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

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 3475

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

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*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ -2 \log \left (\cos \left (\sqrt {x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.00, size = 13, normalized size = 1.44 \[ -\log \left (\frac {1}{\tan \left (\sqrt {x}\right )^{2} + 1}\right ) \]

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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giac [A]  time = 0.14, size = 8, normalized size = 0.89 \[ -2 \, \log \left ({\left | \cos \left (\sqrt {x}\right ) \right |}\right ) \]

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))))

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maple [A]  time = 0.00, size = 8, normalized size = 0.89 \[ -2 \ln \left (\cos \left (\sqrt {x}\right )\right ) \]

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 = 0.30, size = 7, normalized size = 0.78 \[ 2 \, \log \left (\sec \left (\sqrt {x}\right )\right ) \]

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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mupad [B]  time = 3.18, size = 19, normalized size = 2.11 \[ -2\,\ln \left ({\mathrm {e}}^{\sqrt {x}\,2{}\mathrm {i}}+1\right )+\sqrt {x}\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)*2i - 2*log(exp(x^(1/2)*2i) + 1)

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sympy [A]  time = 0.17, size = 10, normalized size = 1.11 \[ \log {\left (\tan ^{2}{\left (\sqrt {x} \right )} + 1 \right )} \]

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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