3.3.15 \(\int \frac {e^{-3 i \text {ArcTan}(a+b x)}}{x^4} \, dx\) [215]

Optimal. Leaf size=339 \[ -\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \sqrt {1+i a+i b x}}-\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {(19-16 i a) b^2 \sqrt {1-i a-i b x}}{6 (i-a)^3 (i+a) x \sqrt {1+i a+i b x}}+\frac {\left (11 i+18 a-6 i a^2\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{9/2} (i+a)^{3/2}} \]

[Out]

(11*I+18*a-6*I*a^2)*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)^(9/2)/(
I+a)^(3/2)-1/6*(52-51*I*a-2*a^2)*b^3*(1-I*a-I*b*x)^(1/2)/(I-a)^4/(I+a)/(1+I*a+I*b*x)^(1/2)-1/3*(I+a)*(1-I*a-I*
b*x)^(1/2)/(I-a)/x^3/(1+I*a+I*b*x)^(1/2)-7/6*I*b*(1-I*a-I*b*x)^(1/2)/(I-a)^2/x^2/(1+I*a+I*b*x)^(1/2)+1/6*(19-1
6*I*a)*b^2*(1-I*a-I*b*x)^(1/2)/(I-a)^3/(I+a)/x/(1+I*a+I*b*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 339, normalized size of antiderivative = 1.00, 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 = {5203, 100, 156, 157, 12, 95, 214} \begin {gather*} -\frac {\left (-2 a^2-51 i a+52\right ) b^3 \sqrt {-i a-i b x+1}}{6 (-a+i)^4 (a+i) \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)^{9/2} (a+i)^{3/2}}+\frac {(19-16 i a) b^2 \sqrt {-i a-i b x+1}}{6 (-a+i)^3 (a+i) x \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}}-\frac {7 i b \sqrt {-i a-i b x+1}}{6 (-a+i)^2 x^2 \sqrt {i a+i b x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^((3*I)*ArcTan[a + b*x])*x^4),x]

[Out]

-1/6*((52 - (51*I)*a - 2*a^2)*b^3*Sqrt[1 - I*a - I*b*x])/((I - a)^4*(I + a)*Sqrt[1 + I*a + I*b*x]) - ((I + a)*
Sqrt[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)^3*(I + a)*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)^(9/2)*(I + a)^(3/2))

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 157

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 214

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]

Rule 5203

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]

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 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}} \, 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 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}} \, 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 (1+i a)^3 (i+a) 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 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}} \, 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)^4 (i+a) \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 (1+i a)^3 (i+a) 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)^4 (i+a)^2 b}\\ &=-\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \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 (1+i a)^3 (i+a) 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)^4 (i+a)}\\ &=-\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \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 (1+i a)^3 (i+a) x \sqrt {1+i a+i b x}}-\frac {\left (\left (11-18 i a-6 a^2\right ) b^3\right ) \text {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)^4 (i+a)}\\ &=-\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \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 (1+i a)^3 (i+a) x \sqrt {1+i a+i b x}}+\frac {\left (11 i+18 a-6 i a^2\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)^{9/2} (i+a)^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.29, size = 275, normalized size = 0.81 \begin {gather*} -\frac {-2 (-1-i a)^{7/2} (1-i a) (-i (i+a+b x))^{5/2}-(-1-i a)^{5/2} (3 i+4 a) b x (-i (i+a+b x))^{5/2}+i \left (-11+18 i a+6 a^2\right ) b^2 x^2 \left (-i \sqrt {-1-i a} \sqrt {-i (i+a+b x)} \left (1+a^2+5 i b x+a b x\right )-6 \sqrt {-1+i a} b x \sqrt {1+i a+i b x} \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )\right )}{6 (-1-i a)^{5/2} \left (1+a^2\right )^2 x^3 \sqrt {1+i a+i b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^((3*I)*ArcTan[a + b*x])*x^4),x]

[Out]

