3.67 \(\int \frac {e^{\frac {1}{2} i \tan ^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=170 \[ \frac {3}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {3}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2} \]

[Out]

-1/3*(1-I*a*x)^(3/4)*(1+I*a*x)^(1/4)/x^3-5/12*I*a*(1-I*a*x)^(3/4)*(1+I*a*x)^(1/4)/x^2+11/24*a^2*(1-I*a*x)^(3/4
)*(1+I*a*x)^(1/4)/x+3/8*I*a^3*arctan((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))+3/8*I*a^3*arctanh((1+I*a*x)^(1/4)/(1-I*a
*x)^(1/4))

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Rubi [A]  time = 0.06, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5062, 99, 151, 12, 93, 212, 206, 203} \[ \frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {3}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {3}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^((I/2)*ArcTan[a*x])/x^4,x]

[Out]

-((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(3*x^3) - (((5*I)/12)*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x^2 + (11*
a^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(24*x) + ((3*I)/8)*a^3*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] +
((3*I)/8)*a^3*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 203

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 5062

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 - I*a*x)^((I*n)/2))/(1 + I*a*x)^((I*n)/2
), x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[(I*n - 1)/2]

Rubi steps

\begin {align*} \int \frac {e^{\frac {1}{2} i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac {\sqrt [4]{1+i a x}}{x^4 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}+\frac {1}{3} \int \frac {\frac {5 i a}{2}-2 a^2 x}{x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}-\frac {1}{6} \int \frac {\frac {11 a^2}{4}+\frac {5}{2} i a^3 x}{x^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {1}{6} \int -\frac {9 i a^3}{8 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac {1}{16} \left (3 i a^3\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac {1}{4} \left (3 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {1}{8} \left (3 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{8} \left (3 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {3}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {3}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 93, normalized size = 0.55 \[ \frac {(1-i a x)^{3/4} \left (6 i a^3 x^3 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i}{i-a x}\right )+11 i a^3 x^3+21 a^2 x^2-18 i a x-8\right )}{24 x^3 (1+i a x)^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I*a*x)^(3/4)*(-8 - (18*I)*a*x + 21*a^2*x^2 + (11*I)*a^3*x^3 + (6*I)*a^3*x^3*Hypergeometric2F1[3/4, 1, 7/
4, (I + a*x)/(I - a*x)]))/(24*x^3*(1 + I*a*x)^(3/4))

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fricas [A]  time = 0.43, size = 183, normalized size = 1.08 \[ \frac {9 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 9 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 9 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 9 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) + {\left (-22 i \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 4 i \, a x - 16\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(9*I*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + 1) - 9*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x +
I)) + I) + 9*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) - I) - 9*I*a^3*x^3*log(sqrt(I*sqrt(a^2*x^2 + 1)/
(a*x + I)) - 1) + (-22*I*a^3*x^3 + 2*a^2*x^2 - 4*I*a*x - 16)*sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)))/x^3

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming 0=[0,0]index.cc index_m operator + Error: Bad Argument ValueWarning, need to choose a branch for
the root of a polynomial with parameters. This might be wrong.The choice was done assuming 0=[0,0]index.cc in
dex_m operator + Error: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with p
arameters. This might be wrong.The choice was done assuming 0=[0,0]index.cc index_m operator + Error: Bad Argu
ment ValueWarning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.T
he choice was done assuming 0=[0,0]index.cc index_m operator + Error: Bad Argument ValueDone

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maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(I*(a*x - I)/sqrt(a**2*x**2 + 1))/x**4, x)

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