3.1.22 \(\int e^{3 i \text {ArcTan}(a x)} \, dx\) [22]

Optimal. Leaf size=60 \[ -\frac {2 i (1+i a x)^2}{a \sqrt {1+a^2 x^2}}-\frac {3 i \sqrt {1+a^2 x^2}}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \]

[Out]

-3*arcsinh(a*x)/a-2*I*(1+I*a*x)^2/a/(a^2*x^2+1)^(1/2)-3*I*(a^2*x^2+1)^(1/2)/a

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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5167, 867, 683, 655, 221} \begin {gather*} -\frac {2 i (1+i a x)^2}{a \sqrt {a^2 x^2+1}}-\frac {3 i \sqrt {a^2 x^2+1}}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*(1 + I*a*x)^2)/(a*Sqrt[1 + a^2*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2])/a - (3*ArcSinh[a*x])/a

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 683

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

Rule 867

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a^
m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && IntegerQ[n]

Rule 5167

Int[E^(ArcTan[(a_.)*(x_)]*(n_)), x_Symbol] :> Int[(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, x] && IntegerQ[(I*n - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.70 \begin {gather*} \frac {\sqrt {1+a^2 x^2} \left (-i+\frac {4}{i+a x}\right )}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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. 127 vs. \(2 (53 ) = 106\).
time = 0.10, size = 128, normalized size = 2.13

method	result	size
risch	\(-\frac {i \sqrt {a^{2} x^{2}+1}}{a}-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {4 \sqrt {\left (x +\frac {i}{a}\right )^{2} a^{2}-2 i a \left (x +\frac {i}{a}\right )}}{a^{2} \left (x +\frac {i}{a}\right )}\)	\(93\)
default	\(\frac {x}{\sqrt {a^{2} x^{2}+1}}-i a^{3} \left (\frac {x^{2}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {a^{2} x^{2}+1}}\right )-3 a^{2} \left (-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {3 i}{a \sqrt {a^{2} x^{2}+1}}\)	\(128\)
meijerg	\(\frac {x}{\sqrt {a^{2} x^{2}+1}}+\frac {3 i \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {3 \left (-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \arcsinh \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {a^{2}}}-\frac {i \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right )}{4 \sqrt {a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\)	\(137\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 57, normalized size = 0.95 \begin {gather*} -\frac {i \, a x^{2}}{\sqrt {a^{2} x^{2} + 1}} + \frac {4 \, x}{\sqrt {a^{2} x^{2} + 1}} - \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a} - \frac {5 i}{\sqrt {a^{2} x^{2} + 1} a} \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, algorithm="maxima")

[Out]

-I*a*x^2/sqrt(a^2*x^2 + 1) + 4*x/sqrt(a^2*x^2 + 1) - 3*arcsinh(a*x)/a - 5*I/(sqrt(a^2*x^2 + 1)*a)

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Fricas [A]
time = 1.87, size = 60, normalized size = 1.00 \begin {gather*} \frac {4 \, a x + 3 \, {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \sqrt {a^{2} x^{2} + 1} {\left (-i \, a x + 5\right )} + 4 i}{a^{2} x + i \, a} \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, algorithm="fricas")

[Out]

(4*a*x + 3*(a*x + I)*log(-a*x + sqrt(a^2*x^2 + 1)) + sqrt(a^2*x^2 + 1)*(-I*a*x + 5) + 4*I)/(a^2*x + I*a)

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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^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \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)

[Out]

-I*(Integral(I/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x**2 + 1)), x) + Integral(-3*a*x/(a**2*x**2*sqrt(a**
2*x**2 + 1) + sqrt(a**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**2*x**2*sqrt(a**2*x**2 + 1) + sqrt(a**2*x**2 +
1)), x) + Integral(-3*I*a**2*x**2/(a**2*x**2*sqrt(a**2*x**2 + 1) + 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, algorithm="giac")

[Out]

undef

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Mupad [B]
time = 0.42, size = 72, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{a}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}+\frac {4\,\sqrt {a^2\,x^2+1}}{\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/(a^2*x^2 + 1)^(3/2),x)

[Out]

(4*(a^2*x^2 + 1)^(1/2))/((((a^2)^(1/2)*1i)/a + x*(a^2)^(1/2))*(a^2)^(1/2)) - (3*asinh(x*(a^2)^(1/2)))/(a^2)^(1
/2) - ((a^2*x^2 + 1)^(1/2)*1i)/a

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