\(\int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx\) [225]

Optimal result

Integrand size = 18, antiderivative size = 211 \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}-\frac {3 i b \arctan \left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{\sqrt [4]{i-a} (i+a)^{7/4}}+\frac {3 i b \text {arctanh}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{\sqrt [4]{i-a} (i+a)^{7/4}} \]

[Out]

-(1-I*a-I*b*x)^(1/4)*(1+I*a+I*b*x)^(3/4)/(1-I*a)/x-3*I*b*arctan((I+a)^(1/4)*(1+I*a+I*b*x)^(1/4)/(I-a)^(1/4)/(1
-I*a-I*b*x)^(1/4))/(I-a)^(1/4)/(I+a)^(7/4)+3*I*b*arctanh((I+a)^(1/4)*(1+I*a+I*b*x)^(1/4)/(I-a)^(1/4)/(1-I*a-I*
b*x)^(1/4))/(I-a)^(1/4)/(I+a)^(7/4)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5203, 96, 95, 304, 211, 214} \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {3 i b \arctan \left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{\sqrt [4]{-a+i} (a+i)^{7/4}}+\frac {3 i b \text {arctanh}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{\sqrt [4]{-a+i} (a+i)^{7/4}}-\frac {\sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{(1-i a) x} \]

[In]

Int[E^(((3*I)/2)*ArcTan[a + b*x])/x^2,x]

[Out]

-(((1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/((1 - I*a)*x)) - ((3*I)*b*ArcTan[((I + a)^(1/4)*(1 + I*a +
I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/((I - a)^(1/4)*(I + a)^(7/4)) + ((3*I)*b*ArcTanh[((I
+ a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/((I - a)^(1/4)*(I + a)^(7/4))

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 96

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

Rule 211

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

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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 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*} \text {integral}& = \int \frac {(1+i a+i b x)^{3/4}}{x^2 (1-i a-i b x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}-\frac {(3 b) \int \frac {1}{x (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}} \, dx}{2 (i+a)} \\ & = -\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}-\frac {(6 b) \text {Subst}\left (\int \frac {x^2}{-1-i a-(-1+i a) x^4} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{i+a} \\ & = -\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}+\frac {(3 i b) \text {Subst}\left (\int \frac {1}{\sqrt {i-a}-\sqrt {i+a} x^2} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/2}}-\frac {(3 i b) \text {Subst}\left (\int \frac {1}{\sqrt {i-a}+\sqrt {i+a} x^2} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/2}} \\ & = -\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}-\frac {3 i b \arctan \left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{\sqrt [4]{i-a} (i+a)^{7/4}}+\frac {3 i b \text {arctanh}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{\sqrt [4]{i-a} (i+a)^{7/4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.50 \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\frac {\sqrt [4]{-i (i+a+b x)} \left (1+a^2+i b x+a b x+6 i b x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )\right )}{(i+a)^2 x \sqrt [4]{1+i a+i b x}} \]

[In]

Integrate[E^(((3*I)/2)*ArcTan[a + b*x])/x^2,x]

[Out]

(((-I)*(I + a + b*x))^(1/4)*(1 + a^2 + I*b*x + a*b*x + (6*I)*b*x*Hypergeometric2F1[1/4, 1, 5/4, (1 + a^2 - I*b
*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/((I + a)^2*x*(1 + I*a + I*b*x)^(1/4))

Maple [F]

\[\int \frac {{\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (137) = 274\).

Time = 0.27 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.29 \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\frac {3 \, \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (-i \, a + 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (a^{6} + 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} - 4 i \, a + 1\right )} \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 3 \, \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (i \, a - 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (a^{6} + 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} - 4 i \, a + 1\right )} \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 3 \, \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a + i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (i \, a^{6} - 4 \, a^{5} - 5 i \, a^{4} - 5 i \, a^{2} + 4 \, a + i\right )} \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) - 3 \, \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a + i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (-i \, a^{6} + 4 \, a^{5} + 5 i \, a^{4} + 5 i \, a^{2} - 4 \, a - i\right )} \left (-\frac {b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}\right )^{\frac {3}{4}}}{b^{3}}\right ) - 2 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{2 \, {\left (a + i\right )} x} \]

[In]

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

[Out]

1/2*(3*(-b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))^(1/4)*(-I*a + 1)*x*log((b^3*
sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)) + (a^6 + 4*I*a^5 - 5*a^4 - 5*a^2 - 4*I*a + 1)*(-b^4/(a
^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))^(3/4))/b^3) + 3*(-b^4/(a^8 + 6*I*a^7 - 14*a
^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))^(1/4)*(I*a - 1)*x*log((b^3*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2
+ 1)/(b*x + a + I)) - (a^6 + 4*I*a^5 - 5*a^4 - 5*a^2 - 4*I*a + 1)*(-b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 -
14*I*a^3 + 14*a^2 + 6*I*a - 1))^(3/4))/b^3) + 3*(-b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 +
6*I*a - 1))^(1/4)*(a + I)*x*log((b^3*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)) - (I*a^6 - 4*a^5
- 5*I*a^4 - 5*I*a^2 + 4*a + I)*(-b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))^(3/
4))/b^3) - 3*(-b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))^(1/4)*(a + I)*x*log((b
^3*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)) - (-I*a^6 + 4*a^5 + 5*I*a^4 + 5*I*a^2 - 4*a - I)*(-
b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))^(3/4))/b^3) - 2*I*sqrt(b^2*x^2 + 2*a*
b*x + a^2 + 1)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/((a + I)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\int { \frac {\left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0,0]Warning, replacing 0 by 14, a substitution variable should perhaps be
purged.War

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\int \frac {{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2}}{x^2} \,d x \]

[In]

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

[Out]

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