt{i a+i b x+1}}{6 (-a+i) (a+i)^3 x \sqrt{-i a-i b x+1}}+\frac{7 i b \sqrt{i a+i b x+1}}{6 (a+i)^2 x^2 \sqrt{-i a-i b x+1}}-\frac{(-a+i) \sqrt{i a+i b x+1}}{3 (a+i) x^3 \sqrt{-i a-i b x+1}} \]

[Out]

((52 + (51*I)*a - 2*a^2)*b^3*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^4*Sqrt[1 - I*a - I*b*x]) - ((I - a)*Sqr
t[1 + I*a + I*b*x])/(3*(I + a)*x^3*Sqrt[1 - I*a - I*b*x]) + (((7*I)/6)*b*Sqrt[1 + I*a + I*b*x])/((I + a)^2*x^2
*Sqrt[1 - I*a - I*b*x]) + ((19 + (16*I)*a)*b^2*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^3*x*Sqrt[1 - I*a - I*
b*x]) - ((11*I - 18*a - (6*I)*a^2)*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a -
I*b*x])])/((I - a)^(3/2)*(I + a)^(9/2))

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Rubi [A]  time = 0.269927, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5095, 98, 151, 152, 12, 93, 208} \[ \frac{\left (-2 a^2+51 i a+52\right ) b^3 \sqrt{i a+i b x+1}}{6 (-a+i) (a+i)^4 \sqrt{-i a-i b x+1}}-\frac{\left (-6 i a^2-18 a+11 i\right ) b^3 \tanh ^{-1}\left (\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right )}{(-a+i)^{3/2} (a+i)^{9/2}}+\frac{(19+16 i a) b^2 \sqrt{i a+i b x+1}}{6 (-a+i) (a+i)^3 x \sqrt{-i a-i b x+1}}+\frac{7 i b \sqrt{i a+i b x+1}}{6 (a+i)^2 x^2 \sqrt{-i a-i b x+1}}-\frac{(-a+i) \sqrt{i a+i b x+1}}{3 (a+i) x^3 \sqrt{-i a-i b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])/x^4,x]

[Out]

((52 + (51*I)*a - 2*a^2)*b^3*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^4*Sqrt[1 - I*a - I*b*x]) - ((I - a)*Sqr
t[1 + I*a + I*b*x])/(3*(I + a)*x^3*Sqrt[1 - I*a - I*b*x]) + (((7*I)/6)*b*Sqrt[1 + I*a + I*b*x])/((I + a)^2*x^2
*Sqrt[1 - I*a - I*b*x]) + ((19 + (16*I)*a)*b^2*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^3*x*Sqrt[1 - I*a - I*
b*x]) - ((11*I - 18*a - (6*I)*a^2)*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a -
I*b*x])])/((I - a)^(3/2)*(I + a)^(9/2))

Rule 5095

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]

Rule 98

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 151

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] && LtQ[m, -1] && IntegerQ[m]

Rule 152

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] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

