3.1.17 \(\int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx\) [17]

Optimal. Leaf size=16 \[ \frac {1}{2} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right ) \]

[Out]

-1/2*I*arcsin(-1+8/3*I*x)

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {633, 221} \begin {gather*} \frac {1}{2} i \text {ArcSin}\left (1-\frac {8 i x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(3*I)*x + 4*x^2],x]

[Out]

(I/2)*ArcSin[1 - ((8*I)/3)*x]

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 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{9}}} \, dx,x,3 i+8 x\right )\\ &=\frac {1}{2} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(16)=32\).
time = 0.03, size = 51, normalized size = 3.19 \begin {gather*} -\frac {\sqrt {x} \sqrt {3 i+4 x} \log \left (-2 \sqrt {x}+\sqrt {3 i+4 x}\right )}{\sqrt {x (3 i+4 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(3*I)*x + 4*x^2],x]

[Out]

-((Sqrt[x]*Sqrt[3*I + 4*x]*Log[-2*Sqrt[x] + Sqrt[3*I + 4*x]])/Sqrt[x*(3*I + 4*x)])

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Maple [A]
time = 0.43, size = 10, normalized size = 0.62

method	result	size
default	\(\frac {\arcsinh \left (i+\frac {8 x}{3}\right )}{2}\)	\(10\)
trager	\(-\frac {\ln \left (-440 x -144-165 i-192 i \sqrt {4 x^{2}+3 i x}+384 i x +220 \sqrt {4 x^{2}+3 i x}\right )}{2}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*I*x+4*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*arcsinh(I+8/3*x)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (8) = 16\).
time = 0.49, size = 21, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*log(8*x + 4*sqrt(4*x^2 + 3*I*x) + 3*I)

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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. 19 vs. \(2 (8) = 16\).
time = 1.35, size = 19, normalized size = 1.19 \begin {gather*} -\frac {1}{2} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {4 x^{2} + 3 i x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x**2)**(1/2),x)

[Out]

Integral(1/sqrt(4*x**2 + 3*I*x), x)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (8) = 16\).
time = 1.58, size = 110, normalized size = 6.88 \begin {gather*} \frac {1}{32} \, \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (8 \, x + 3 i\right )} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - \frac {9}{64} \, \log \left (2 \, \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - 8 \, x - 3 i\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*I*x+4*x^2)^(1/2),x, algorithm="giac")

[Out]

1/32*sqrt(8*x^2 + 2*sqrt(16*x^2 + 9)*x)*(8*x + 3*I)*(3*I*x/(4*x^2 + sqrt(16*x^4 + 9*x^2)) + 1) - 9/64*log(2*sq
rt(8*x^2 + 2*sqrt(16*x^2 + 9)*x)*(3*I*x/(4*x^2 + sqrt(16*x^4 + 9*x^2)) + 1) - 8*x - 3*I)

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Mupad [B]
time = 0.28, size = 19, normalized size = 1.19 \begin {gather*} \frac {\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*3i + 4*x^2)^(1/2),x)

[Out]

log(x + (x*(4*x + 3i))^(1/2)/2 + 3i/8)/2

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