3.87 \(\int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=233 \[ \frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}-\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}} \]

[Out]

(2467*a^4*(1 + I*a*x)^(1/4))/(192*(1 - I*a*x)^(1/4)) - (1 + I*a*x)^(1/4)/(4*x^4*(1 - I*a*x)^(1/4)) - (((17*I)/
24)*a*(1 + I*a*x)^(1/4))/(x^3*(1 - I*a*x)^(1/4)) + (113*a^2*(1 + I*a*x)^(1/4))/(96*x^2*(1 - I*a*x)^(1/4)) + ((
(521*I)/192)*a^3*(1 + I*a*x)^(1/4))/(x*(1 - I*a*x)^(1/4)) - (475*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4
)])/64 - (475*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)])/64

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Rubi [A]  time = 0.0996443, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5062, 98, 151, 155, 12, 93, 212, 206, 203} \[ \frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}-\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}} \]

Antiderivative was successfully verified.

[In]

Int[E^(((5*I)/2)*ArcTan[a*x])/x^5,x]

[Out]

(2467*a^4*(1 + I*a*x)^(1/4))/(192*(1 - I*a*x)^(1/4)) - (1 + I*a*x)^(1/4)/(4*x^4*(1 - I*a*x)^(1/4)) - (((17*I)/
24)*a*(1 + I*a*x)^(1/4))/(x^3*(1 - I*a*x)^(1/4)) + (113*a^2*(1 + I*a*x)^(1/4))/(96*x^2*(1 - I*a*x)^(1/4)) + ((
(521*I)/192)*a^3*(1 + I*a*x)^(1/4))/(x*(1 - I*a*x)^(1/4)) - (475*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4
)])/64 - (475*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)])/64

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]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 155

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] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

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 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 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 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])

Rubi steps

\begin{align*} \int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x^5} \, dx &=\int \frac{(1+i a x)^{5/4}}{x^5 (1-i a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{1}{4} \int \frac{-\frac{17 i a}{2}+8 a^2 x}{x^4 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{1}{12} \int \frac{-\frac{113 a^2}{4}-\frac{51}{2} i a^3 x}{x^3 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}-\frac{1}{24} \int \frac{\frac{521 i a^3}{8}-\frac{113 a^4 x}{2}}{x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac{1}{24} \int \frac{\frac{1425 a^4}{16}+\frac{521}{8} i a^5 x}{x (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac{i \int -\frac{1425 i a^5}{32 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{12 a}\\ &=\frac{2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac{1}{128} \left (475 a^4\right ) \int \frac{1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}+\frac{1}{32} \left (475 a^4\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{2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}-\frac{1}{64} \left (475 a^4\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}{64} \left (475 a^4\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{2467 a^4 \sqrt [4]{1+i a x}}{192 \sqrt [4]{1-i a x}}-\frac{\sqrt [4]{1+i a x}}{4 x^4 \sqrt [4]{1-i a x}}-\frac{17 i a \sqrt [4]{1+i a x}}{24 x^3 \sqrt [4]{1-i a x}}+\frac{113 a^2 \sqrt [4]{1+i a x}}{96 x^2 \sqrt [4]{1-i a x}}+\frac{521 i a^3 \sqrt [4]{1+i a x}}{192 x \sqrt [4]{1-i a x}}-\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end{align*}

Mathematica [C]  time = 0.034015, size = 118, normalized size = 0.51 \[ \frac{950 i a^4 x^4 (a x+i) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},\frac{a x+i}{-a x+i}\right )+2467 i a^5 x^5+1946 a^4 x^4+747 i a^3 x^3+362 a^2 x^2-184 i a x-48}{192 x^4 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(((5*I)/2)*ArcTan[a*x])/x^5,x]

[Out]

(-48 - (184*I)*a*x + 362*a^2*x^2 + (747*I)*a^3*x^3 + 1946*a^4*x^4 + (2467*I)*a^5*x^5 + (950*I)*a^4*x^4*(I + a*
x)*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)])/(192*x^4*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))

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Maple [F]  time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x)

[Out]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm="maxima")

[Out]

integrate(((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(5/2)/x^5, x)

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Fricas [A]  time = 2.07907, size = 482, normalized size = 2.07 \begin{align*} -\frac{1425 \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 1425 i \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 1425 i \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 1425 \, a^{4} x^{4} \log \left (\sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}} - 1\right ) -{\left (4934 \, a^{4} x^{4} + 1042 i \, a^{3} x^{3} + 452 \, a^{2} x^{2} - 272 i \, a x - 96\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{384 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm="fricas")

[Out]

-1/384*(1425*a^4*x^4*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + 1) + 1425*I*a^4*x^4*log(sqrt(I*sqrt(a^2*x^2 + 1
)/(a*x + I)) + I) - 1425*I*a^4*x^4*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) - I) - 1425*a^4*x^4*log(sqrt(I*sqrt
(a^2*x^2 + 1)/(a*x + I)) - 1) - (4934*a^4*x^4 + 1042*I*a^3*x^3 + 452*a^2*x^2 - 272*I*a*x - 96)*sqrt(I*sqrt(a^2
*x^2 + 1)/(a*x + I)))/x^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(5/2)/x**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(5/2)/x^5,x, algorithm="giac")

[Out]

integrate(((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(5/2)/x^5, x)