3.1.23 \(\int \frac {e^{3 i \text {ArcTan}(a x)}}{x} \, dx\) [23]

Optimal. Leaf size=51 \[ \frac {4 i \sqrt {1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]

[Out]

-I*arcsinh(a*x)-arctanh((a^2*x^2+1)^(1/2))+4*I*(a^2*x^2+1)^(1/2)/(I+a*x)

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Rubi [A]
time = 0.55, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5168, 6874, 221, 272, 65, 214, 665} \begin {gather*} \frac {4 i \sqrt {a^2 x^2+1}}{a x+i}-\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-i \sinh ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*I)*Sqrt[1 + a^2*x^2])/(I + a*x) - I*ArcSinh[a*x] - ArcTanh[Sqrt[1 + a^2*x^2]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 221

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a + c*x^2)^(p + 1)/
(2*c*d*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 5168

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

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {e^{3 i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac {(1+i a x)^2}{x (1-i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \left (-\frac {i a}{\sqrt {1+a^2 x^2}}+\frac {1}{x \sqrt {1+a^2 x^2}}-\frac {4 a}{(i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx\\ &=-\left ((i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\right )-(4 a) \int \frac {1}{(i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=\frac {4 i \sqrt {1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac {4 i \sqrt {1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{a^2}\\ &=\frac {4 i \sqrt {1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 55, normalized size = 1.08 \begin {gather*} \frac {4 i \sqrt {1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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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. 100 vs. \(2 (44 ) = 88\).
time = 0.08, size = 101, normalized size = 1.98

method	result	size
default	\(-i a^{3} \left (-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}\right )+\frac {4}{\sqrt {a^{2} x^{2}+1}}+\frac {3 i a x}{\sqrt {a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\)	\(101\)
meijerg	\(\frac {-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{2}}{\sqrt {\pi }}+\frac {3 i a x}{\sqrt {a^{2} x^{2}+1}}-\frac {3 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}-\frac {i a \left (-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \arcsinh \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {a^{2}}}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)^3/(a^2*x^2+1)^(3/2)/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 46, normalized size = 0.90 \begin {gather*} \frac {4 i \, a x}{\sqrt {a^{2} x^{2} + 1}} + \frac {4}{\sqrt {a^{2} x^{2} + 1}} - i \, \operatorname {arsinh}\left (a x\right ) - \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*I*a*x/sqrt(a^2*x^2 + 1) + 4/sqrt(a^2*x^2 + 1) - I*arcsinh(a*x) - arcsinh(1/(a*abs(x)))

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (41) = 82\).
time = 2.17, size = 100, normalized size = 1.96 \begin {gather*} \frac {4 i \, a x - {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + {\left (i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 4 i \, \sqrt {a^{2} x^{2} + 1} - 4}{a x + i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*I*a*x - (a*x + I)*log(-a*x + sqrt(a^2*x^2 + 1) + 1) + (I*a*x - 1)*log(-a*x + sqrt(a^2*x^2 + 1)) + (a*x + I)
*log(-a*x + sqrt(a^2*x^2 + 1) - 1) + 4*I*sqrt(a^2*x^2 + 1) - 4)/(a*x + I)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \left (\int \frac {i}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*(Integral(I/(a**2*x**3*sqrt(a**2*x**2 + 1) + x*sqrt(a**2*x**2 + 1)), x) + Integral(-3*a*x/(a**2*x**3*sqrt(a
**2*x**2 + 1) + x*sqrt(a**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**2*x**3*sqrt(a**2*x**2 + 1) + x*sqrt(a**2*x
**2 + 1)), x) + Integral(-3*I*a**2*x**2/(a**2*x**3*sqrt(a**2*x**2 + 1) + x*sqrt(a**2*x**2 + 1)), x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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Mupad [B]
time = 0.43, size = 73, normalized size = 1.43 \begin {gather*} -\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*(a^2*x^2 + 1)^(1/2)*4i)/((((a^2)^(1/2)*1i)/a + x*(a^2)^(1/2))*(a^2)^(1/2)) - (a*asinh(x*(a^2)^(1/2))*1i)/(a
^2)^(1/2) - atanh((a^2*x^2 + 1)^(1/2))

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