3.2.79 \(\int \frac {e^{2 i \text {ArcTan}(a+b x)}}{x^4} \, dx\) [179]

Optimal. Leaf size=93 \[ -\frac {i-a}{3 (i+a) x^3}+\frac {i b}{(i+a)^2 x^2}+\frac {2 b^2}{(1-i a)^3 x}-\frac {2 i b^3 \log (x)}{(i+a)^4}+\frac {2 i b^3 \log (i+a+b x)}{(i+a)^4} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5203, 78} \begin {gather*} -\frac {2 i b^3 \log (x)}{(a+i)^4}+\frac {2 i b^3 \log (a+b x+i)}{(a+i)^4}+\frac {2 b^2}{(1-i a)^3 x}+\frac {i b}{(a+i)^2 x^2}-\frac {-a+i}{3 (a+i) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((2*I)*ArcTan[a + b*x])/x^4,x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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^{2 i \tan ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {1+i a+i b x}{x^4 (1-i a-i b x)} \, dx\\ &=\int \left (\frac {i-a}{(i+a) x^4}-\frac {2 i b}{(i+a)^2 x^3}+\frac {2 i b^2}{(i+a)^3 x^2}-\frac {2 i b^3}{(i+a)^4 x}+\frac {2 i b^4}{(i+a)^4 (i+a+b x)}\right ) \, dx\\ &=-\frac {i-a}{3 (i+a) x^3}+\frac {i b}{(i+a)^2 x^2}+\frac {2 b^2}{(1-i a)^3 x}-\frac {2 i b^3 \log (x)}{(i+a)^4}+\frac {2 i b^3 \log (i+a+b x)}{(i+a)^4}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 88, normalized size = 0.95 \begin {gather*} \frac {(i+a) \left (i+a+i a^2+a^3-3 b x+3 i a b x-6 i b^2 x^2\right )-6 i b^3 x^3 \log (x)+6 i b^3 x^3 \log (i+a+b x)}{3 (i+a)^4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((2*I)*ArcTan[a + b*x])/x^4,x]

[Out]

((I + a)*(I + a + I*a^2 + a^3 - 3*b*x + (3*I)*a*b*x - (6*I)*b^2*x^2) - (6*I)*b^3*x^3*Log[x] + (6*I)*b^3*x^3*Lo
g[I + a + b*x])/(3*(I + a)^4*x^3)

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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. 266 vs. \(2 (79 ) = 158\).
time = 0.16, size = 267, normalized size = 2.87