-1/6*(-2*(-1 - I*a)^(7/2)*(1 - I*a)*((-I)*(I + a + b*x))^(5/2) - (-1 - I*a)^(5/2)*(3*I + 4*a)*b*x*((-I)*(I + a
+ b*x))^(5/2) + I*(-11 + (18*I)*a + 6*a^2)*b^2*x^2*((-I)*Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)]*(1 + a^2 + (
5*I)*b*x + a*b*x) - 6*Sqrt[-1 + I*a]*b*x*Sqrt[1 + I*a + I*b*x]*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)^(5/2)*(1 + a^2)^2*x^3*Sqrt[1 + I*a + I*b*x])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3636 vs. \(2 (267 ) = 534\).
time = 0.30, size = 3637, normalized size = 10.73

method	result	size
risch	\(-\frac {i \left (2 a^{2} b^{4} x^{4}+27 i a \,b^{4} x^{4}+2 a^{3} b^{3} x^{3}+45 i a^{2} b^{3} x^{3}+9 i a^{3} b^{2} x^{2}-28 x^{4} b^{4}+2 a^{5} b x -9 i a^{4} b x -58 a \,b^{3} x^{3}-9 i b^{3} x^{3}+2 a^{6}-26 a^{2} b^{2} x^{2}+9 i a \,b^{2} x^{2}+4 a^{3} b x -18 i x \,a^{2} b +6 a^{4}-26 b^{2} x^{2}+2 a b x -9 i b x +6 a^{2}+2\right )}{6 x^{3} \left (i+a \right ) \left (a -i\right )^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {6 i b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{2}}{\left (i+a \right ) \left (a^{4}-4 i a^{3}-6 a^{2}+4 i a +1\right ) \left (i-a \right ) \sqrt {a^{2}+1}}+\frac {3 b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{3}}{\left (i+a \right ) \left (a^{4}-4 i a^{3}-6 a^{2}+4 i a +1\right ) \left (i-a \right ) \sqrt {a^{2}+1}}+\frac {11 i b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (i+a \right ) \left (a^{4}-4 i a^{3}-6 a^{2}+4 i a +1\right ) \left (i-a \right ) \sqrt {a^{2}+1}}+\frac {7 b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a}{2 \left (i+a \right ) \left (a^{4}-4 i a^{3}-6 a^{2}+4 i a +1\right ) \left (i-a \right ) \sqrt {a^{2}+1}}-\frac {4 b^{2} \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{2}}{\left (i+a \right ) \left (a^{4}-4 i a^{3}-6 a^{2}+4 i a +1\right ) \left (i-a \right ) \left (x -\frac {i}{b}+\frac {a}{b}\right )}-\frac {4 b^{2} \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{\left (i+a \right ) \left (a^{4}-4 i a^{3}-6 a^{2}+4 i a +1\right ) \left (i-a \right ) \left (x -\frac {i}{b}+\frac {a}{b}\right )}\)	\(781\)
default	\(\text {Expression too large to display}\)	\(3637\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I/(I-a)^3*(-1/3/(a^2+1)/x^3*(b^2*x^2+2*a*b*x+a^2+1)^(5/2)-1/3*a*b/(a^2+1)*(-1/2/(a^2+1)/x^2*(b^2*x^2+2*a*b*x+
a^2+1)^(5/2)+1/2*a*b/(a^2+1)*(-1/(a^2+1)/x*(b^2*x^2+2*a*b*x+a^2+1)^(5/2)+3*a*b/(a^2+1)*(1/3*(b^2*x^2+2*a*b*x+a
^2+1)^(3/2)+a*b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b
^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*l
n((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*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)))+4*b^2/(a^2+1)*(1/8*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(3
/2)+3/16*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+
1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))))+3/2*b^2/(a^2+1)*(1/
3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+a*b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-
4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+(a^2+1)*((b^2*x^2+2*a*b*
x+a^2+1)^(1/2)+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*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))))+2/3*b^2/(a^2+1)*(-1/(a^2+1)/x*(b^2*x^2+2*a*b
*x+a^2+1)^(5/2)+3*a*b/(a^2+1)*(1/3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+a*b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x
+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2
)^(1/2))+(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/
(b^2)^(1/2)-(a^2+1)^(1/2)*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)))+4*b^2/(a^2+1
)*(1/8*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+3/16*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*(1/4*(2*b^2*x+2*a*
b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a
*b*x+a^2+1)^(1/2))/(b^2)^(1/2)))))-3*I/(I-a)^4*b*(-1/2/(a^2+1)/x^2*(b^2*x^2+2*a*b*x+a^2+1)^(5/2)+1/2*a*b/(a^2+
1)*(-1/(a^2+1)/x*(b^2*x^2+2*a*b*x+a^2+1)^(5/2)+3*a*b/(a^2+1)*(1/3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+a*b*(1/4*(2*b^
2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2
*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+
(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*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)))+4*b^2/(a^2+1)*(1/8*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+3/16*(4*b^2*(a^2+1)-4*
a^2*b^2)/b^2*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*
x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))))+3/2*b^2/(a^2+1)*(1/3*(b^2*x^2+2*a*b*x+a^2+1)^
(3/2)+a*b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a
*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*ln((b^2
*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*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))))+10*I*b^3/(I-a)^6*(1/3*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(3/2)+I*b*
(1/4*(2*(x-(I-a)/b)*b^2+2*I*b)/b^2*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(1/2)+1/2*ln((I*b+(x-(I-a)/b)*b^2)/(b
^2)^(1/2)+((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(1/2))/(b^2)^(1/2)))-10*I/(I-a)^6*b^3*(1/3*(b^2*x^2+2*a*b*x+a^
2+1)^(3/2)+a*b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^
2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2)+a*b*ln
((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*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)))-4*I*b^2/(I-a)^5*(-I/b/(x-(I-a)/b)^2*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I
-a)/b))^(5/2)+3*I*b*(1/3*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(3/2)+I*b*(1/4*(2*(x-(I-a)/b)*b^2+2*I*b)/b^2*((
x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(1/2)+1/2*ln((I*b+(x-(I-a)/b)*b^2)/(b^2)^(1/2)+((x-(I-a)/b)^2*b^2+2*I*b*(x
-(I-a)/b))^(1/2))/(b^2)^(1/2))))-6*I/(I-a)^5*b^2*(-1/(a^2+1)/x*(b^2*x^2+2*a*b*x+a^2+1)^(5/2)+3*a*b/(a^2+1)*(1/
3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+a*b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-
4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+(a^2+1)*((b^2*x^2+2*a*b*
x+a^2+1)^(1/2)+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-(a^2+1)^(1/2)*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)))+4*b^2/(a^2+1)*(1/8*(2*b^2*x+2*a*b)/b^2*(b^2*x
^2+2*a*b*x+a^2+1)^(3/2)+3/16*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1
/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))))
+I*b/(I-a)^4*(I/b/(x-(I-a)/b)^3*((x-(I-a)/b)^2*...

