3.2.83 \(\int e^{3 i \text {ArcTan}(a+b x)} x \, dx\) [183]

Optimal. Leaf size=163 \[ -\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {3 (3 i+2 a) \sinh ^{-1}(a+b x)}{2 b^2} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5203, 79, 52, 55, 633, 221} \begin {gather*} -\frac {(1-i a) (i a+i b x+1)^{5/2}}{b^2 \sqrt {-i a-i b x+1}}-\frac {(3-2 i a) \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}-\frac {3 (3-2 i a) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^2}+\frac {3 (2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])*x,x]

[Out]

(-3*(3 - (2*I)*a)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^2) - ((3 - (2*I)*a)*Sqrt[1 - I*a - I*b*x]*
(1 + I*a + I*b*x)^(3/2))/(2*b^2) - ((1 - I*a)*(1 + I*a + I*b*x)^(5/2))/(b^2*Sqrt[1 - I*a - I*b*x]) + (3*(3*I +
2*a)*ArcSinh[a + b*x])/(2*b^2)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 55

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 221

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 633

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 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 e^{3 i \tan ^{-1}(a+b x)} x \, dx &=\int \frac {x (1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 i+2 a) \int \frac {(1+i a+i b x)^{3/2}}{\sqrt {1-i a-i b x}} \, dx}{b}\\ &=-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{2 b}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{4 b^3}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {3 (3 i+2 a) \sinh ^{-1}(a+b x)}{2 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 132, normalized size = 0.81 \begin {gather*} \frac {\sqrt {1+i a+i b x} \left (-14+15 i a+a^2+5 i b x-b^2 x^2\right )}{2 b^2 \sqrt {-i (i+a+b x)}}+\frac {3 \sqrt [4]{-1} (3 i+2 a) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + I*a + I*b*x]*(-14 + (15*I)*a + a^2 + (5*I)*b*x - b^2*x^2))/(2*b^2*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1
)^(1/4)*(3*I + 2*a)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(5/2)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (129 ) = 258\).
time = 0.15, size = 1052, normalized size = 6.45

method	result	size
risch	\(\frac {i \left (-b x +a +6 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}+\frac {9 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b \sqrt {b^{2}}}+\frac {3 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}-\frac {4 \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a}{b^{3} \left (x +\frac {i}{b}+\frac {a}{b}\right )}-\frac {4 i \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{3} \left (x +\frac {i}{b}+\frac {a}{b}\right )}\)	\(237\)
default	\(-i b^{3} \left (\frac {x^{3}}{2 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {5 a \left (\frac {x^{2}}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \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}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )}{b}-\frac {2 \left (a^{2}+1\right ) \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \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}}\right )}{b^{2}}\right )}{2 b}-\frac {3 \left (a^{2}+1\right ) \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \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}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )}{2 b^{2}}\right )-3 \left (i a +1\right ) b^{2} \left (\frac {x^{2}}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \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}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )}{b}-\frac {2 \left (a^{2}+1\right ) \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \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}}\right )}{b^{2}}\right )+3 i \left (i a +1\right )^{2} b \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \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}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )+\left (i a +1\right )^{3} \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \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}}\right )\)	\(1052\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (113) = 226\).
time = 0.28, size = 1108, normalized size = 6.80 \begin {gather*} \frac {15 i \, a^{4} 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 {31 i \, {\left (a^{2} + 1\right )} a^{2} b x}{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 {i \, b x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {5 i \, {\left (a^{2} + 1\right )} a^{3}}{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 {6 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} a^{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 {18 \, {\left (i \, a b^{2} + b^{2}\right )} a^{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} b} + \frac {3 i \, {\left (a^{2} + 1\right )}^{2} b x}{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 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 {5 i \, a x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 i \, {\left (a^{2} + 1\right )}^{2} 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 (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\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 {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} {\left (a^{2} + 1\right )} 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 {15 \, {\left (i \, a b^{2} + b^{2}\right )} {\left (a^{2} + 1\right )} a 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} b} - \frac {15 i \, a^{2} \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 )}{2 \, b^{2}} - \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} {\left (a^{2} + 1\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} b^{2}} + \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} {\left (a^{2} + 1\right )} a}{{\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} b} - \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}} - \frac {3 \, {\left (-i \, a^{2} - i\right )} \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 )}{2 \, b^{2}} + \frac {5 i \, {\left (a^{2} + 1\right )} a}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}} + \frac {9 \, {\left (i \, a b^{2} + b^{2}\right )} a \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 )}{b^{4}} - \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} \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 )}{b^{3}} + \frac {i \, a^{3} + 3 \, a^{2} - 3 i \, a - 1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}} - \frac {6 \, {\left (i \, a b^{2} + b^{2}\right )} {\left (a^{2} + 1\right )}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{4}} \end {gather*}