method	result	size
default	\(\frac {2 b^{4} \left (\frac {\left (i a^{4} b +4 a^{3} b -6 i a^{2} b -4 a b +i b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (i a^{5}-10 i a^{3}+5 a^{4}+5 i a -10 a^{2}+1-\frac {\left (i a^{4} b +4 a^{3} b -6 i a^{2} b -4 a b +i b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\right )}{\left (a^{2}+1\right )^{4}}-\frac {-a^{2}+2 i a +1}{3 \left (a^{2}+1\right ) x^{3}}+\frac {b \left (i a^{2}+2 a -i\right )}{\left (a^{2}+1\right )^{2} x^{2}}-\frac {2 b^{2} \left (i a^{3}+3 a^{2}-3 i a -1\right )}{\left (a^{2}+1\right )^{3} x}-\frac {2 b^{3} \left (i a^{4}+4 a^{3}-6 i a^{2}-4 a +i\right ) \ln \left (x \right )}{\left (a^{2}+1\right )^{4}}\)	\(267\)
risch	\(\frac {-\frac {2 i b^{2} x^{2}}{\left (a^{2}+2 i a -1\right ) \left (i+a \right )}+\frac {i b x}{a^{2}+2 i a -1}+\frac {a -i}{3 i+3 a}}{x^{3}}-\frac {b^{3} \ln \left (4 a^{12} b^{2} x^{2}+8 a^{13} b x +4 a^{14}+24 a^{10} b^{2} x^{2}+48 a^{11} b x +28 a^{12}+60 a^{8} b^{2} x^{2}+120 a^{9} b x +84 a^{10}+80 a^{6} b^{2} x^{2}+160 a^{7} b x +140 a^{8}+60 a^{4} b^{2} x^{2}+120 a^{5} b x +140 a^{6}+24 a^{2} b^{2} x^{2}+48 a^{3} b x +84 a^{4}+4 b^{2} x^{2}+8 a b x +28 a^{2}+4\right )}{i a^{4}-4 a^{3}-6 i a^{2}+4 a +i}+\frac {2 i b^{3} \arctan \left (\frac {\left (2 a^{6} b +6 a^{4} b +6 a^{2} b +2 b \right ) x +2 a^{7}+6 a^{5}+6 a^{3}+2 a}{2 a^{6}+6 a^{4}+6 a^{2}+2}\right )}{i a^{4}-4 a^{3}-6 i a^{2}+4 a +i}+\frac {2 b^{3} \ln \left (\left (-2 a^{6} b -6 a^{4} b -6 a^{2} b -2 b \right ) x \right )}{i a^{4}-4 a^{3}-6 i a^{2}+4 a +i}\)	\(400\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^4/(a^2+1)^4*(1/2*(I*a^4*b-6*I*b*a^2+4*a^3*b+I*b-4*a*b)/b^2*ln(b^2*x^2+2*a*b*x+a^2+1)+(I*a^5-10*I*a^3+5*a^4
+5*I*a-10*a^2+1-(I*a^4*b-6*I*b*a^2+4*a^3*b+I*b-4*a*b)*a/b)/b*arctan(1/2*(2*b^2*x+2*a*b)/b))-1/3*(2*I*a-a^2+1)/
(a^2+1)/x^3+b*(I*a^2-I+2*a)/(a^2+1)^2/x^2-2*b^2*(I*a^3-3*I*a+3*a^2-1)/(a^2+1)^3/x-2*b^3*(I*a^4-6*I*a^2+4*a^3+I
-4*a)/(a^2+1)^4*ln(x)

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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. 263 vs. \(2 (69) = 138\).
time = 0.49, size = 263, normalized size = 2.83 \begin {gather*} \frac {2 \, {\left (a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1\right )} b^{3} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} + \frac {{\left (i \, a^{4} + 4 \, a^{3} - 6 i \, a^{2} - 4 \, a + i\right )} b^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} - \frac {2 \, {\left (i \, a^{4} + 4 \, a^{3} - 6 i \, a^{2} - 4 \, a + i\right )} b^{3} \log \left (x\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} + \frac {a^{6} - 2 i \, a^{5} + 6 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} b^{2} x^{2} + a^{4} - 4 i \, a^{3} + 3 \, {\left (i \, a^{4} + 2 \, a^{3} + 2 \, a - i\right )} b x - a^{2} - 2 i \, a - 1}{3 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.25, size = 94, normalized size = 1.01 \begin {gather*} \frac {-6 i \, b^{3} x^{3} \log \left (x\right ) + 6 i \, b^{3} x^{3} \log \left (\frac {b x + a + i}{b}\right ) - 6 \, {\left (i \, a - 1\right )} b^{2} x^{2} + a^{4} + 2 i \, a^{3} - 3 \, {\left (-i \, a^{2} + 2 \, a + i\right )} b x + 2 i \, a - 1}{3 \, {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (73) = 146\).
time = 0.62, size = 286, normalized size = 3.08 \begin {gather*} - \frac {2 i b^{3} \log {\left (- \frac {2 a^{5} b^{3}}{\left (a + i\right )^{4}} - \frac {10 i a^{4} b^{3}}{\left (a + i\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a + i\right )^{4}} + \frac {20 i a^{2} b^{3}}{\left (a + i\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a + i\right )^{4}} + 4 b^{4} x + 2 i b^{3} - \frac {2 i b^{3}}{\left (a + i\right )^{4}} \right )}}{\left (a + i\right )^{4}} + \frac {2 i b^{3} \log {\left (\frac {2 a^{5} b^{3}}{\left (a + i\right )^{4}} + \frac {10 i a^{4} b^{3}}{\left (a + i\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a + i\right )^{4}} - \frac {20 i a^{2} b^{3}}{\left (a + i\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a + i\right )^{4}} + 4 b^{4} x + 2 i b^{3} + \frac {2 i b^{3}}{\left (a + i\right )^{4}} \right )}}{\left (a + i\right )^{4}} - \frac {- a^{3} - i a^{2} - a + 6 i b^{2} x^{2} + x \left (- 3 i a b + 3 b\right ) - i}{x^{3} \cdot \left (3 a^{3} + 9 i a^{2} - 9 a - 3 i\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*b**3*log(-2*a**5*b**3/(a + I)**4 - 10*I*a**4*b**3/(a + I)**4 + 20*a**3*b**3/(a + I)**4 + 20*I*a**2*b**3/(
a + I)**4 + 2*a*b**3 - 10*a*b**3/(a + I)**4 + 4*b**4*x + 2*I*b**3 - 2*I*b**3/(a + I)**4)/(a + I)**4 + 2*I*b**3
*log(2*a**5*b**3/(a + I)**4 + 10*I*a**4*b**3/(a + I)**4 - 20*a**3*b**3/(a + I)**4 - 20*I*a**2*b**3/(a + I)**4
+ 2*a*b**3 + 10*a*b**3/(a + I)**4 + 4*b**4*x + 2*I*b**3 + 2*I*b**3/(a + I)**4)/(a + I)**4 - (-a**3 - I*a**2 -
a + 6*I*b**2*x**2 + x*(-3*I*a*b + 3*b) - I)/(x**3*(3*a**3 + 9*I*a**2 - 9*a - 3*I))