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (223) = 446\).
time = 2.53, size = 839, normalized size = 2.47 \begin {gather*} \frac {{\left (-2 i \, a^{2} + 51 \, a + 52 i\right )} b^{4} x^{4} + {\left (-2 i \, a^{3} + 49 \, a^{2} + i \, a + 52\right )} b^{3} x^{3} + 3 \, \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} 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}} {\left ({\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} b x^{4} + {\left (a^{6} - 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} + 4 i \, a + 1\right )} x^{3}\right )} \log \left (-\frac {{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3} + {\left (a^{7} - 3 i \, a^{6} - a^{5} - 5 i \, a^{4} - 5 \, a^{3} - i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} 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}}}{{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3}}\right ) - 3 \, \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} 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}} {\left ({\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} b x^{4} + {\left (a^{6} - 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} + 4 i \, a + 1\right )} x^{3}\right )} \log \left (-\frac {{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3} - {\left (a^{7} - 3 i \, a^{6} - a^{5} - 5 i \, a^{4} - 5 \, a^{3} - i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} 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}}}{{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3}}\right ) + {\left ({\left (-2 i \, a^{2} + 51 \, a + 52 i\right )} b^{3} x^{3} - 2 i \, a^{5} + {\left (16 \, a^{2} + 3 i \, a + 19\right )} b^{2} x^{2} - 2 \, a^{4} - 4 i \, a^{3} - 7 \, {\left (a^{3} - i \, a^{2} + a - i\right )} b x - 4 \, a^{2} - 2 i \, a - 2\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, {\left ({\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} b x^{4} + {\left (a^{6} - 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} + 4 i \, a + 1\right )} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((-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 + 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))*((a^5 - 3*I*a^4 - 2*a^3 - 2*I*a^2 - 3*a + I)*b*x^4 + (a^6 - 4*I*a^5 - 5*a^4 - 5*a
^2 + 4*I*a + 1)*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 - 1
1)*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)) - 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))*((a^5 -
3*I*a^4 - 2*a^3 - 2*I*a^2 - 3*a + I)*b*x^4 + (a^6 - 4*I*a^5 - 5*a^4 - 5*a^2 + 4*I*a + 1)*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 - I*a^2 + a - I)*b*x - 4*a^2 - 2*I*a - 2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((a^5 - 3*I*a^4 - 2*a^3 - 2*I*a
^2 - 3*a + I)*b*x^4 + (a^6 - 4*I*a^5 - 5*a^4 - 5*a^2 + 4*I*a + 1)*x^3)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{x^4\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________