3.3.75 \(\int \frac {1}{a^2+\sqrt {-a} x} \, dx\) [275]

Optimal. Leaf size=22 \[ \frac {\log \left (a^2+\sqrt {-a} x\right )}{\sqrt {-a}} \]

[Out]

ln(a^2+x*(-a)^(1/2))/(-a)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {31} \begin {gather*} \frac {\log \left (a^2+\sqrt {-a} x\right )}{\sqrt {-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a^2 + Sqrt[-a]*x]/Sqrt[-a]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {align*} \int \frac {1}{a^2+\sqrt {-a} x} \, dx &=\frac {\log \left (a^2+\sqrt {-a} x\right )}{\sqrt {-a}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log \left (a^2+\sqrt {-a} x\right )}{\sqrt {-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a^2 + Sqrt[-a]*x]/Sqrt[-a]

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Maple [A]
time = 0.09, size = 19, normalized size = 0.86

method	result	size
default	\(\frac {\ln \left (a^{2}+x \sqrt {-a}\right )}{\sqrt {-a}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(a^2+x*(-a)^(1/2))/(-a)^(1/2)

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Maxima [A]
time = 0.32, size = 18, normalized size = 0.82 \begin {gather*} \frac {\log \left (a^{2} + \sqrt {-a} x\right )}{\sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a^2 + sqrt(-a)*x)/sqrt(-a)

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Fricas [A]
time = 0.40, size = 21, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-a} \log \left (-\sqrt {-a} a + x\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a)*log(-sqrt(-a)*a + x)/a

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Sympy [A]
time = 0.01, size = 19, normalized size = 0.86 \begin {gather*} \frac {\log {\left (a^{2} + x \sqrt {- a} \right )}}{\sqrt {- a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2+x*(-a)**(1/2)),x)

[Out]

log(a**2 + x*sqrt(-a))/sqrt(-a)

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Giac [A]
time = 0.56, size = 19, normalized size = 0.86 \begin {gather*} \frac {\log \left ({\left | a^{2} + \sqrt {-a} x \right |}\right )}{\sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a^2 + sqrt(-a)*x))/sqrt(-a)

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Mupad [B]
time = 0.05, size = 14, normalized size = 0.64 \begin {gather*} \frac {\ln \left (x+{\left (-a\right )}^{3/2}\right )}{\sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 + (-a)^(1/2)*x),x)

[Out]

log(x + (-a)^(3/2))/(-a)^(1/2)

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