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

Rule 93

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 208

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

Rubi steps

\begin{align*} \int \frac{e^{3 i \tan ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac{(1+i a+i b x)^{3/2}}{x^4 (1-i a-i b x)^{3/2}} \, dx\\ &=-\frac{(i-a) \sqrt{1+i a+i b x}}{3 (i+a) x^3 \sqrt{1-i a-i b x}}-\frac{\int \frac{-7 (i-a) b+6 b^2 x}{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}} \, dx}{3 (1-i a)}\\ &=-\frac{(i-a) \sqrt{1+i a+i b x}}{3 (i+a) x^3 \sqrt{1-i a-i b x}}+\frac{7 i b \sqrt{1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt{1-i a-i b x}}+\frac{\int \frac{-\left (19+35 i a-16 a^2\right ) b^2-14 (i-a) b^3 x}{x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}} \, dx}{6 (1-i a) \left (1+a^2\right )}\\ &=-\frac{(i-a) \sqrt{1+i a+i b x}}{3 (i+a) x^3 \sqrt{1-i a-i b x}}+\frac{7 i b \sqrt{1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt{1-i a-i b x}}+\frac{(19 i-16 a) b^2 \sqrt{1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt{1-i a-i b x}}-\frac{\int \frac{3 (i-a) \left (11+18 i a-6 a^2\right ) b^3-\left (19+35 i a-16 a^2\right ) b^4 x}{x (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}} \, dx}{6 (1-i a) \left (1+a^2\right )^2}\\ &=\frac{\left (52+51 i a-2 a^2\right ) b^3 \sqrt{1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt{1-i a-i b x}}-\frac{(i-a) \sqrt{1+i a+i b x}}{3 (i+a) x^3 \sqrt{1-i a-i b x}}+\frac{7 i b \sqrt{1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt{1-i a-i b x}}+\frac{(19 i-16 a) b^2 \sqrt{1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt{1-i a-i b x}}+\frac{i \int \frac{3 \left (11+29 i a-24 a^2-6 i a^3\right ) b^4}{x \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{6 (i-a)^2 (i+a)^4 b}\\ &=\frac{\left (52+51 i a-2 a^2\right ) b^3 \sqrt{1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt{1-i a-i b x}}-\frac{(i-a) \sqrt{1+i a+i b x}}{3 (i+a) x^3 \sqrt{1-i a-i b x}}+\frac{7 i b \sqrt{1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt{1-i a-i b x}}+\frac{(19 i-16 a) b^2 \sqrt{1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt{1-i a-i b x}}+\frac{\left (\left (11+18 i a-6 a^2\right ) b^3\right ) \int \frac{1}{x \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{2 (i-a) (i+a)^4}\\ &=\frac{\left (52+51 i a-2 a^2\right ) b^3 \sqrt{1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt{1-i a-i b x}}-\frac{(i-a) \sqrt{1+i a+i b x}}{3 (i+a) x^3 \sqrt{1-i a-i b x}}+\frac{7 i b \sqrt{1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt{1-i a-i b x}}+\frac{(19 i-16 a) b^2 \sqrt{1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt{1-i a-i b x}}+\frac{\left (\left (11+18 i a-6 a^2\right ) b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac{\sqrt{1+i a+i b x}}{\sqrt{1-i a-i b x}}\right )}{(i-a) (i+a)^4}\\ &=\frac{\left (52+51 i a-2 a^2\right ) b^3 \sqrt{1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt{1-i a-i b x}}-\frac{(i-a) \sqrt{1+i a+i b x}}{3 (i+a) x^3 \sqrt{1-i a-i b x}}+\frac{7 i b \sqrt{1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt{1-i a-i b x}}+\frac{(19 i-16 a) b^2 \sqrt{1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt{1-i a-i b x}}+\frac{\left (18 a-i \left (11-6 a^2\right )\right ) b^3 \tanh ^{-1}\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)^{3/2} (i+a)^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.420622, size = 277, normalized size = 0.82 \[ -\frac{-i \left (6 a^2-18 i a-11\right ) b^2 x^2 \left (i \sqrt{-1+i a} \sqrt{i a+i b x+1} \left (a^2+a b x-5 i b x+1\right )+6 \sqrt{1+i a} b x \sqrt{-i (a+b x+i)} \tan ^{-1}\left (\frac{\sqrt{-i (a+b x+i)}}{\sqrt{\frac{a+i}{a-i}} \sqrt{i a+i b x+1}}\right )\right )+2 (-1+i a)^{3/2} (1+i a) (a+i)^2 (i a+i b x+1)^{5/2}+(-4 a+3 i) (-1+i a)^{5/2} b x (i a+i b x+1)^{5/2}}{6 (-1+i a)^{5/2} \left (a^2+1\right )^2 x^3 \sqrt{-i (a+b x+i)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((3*I)*ArcTan[a + b*x])/x^4,x]

[Out]