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Giac [A]
time = 0.42, size = 126, normalized size = 1.35 \begin {gather*} \frac {2 \, b^{4} \log \left (b x + a + i\right )}{-i \, a^{4} b + 4 \, a^{3} b + 6 i \, a^{2} b - 4 \, a b - i \, b} + \frac {2 \, b^{3} \log \left ({\left | x \right |}\right )}{i \, a^{4} - 4 \, a^{3} - 6 i \, a^{2} + 4 \, a + i} + \frac {a^{4} + 2 i \, a^{3} - 6 i \, {\left (a b^{2} + i \, b^{2}\right )} x^{2} + 3 i \, {\left (a^{2} b + 2 i \, a b - b\right )} x + 2 i \, a - 1}{3 \, {\left (a + i\right )}^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.72, size = 199, normalized size = 2.14 \begin {gather*} \frac {\frac {a-\mathrm {i}}{3\,\left (a+1{}\mathrm {i}\right )}-\frac {b^2\,x^2\,2{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^3}+\frac {b\,x\,1{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^2}}{x^3}+\frac {b^3\,\mathrm {atanh}\left (\frac {a^4+a^3\,4{}\mathrm {i}-6\,a^2-a\,4{}\mathrm {i}+1}{{\left (a+1{}\mathrm {i}\right )}^4}-\frac {x\,\left (2\,a^{12}\,b^2+12\,a^{10}\,b^2+30\,a^8\,b^2+40\,a^6\,b^2+30\,a^4\,b^2+12\,a^2\,b^2+2\,b^2\right )}{{\left (a+1{}\mathrm {i}\right )}^4\,\left (-b\,a^9+3{}\mathrm {i}\,b\,a^8+8{}\mathrm {i}\,b\,a^6+6\,b\,a^5+6{}\mathrm {i}\,b\,a^4+8\,b\,a^3+3\,b\,a-b\,1{}\mathrm {i}\right )}\right )\,4{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a - 1i)/(3*(a + 1i)) - (b^2*x^2*2i)/(a + 1i)^3 + (b*x*1i)/(a + 1i)^2)/x^3 + (b^3*atanh((a^3*4i - 6*a^2 - a*4
i + a^4 + 1)/(a + 1i)^4 - (x*(2*b^2 + 12*a^2*b^2 + 30*a^4*b^2 + 40*a^6*b^2 + 30*a^8*b^2 + 12*a^10*b^2 + 2*a^12
*b^2))/((a + 1i)^4*(3*a*b - b*1i + 8*a^3*b + a^4*b*6i + 6*a^5*b + a^6*b*8i + a^8*b*3i - a^9*b)))*4i)/(a + 1i)^
4

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