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

Optimal. Leaf size=134 \[ \frac {4 \sqrt {i a+i b x+1}}{(1-i a) \sqrt {-i a-i b x+1}}-i \sinh ^{-1}(a+b x)-\frac {2 (-a+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(a+i)^{3/2}} \]

[Out]

-I*arcsinh(b*x+a)-2*(I-a)^(3/2)*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)
^(3/2)+4*(1+I*a+I*b*x)^(1/2)/(1-I*a)/(1-I*a-I*b*x)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5095, 98, 157, 53, 619, 215, 93, 208} \[ \frac {4 \sqrt {i a+i b x+1}}{(1-i a) \sqrt {-i a-i b x+1}}-i \sinh ^{-1}(a+b x)-\frac {2 (-a+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(a+i)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])/x,x]

[Out]

(4*Sqrt[1 + I*a + I*b*x])/((1 - I*a)*Sqrt[1 - I*a - I*b*x]) - I*ArcSinh[a + b*x] - (2*(I - a)^(3/2)*ArcTanh[(S
qrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(I + a)^(3/2)

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

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 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 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

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]

Rubi steps

\begin {align*} \int \frac {e^{3 i \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac {(1+i a+i b x)^{3/2}}{x (1-i a-i b x)^{3/2}} \, dx\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {2 \int \frac {\frac {1}{2} i (i-a)^2 b-\frac {1}{2} (1-i a) b^2 x}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i+a) b}\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {(i-a)^2 \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{1-i a}-(i b) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {\left (2 (i-a)^2\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 )}{1-i a}-(i b) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {2 (i-a)^{3/2} \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}}-\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{2 b}\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-i \sinh ^{-1}(a+b x)-\frac {2 (i-a)^{3/2} \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}}\\ \end {align*}

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Mathematica [A]  time = 0.80, size = 196, normalized size = 1.46 \[ \frac {2 \left (\frac {\sqrt [4]{-1} (a+i) (-i b)^{3/2} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{b^{3/2}}+\frac {2 i \sqrt {i a+i b x+1}}{\sqrt {-i (a+b x+i)}}+\frac {\sqrt {-1-i a} (a-i) \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (a+b x+i)}}{\sqrt {-1+i a} \sqrt {i a+i b x+1}}\right )}{\sqrt {-1+i a}}\right )}{a+i} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(((2*I)*Sqrt[1 + I*a + I*b*x])/Sqrt[(-I)*(I + a + b*x)] + ((-1)^(1/4)*(I + a)*((-I)*b)^(3/2)*ArcSinh[((1/2
+ I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(3/2) + (Sqrt[-1 - I*a]*(-I + a)*ArcTanh[(Sqrt[-1 -
I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/Sqrt[-1 + I*a]))/(I + a)

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fricas [B]  time = 0.51, size = 381, normalized size = 2.84 \[ -\frac {{\left ({\left (a + i\right )} b x + a^{2} + 2 i \, a - 1\right )} \sqrt {-\frac {4 \, a^{3} - 12 i \, a^{2} - 12 \, a + 4 i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}} \log \left (-\frac {{\left (2 \, a - 2 i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 2 i\right )} - {\left (i \, a^{2} - 2 \, a - i\right )} \sqrt {-\frac {4 \, a^{3} - 12 i \, a^{2} - 12 \, a + 4 i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}}}{2 \, a - 2 i}\right ) - {\left ({\left (a + i\right )} b x + a^{2} + 2 i \, a - 1\right )} \sqrt {-\frac {4 \, a^{3} - 12 i \, a^{2} - 12 \, a + 4 i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}} \log \left (-\frac {{\left (2 \, a - 2 i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 2 i\right )} - {\left (-i \, a^{2} + 2 \, a + i\right )} \sqrt {-\frac {4 \, a^{3} - 12 i \, a^{2} - 12 \, a + 4 i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}}}{2 \, a - 2 i}\right ) + 8 \, b x + {\left (2 \, {\left (-i \, a + 1\right )} b x - 2 i \, a^{2} + 4 \, a + 2 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + 8 \, a + 8 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 8 i}{{\left (2 \, a + 2 i\right )} b x + 2 \, a^{2} + 4 i \, a - 2} \]

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,x, algorithm="fricas")

[Out]

-(((a + I)*b*x + a^2 + 2*I*a - 1)*sqrt(-(4*a^3 - 12*I*a^2 - 12*a + 4*I)/(a^3 + 3*I*a^2 - 3*a - I))*log(-((2*a
- 2*I)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a - 2*I) - (I*a^2 - 2*a - I)*sqrt(-(4*a^3 - 12*I*a^2 - 12*a
+ 4*I)/(a^3 + 3*I*a^2 - 3*a - I)))/(2*a - 2*I)) - ((a + I)*b*x + a^2 + 2*I*a - 1)*sqrt(-(4*a^3 - 12*I*a^2 - 12
*a + 4*I)/(a^3 + 3*I*a^2 - 3*a - I))*log(-((2*a - 2*I)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a - 2*I) - (
-I*a^2 + 2*a + I)*sqrt(-(4*a^3 - 12*I*a^2 - 12*a + 4*I)/(a^3 + 3*I*a^2 - 3*a - I)))/(2*a - 2*I)) + 8*b*x + (2*
(-I*a + 1)*b*x - 2*I*a^2 + 4*a + 2*I)*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 8*a + 8*sqrt(b^2*x^2
+ 2*a*b*x + a^2 + 1) + 8*I)/((2*a + 2*I)*b*x + 2*a^2 + 4*I*a - 2)