-(2*(-1 + I*a)^(3/2)*(1 + I*a)*(I + a)^2*(1 + I*a + I*b*x)^(5/2) + (3*I - 4*a)*(-1 + I*a)^(5/2)*b*x*(1 + I*a +
I*b*x)^(5/2) - I*(-11 - (18*I)*a + 6*a^2)*b^2*x^2*(I*Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x]*(1 + a^2 - (5*I)*b*
x + a*b*x) + 6*Sqrt[1 + I*a]*b*x*Sqrt[(-I)*(I + a + b*x)]*ArcTan[Sqrt[(-I)*(I + a + b*x)]/(Sqrt[(I + a)/(-I +
a)]*Sqrt[1 + I*a + I*b*x])]))/(6*(-1 + I*a)^(5/2)*(1 + a^2)^2*x^3*Sqrt[(-I)*(I + a + b*x)])

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Maple [B]  time = 0.119, size = 2624, normalized size = 7.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x)

[Out]

-1/3/(a^2+1)/x^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3/2*I*b/(a^2+1)/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+3*I*b^2/(a^2+
1)/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a-260/3*a^3*b^3/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+205/2*a^5*b^3/(a^2+
1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-56*a^3*b^3/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+62/3*b^3/(a^2+1)^2/(b^2*
x^2+2*a*b*x+a^2+1)^(1/2)*a-9/2*I*b^3/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+9/2*I*b^3/(a^2+1)^(5/2)*ln((2*a^2
+2+2*x*a*b+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)+35/2*a^3*b^3/(a^2+1)^(9/2)*ln((2*a^2+2+2*x*a*b+2*
(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)-15/2*a*b^3/(a^2+1)^(7/2)*ln((2*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*(
b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)+3*b^2/(a^2+1)/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+6*b^4/(a^2+1)/(b^2*x^2+2*a*b*x+
a^2+1)^(1/2)*x+6*b^3/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+I*b^3/(a^2+1)^(3/2)*ln((2*a^2+2+2*x*a*b+2*(a^2+1)
^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)-I*b^3/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/(a^2+1)/x^3/(b^2*x^2+2*
a*b*x+a^2+1)^(1/2)*a^2-105/2*a^7*b^3/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-105/2*a^5*b^3/(a^2+1)^(9/2)*ln((2
*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)+4/3*b^2/(a^2+1)^2/x/(b^2*x^2+2*a*b*x+a^2+1)^(
1/2)+8/3*b^4/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-56*a^2*b^4/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-19
*a^2*b^2/(a^2+1)^2/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+205/2*a^4*b^4/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x+3*b
/(a^2+1)/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a-110*I*a^4*b^3/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+125/3*I*a^6
*b^3/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+7*I*b^3/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2-27/2*I*b^3/(a^2
+1)^(5/2)*ln((2*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)*a^2-45*I*a^2*b^3/(a^2+1)^(7/2)
*ln((2*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)+45*I*a^2*b^3/(a^2+1)^3/(b^2*x^2+2*a*b*x
+a^2+1)^(1/2)+30*I*a^4*b^3/(a^2+1)^(7/2)*ln((2*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)
-187/6*I*a^4*b^3/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+41*I*a^2*b^3/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-
35/2*I*a^8*b^3/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-35/2*I*a^6*b^3/(a^2+1)^(9/2)*ln((2*a^2+2+2*x*a*b+2*(a^2
+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)+105/2*I*a^4*b^3/(a^2+1)^(9/2)*ln((2*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*
(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)+1/3*I/(a^2+1)/x^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^3-I/(a^2+1)/x^3/(b^2*x^2+2
*a*b*x+a^2+1)^(1/2)*a+70*I*a^6*b^3/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-105/2*I*a^4*b^3/(a^2+1)^4/(b^2*x^2+
2*a*b*x+a^2+1)^(1/2)-18*a*b^3/(a^2+1)^(5/2)*ln((2*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))
/x)-35/2*a^3*b^3/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+70*a^5*b^3/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+15
/2*a*b^3/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-35/6*a^2*b^2/(a^2+1)^3/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+35/2*a
^4*b^4/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-115/6*a^2*b^4/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x+3/2*I
*b/(a^2+1)/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2-53/6*I*a^3*b^2/(a^2+1)^2/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+23/2
*I*a*b^2/(a^2+1)^2/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+125/3*I*a^5*b^4/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-8
0*I*a^3*b^4/(a^2+1)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-7/6*I*a^4*b/(a^2+1)^2/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+
7/2*I*a^2*b/(a^2+1)^2/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+35/6*I*a^5*b^2/(a^2+1)^3/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/
2)-35/2*I*a^3*b^2/(a^2+1)^3/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-35/2*I*a^7*b^4/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(
1/2)*x+105/2*I*a^5*b^4/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x+7*I*b^4/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)
*x*a+55/2*I*a*b^4/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-187/6*I*a^3*b^4/(a^2+1)^2/(b^2*x^2+2*a*b*x+a^2+1)^
(1/2)*x+135/2*a^3*b^3/(a^2+1)^(7/2)*ln((2*a^2+2+2*x*a*b+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)-105/
2*a^6*b^4/(a^2+1)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-7/2*a^3*b/(a^2+1)^2/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+35/2
*a^4*b^2/(a^2+1)^3/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+7/6*a*b/(a^2+1)^2/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.62691, size = 2215, normalized size = 6.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x, algorithm="fricas")

