3.1.53 \(\int \frac {1}{a+c x^2} \, dx\) [53]

Optimal. Leaf size=24 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}} \]

[Out]

arctan(x*c^(1/2)/a^(1/2))/a^(1/2)/c^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(-1),x]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[c])

Rule 211

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(-1),x]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[c])

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Maple [A]
time = 0.40, size = 16, normalized size = 0.67

method	result	size
default	\(\frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}\)	\(16\)
risch	\(-\frac {\ln \left (c x +\sqrt {-a c}\right )}{2 \sqrt {-a c}}+\frac {\ln \left (-c x +\sqrt {-a c}\right )}{2 \sqrt {-a c}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))

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Maxima [A]
time = 0.52, size = 15, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(c*x/sqrt(a*c))/sqrt(a*c)

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Fricas [A]
time = 1.75, size = 67, normalized size = 2.79 \begin {gather*} \left [-\frac {\sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, \frac {\sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{a c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a))/(a*c), sqrt(a*c)*arctan(sqrt(a*c)*x/a)/(a*c)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
time = 0.05, size = 53, normalized size = 2.21 \begin {gather*} - \frac {\sqrt {- \frac {1}{a c}} \log {\left (- a \sqrt {- \frac {1}{a c}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a c}} \log {\left (a \sqrt {- \frac {1}{a c}} + x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(a*c))*log(-a*sqrt(-1/(a*c)) + x)/2 + sqrt(-1/(a*c))*log(a*sqrt(-1/(a*c)) + x)/2

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Giac [A]
time = 1.44, size = 15, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(c*x/sqrt(a*c))/sqrt(a*c)

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Mupad [B]
time = 0.07, size = 16, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + c*x^2),x)

[Out]

atan((c^(1/2)*x)/a^(1/2))/(a^(1/2)*c^(1/2))

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