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giac [B]  time = 0.36, size = 259, normalized size = 1.93 \[ -\frac {{\left (a^{2} i + 2 \, a - i\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1} {\left (a + i\right )}} - \frac {b \log \left (-3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b - a^{3} b - 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i - 2 \, a^{2} b i - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} - 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a i {\left | b \right |} + a b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{3 \, i {\left | b \right |}} \]

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,x, algorithm="giac")

[Out]

-(a^2*i + 2*a - i)*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)))/(sqrt(a^2 + 1)*(a + i)) - 1/3*b*log(-3*(x*abs(b) - sqrt((b*x + a)^2 + 1))^
2*a*b - a^3*b - 2*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*b*i - 2*a^2*b*i - (x*abs(b) - sqrt((b*x + a)^2 + 1))^3*
abs(b) - 3*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^2*abs(b) - 4*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a*i*abs(b) + a
*b + (x*abs(b) - sqrt((b*x + a)^2 + 1))*abs(b))/(i*abs(b))

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maple [B]  time = 0.17, size = 818, normalized size = 6.10 \[ \frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {i a^{5}}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {i a^{4} b x}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {6 i \left (i a +1\right )^{2} b \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {2 i a}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {3 b a x}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {4 i a^{3}}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\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 )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {3 a^{2}}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {3}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {3 a^{3} b x}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a b x}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 i a^{2} b x}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {i \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 (a^{2}+1\right )^{\frac {3}{2}}}+\frac {2 i a^{3}}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {i b x}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 i \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}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {3 i a}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {2 i b \,a^{2} x}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {4 a^{2}}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {3 a^{4}}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {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 (a^{2}+1\right )^{\frac {3}{2}}} \]

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,x)

[Out]

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

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maxima [B]  time = 0.34, size = 738, normalized size = 5.51 \[ \frac {2 i \, a^{2} b^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {{\left (4 i \, a^{2} + 4 i\right )} b^{3} x}{4 \, {\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {{\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a b^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} + \frac {i \, {\left (a^{2} + 1\right )} a b^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {{\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a^{2} b^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} + \frac {{\left (-3 i \, a b^{2} - 3 \, b^{2}\right )} a b x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {{\left (-3 i \, a^{2} b - 6 \, a b + 3 i \, b\right )} b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {{\left (-3 i \, a b^{2} - 3 \, b^{2}\right )} a^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {{\left (-3 i \, a^{2} b - 6 \, a b + 3 i \, b\right )} a b}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {{\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} + \frac {-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} - \frac {-3 i \, a b^{2} - 3 \, b^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}} - i \, \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right ) \]

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,x, algorithm="maxima")

[Out]

2*I*a^2*b^3*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 1/4*(4*I*a^2 + 4*I)*b^3*x/((a^2*
b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*a*b^3*x/((a^2*b^2 - (a^
2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) + I*(a^2 + 1)*a*b^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b
^2*x^2 + 2*a*b*x + a^2 + 1)) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 +
2*a*b*x + a^2 + 1)*(a^2 + 1)) + (-3*I*a*b^2 - 3*b^2)*a*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x +
a^2 + 1)) - (-3*I*a^2*b - 6*a*b + 3*I*b)*b^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))
+ (-3*I*a*b^2 - 3*b^2)*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - (-3*I*a^2*b - 6*a*b
+ 3*I*b)*a*b/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - (-I*a^3 - 3*a^2 + 3*I*a + 1)*arc
sinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) +
2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(3/2) + (-I*a^3 - 3*a^2 + 3*I*a + 1)/(sqrt(b^2*x^2 +
2*a*b*x + a^2 + 1)*(a^2 + 1)) - (-3*I*a*b^2 - 3*b^2)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - I*arcsinh(2*(b
^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{x\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - i \left (\int \frac {i}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 b x}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{3}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i b^{2} x^{2}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{2}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx\right ) \]

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,x)

[Out]

-I*(Integral(I/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*a
/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3/(a**2*x*sqrt(
a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*
b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2/(a**2*x*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**
2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2
*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**3/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1
) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(
a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**2/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) +
2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a*b**2*x**2/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b
*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1)), x) + Integral(3*a**2*b*x/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1)), x) + Integral(-6*I*a*b*x/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**
2 + 1)), x))

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