[Out]

((2*I*a^2 + 51*a - 52*I)*b^4*x^4 + (2*I*a^3 + 49*a^2 - I*a + 52)*b^3*x^3 + sqrt((36*a^4 - 216*I*a^3 - 456*a^2
+ 396*I*a + 121)*b^6/(a^12 + 6*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 +
12*a^2 + 6*I*a - 1))*((3*a^5 + 9*I*a^4 - 6*a^3 + 6*I*a^2 - 9*a - 3*I)*b*x^4 + (3*a^6 + 12*I*a^5 - 15*a^4 - 15*
a^2 - 12*I*a + 3)*x^3)*log(-((6*a^2 - 18*I*a - 11)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(6*a^2 - 18*I*a -
11)*b^3 + (a^7 + 3*I*a^6 - a^5 + 5*I*a^4 - 5*a^3 + I*a^2 - 3*a - I)*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*
I*a + 121)*b^6/(a^12 + 6*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2
+ 6*I*a - 1)))/((6*a^2 - 18*I*a - 11)*b^3)) - sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*I*a + 121)*b^6/(a^12 +
6*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2 + 6*I*a - 1))*((3*a^5
+ 9*I*a^4 - 6*a^3 + 6*I*a^2 - 9*a - 3*I)*b*x^4 + (3*a^6 + 12*I*a^5 - 15*a^4 - 15*a^2 - 12*I*a + 3)*x^3)*log(-
((6*a^2 - 18*I*a - 11)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(6*a^2 - 18*I*a - 11)*b^3 - (a^7 + 3*I*a^6 -
a^5 + 5*I*a^4 - 5*a^3 + I*a^2 - 3*a - I)*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*I*a + 121)*b^6/(a^12 + 6*I*a
^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2 + 6*I*a - 1)))/((6*a^2 - 18
*I*a - 11)*b^3)) + ((2*I*a^2 + 51*a - 52*I)*b^3*x^3 + 2*I*a^5 + (16*a^2 - 3*I*a + 19)*b^2*x^2 - 2*a^4 + 4*I*a^
3 - (7*a^3 + 7*I*a^2 + 7*a + 7*I)*b*x - 4*a^2 + 2*I*a - 2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((6*a^5 + 18*I*a
^4 - 12*a^3 + 12*I*a^2 - 18*a - 6*I)*b*x^4 + (6*a^6 + 24*I*a^5 - 30*a^4 - 30*a^2 - 24*I*a + 6)*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))**3/(1+(b*x+a)**2)**(3/2)/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x, algorithm="giac")

[Out]

undef