3.58 \(\int \frac{\tan (\sqrt{x})}{\sqrt{x}} \, dx\)

Optimal. Leaf size=9 \[ -2 \log \left (\cos \left (\sqrt{x}\right )\right ) \]

[Out]

-2*Log[Cos[Sqrt[x]]]

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Rubi [A]  time = 0.0097141, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3747, 3475} \[ -2 \log \left (\cos \left (\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[Tan[Sqrt[x]]/Sqrt[x],x]

[Out]

-2*Log[Cos[Sqrt[x]]]

Rule 3747

Int[(x_)^(m_.)*((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\tan \left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \tan (x) \, dx,x,\sqrt{x}\right )\\ &=-2 \log \left (\cos \left (\sqrt{x}\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0120247, size = 9, normalized size = 1. \[ -2 \log \left (\cos \left (\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[Sqrt[x]]/Sqrt[x],x]

[Out]

-2*Log[Cos[Sqrt[x]]]

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Maple [A]  time = 0.003, size = 8, normalized size = 0.9 \begin{align*} -2\,\ln \left ( \cos \left ( \sqrt{x} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x^(1/2))/x^(1/2),x)

[Out]

-2*ln(cos(x^(1/2)))

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Maxima [A]  time = 1.01361, size = 9, normalized size = 1. \begin{align*} 2 \, \log \left (\sec \left (\sqrt{x}\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x^(1/2))/x^(1/2),x, algorithm="maxima")

[Out]

2*log(sec(sqrt(x)))

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Fricas [A]  time = 2.3592, size = 41, normalized size = 4.56 \begin{align*} -\log \left (\frac{1}{\tan \left (\sqrt{x}\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x^(1/2))/x^(1/2),x, algorithm="fricas")

[Out]

-log(1/(tan(sqrt(x))^2 + 1))

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Sympy [A]  time = 0.991418, size = 10, normalized size = 1.11 \begin{align*} \log{\left (\tan ^{2}{\left (\sqrt{x} \right )} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x**(1/2))/x**(1/2),x)

[Out]

log(tan(sqrt(x))**2 + 1)

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Giac [A]  time = 1.10323, size = 11, normalized size = 1.22 \begin{align*} -2 \, \log \left ({\left | \cos \left (\sqrt{x}\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x^(1/2))/x^(1/2),x, algorithm="giac")

[Out]

-2*log(abs(cos(sqrt(x))))