3.1.87 \(\int \frac {e^{\frac {5}{2} i \text {ArcTan}(a x)}}{x^5} \, dx\) [87]

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

[Out]

2467/192*a^4*(1+I*a*x)^(1/4)/(1-I*a*x)^(1/4)-1/4*(1+I*a*x)^(1/4)/x^4/(1-I*a*x)^(1/4)-17/24*I*a*(1+I*a*x)^(1/4)
/x^3/(1-I*a*x)^(1/4)+113/96*a^2*(1+I*a*x)^(1/4)/x^2/(1-I*a*x)^(1/4)+521/192*I*a^3*(1+I*a*x)^(1/4)/x/(1-I*a*x)^
(1/4)-475/64*a^4*arctan((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))-475/64*a^4*arctanh((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))

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Rubi [A]
time = 0.07, antiderivative size = 233, normalized size of antiderivative = 1.00, 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 = {5170, 100, 156, 160, 12, 95, 218, 212, 209} \begin {gather*} -\frac {475}{64} a^4 \text {ArcTan}\left (\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 {475}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {521 i a^3 \sqrt [4]{1+i a x}}{192 x \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 {\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}} \end {gather*}

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 12

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

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 100

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 156

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, -1]

Rule 160

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 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 5170

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 {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.03, size = 118, normalized size = 0.51 \begin {gather*} \frac {-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) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {i+a x}{i-a x}\right )}{192 x^4 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}{x^{5}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 1.93, size = 192, normalized size = 0.82 \begin {gather*} -\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 ) - 2 \, {\left (2467 \, a^{4} x^{4} + 521 i \, a^{3} x^{3} + 226 \, a^{2} x^{2} - 136 i \, a x - 48\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{384 \, x^{4}} \end {gather*}

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) - 2*(2467*a^4*x^4 + 521*I*a^3*x^3 + 226*a^2*x^2 - 136*I*a*x - 48)*sqrt(I*sqrt(a^
2*x^2 + 1)/(a*x + I)))/x^4

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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