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

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

[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)

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Rubi [A]  time = 0.119658, antiderivative size = 163, normalized size of antiderivative = 1., 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 = {5095, 78, 50, 53, 619, 215} \[ -\frac{(1-i a) (i a+i b x+1)^{5/2}}{b^2 \sqrt{-i a-i b x+1}}-\frac{(3-2 i a) \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}-\frac{3 (3-2 i a) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 b^2}+\frac{3 (2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2} \]

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 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 78

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] || LtQ
[p, n]))))

Rule 50

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 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 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 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]

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)) \operatorname{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*}

Mathematica [A]  time = 0.163109, size = 132, normalized size = 0.81 \[ \frac{\sqrt{i a+i b x+1} \left (a^2+15 i a-b^2 x^2+5 i b x-14\right )}{2 b^2 \sqrt{-i (a+b x+i)}}+\frac{3 \sqrt [4]{-1} (2 a+3 i) \sqrt{-i b} \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{b^{5/2}} \]

Warning: Unable to verify antiderivative.

[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]  time = 0.116, size = 358, normalized size = 2.2 \begin{align*} -7\,{\frac{1}{{b}^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}-7\,{\frac{{a}^{2}}{{b}^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}+{\frac{{\frac{i}{2}}{a}^{2}x}{b}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-3\,{\frac{{x}^{2}}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}-{{\frac{i}{2}}a{x}^{2}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{{\frac{i}{2}}a}{{b}^{2}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-{\frac{{\frac{9\,i}{2}}x}{b}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+{\frac{{\frac{9\,i}{2}}}{b}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{{\frac{i}{2}}{a}^{3}}{{b}^{2}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}-10\,{\frac{ax}{b\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}-{{\frac{i}{2}}b{x}^{3}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}+3\,{\frac{a}{b\sqrt{{b}^{2}}}\ln \left ({\frac{{b}^{2}x+ab}{\sqrt{{b}^{2}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ) } \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,x)

[Out]

-7/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-7*a^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/2*I/b*a^2*x/(b^2*x^2+2*a*b*x+a^
2+1)^(1/2)-3*x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/2*I*a*x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/2*I/b^2*a/(b^2*x^2+
2*a*b*x+a^2+1)^(1/2)-9/2*I/b*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+9/2*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)+1/2*I/b^2*a^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-10*a/b/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*
x-1/2*I*b*x^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+3*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)

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.03503, size = 366, normalized size = 2.25 \begin{align*} \frac{3 i \, a^{3} +{\left (3 i \, a^{2} - 44 \, a - 32 i\right )} b x - 47 \, a^{2} -{\left ({\left (24 \, a + 36 i\right )} b x + 24 \, a^{2} + 60 i \, a - 36\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (-4 i \, b^{2} x^{2} + 4 i \, a^{2} - 20 \, b x - 60 \, a - 56 i\right )} - 76 i \, a + 32}{8 \, b^{3} x +{\left (8 \, a + 8 i\right )} b^{2}} \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,x, algorithm="fricas")

[Out]

(3*I*a^3 + (3*I*a^2 - 44*a - 32*I)*b*x - 47*a^2 - ((24*a + 36*I)*b*x + 24*a^2 + 60*I*a - 36)*log(-b*x - a + sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(-4*I*b^2*x^2 + 4*I*a^2 - 20*b*x - 60*a -
56*I) - 76*I*a + 32)/(8*b^3*x + (8*a + 8*I)*b^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (i a + i b x + 1\right )^{3}}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac{3}{2}}}\, dx \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,x)

[Out]

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

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Giac [B]  time = 1.15689, size = 319, normalized size = 1.96 \begin{align*} -\frac{1}{2} \, \sqrt{{\left (b x + a\right )}^{2} + 1}{\left (\frac{i x}{b} - \frac{a b^{2} i - 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 + 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 )}{2 \, b{\left | b \right |}} - \frac{{\left (2 \, a{\left | b \right |} + 3 \, i{\left | b \right |}\right )} \log \left (24 \, b^{2}\right )}{b^{3}} \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,x, algorithm="giac")

[Out]

-1/2*sqrt((b*x + a)^2 + 1)*(i*x/b - (a*b^2*i - 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 + 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*ab
s(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))*abs(b))/(b*abs(b)) - (2*a*abs(b) + 3*i*abs(b))*log(24*b^2)/b^3