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]

15*I*a^4*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 31/2*I*(a^2 + 1)*a^2*b*x/((a^2*b^
2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 1/2*I*b*x^3/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 5/2*I*
(a^2 + 1)*a^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 6*(I*a^2*b + 2*a*b - I*b)*a^2*x/
((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 18*(I*a*b^2 + b^2)*a^3*x/((a^2*b^2 - (a^2 + 1)
*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 3/2*I*(a^2 + 1)^2*b*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*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)) + 5/2*I*a*x^2/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 3/2*I*(a^2 + 1)^2*a/((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/((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 + 2*a*b - I*b)*(a^2 + 1)*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a
^2 + 1)) + 15*(I*a*b^2 + b^2)*(a^2 + 1)*a*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) -
15/2*I*a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^2 - 3*(I*a*b^2 + b^2)*(a^2 + 1)*a^2/(
(a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 3*(I*a^2*b + 2*a*b - I*b)*(a^2 + 1)*a/((a^2
*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 3*(I*a*b^2 + b^2)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a
^2 + 1)*b^2) - 3/2*(-I*a^2 - I)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^2 + 5*I*(a^2 + 1
)*a/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 9*(I*a*b^2 + b^2)*a*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*
(a^2 + 1)*b^2))/b^4 - 3*(I*a^2*b + 2*a*b - I*b)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^
3 + (I*a^3 + 3*a^2 - 3*I*a - 1)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 6*(I*a*b^2 + b^2)*(a^2 + 1)/(sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1)*b^4)

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Fricas [A]
time = 3.04, size = 136, normalized size = 0.83 \begin {gather*} \frac {3 i \, a^{3} + {\left (3 i \, a^{2} - 44 \, a - 32 i\right )} b x - 47 \, a^{2} - 12 \, {\left ({\left (2 \, a + 3 i\right )} b x + 2 \, a^{2} + 5 i \, a - 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, b^{2} x^{2} - i \, a^{2} + 5 \, b x + 15 \, a + 14 i\right )} - 76 i \, a + 32}{8 \, {\left (b^{3} x + {\left (a + i\right )} b^{2}\right )}} \end {gather*}

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]

1/8*(3*I*a^3 + (3*I*a^2 - 44*a - 32*I)*b*x - 47*a^2 - 12*((2*a + 3*I)*b*x + 2*a^2 + 5*I*a - 3)*log(-b*x - a +
sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 4*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(I*b^2*x^2 - I*a^2 + 5*b*x + 15*a + 1
4*I) - 76*I*a + 32)/(b^3*x + (a + I)*b^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \left (\int \frac {i x}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 b x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 i b^{2} x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx\right ) \end {gather*}

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*x/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b
**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*a*x/(a
**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3*x/(a**2*sqrt(a**2 + 2*a
*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x*
*2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x**2/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2
+ 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a
**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**4/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*
sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1)), x) + Integral(-3*I*a**2*x/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
), x) + Integral(-3*I*b**2*x**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**
2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Int
egral(3*a*b**2*x**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a**2
*b*x**2/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*
sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-6*I*a*b*x**2/(a**
2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 +
2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x))

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Giac [A]
time = 0.45, size = 209, normalized size = 1.28 \begin {gather*} -\frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (\frac {i \, x}{b} + \frac {-i \, a b^{2} + 6 \, b^{2}}{b^{4}}\right )} - \frac {{\left (2 \, a + 3 i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\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 |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \end {gather*}

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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