3.3.88 \(\int \sqrt {1+\tan ^2(x)} \, dx\) [288]

Optimal. Leaf size=3 \[ \sinh ^{-1}(\tan (x)) \]

[Out]

arcsinh(tan(x))

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Rubi [A]
time = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3738, 4207, 221} \begin {gather*} \sinh ^{-1}(\tan (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Tan[x]^2],x]

[Out]

ArcSinh[Tan[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 3738

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(3)=6\).
time = 0.01, size = 44, normalized size = 14.67 \begin {gather*} \cos (x) \left (-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )\right ) \sqrt {\sec ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Tan[x]^2],x]

[Out]

Cos[x]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])*Sqrt[Sec[x]^2]

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Maple [A]
time = 0.04, size = 4, normalized size = 1.33

method	result	size
derivativedivides	\(\arcsinh \left (\tan \left (x \right )\right )\)	\(4\)
default	\(\arcsinh \left (\tan \left (x \right )\right )\)	\(4\)
risch	\(2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )-2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+tan(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsinh(tan(x))

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Maxima [A]
time = 0.51, size = 3, normalized size = 1.00 \begin {gather*} \operatorname {arsinh}\left (\tan \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(tan(x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (3) = 6\).
time = 1.88, size = 60, normalized size = 20.00 \begin {gather*} \frac {1}{2} \, \log \left (\frac {\tan \left (x\right )^{2} + \sqrt {\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {\tan \left (x\right )^{2} - \sqrt {\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log((tan(x)^2 + sqrt(tan(x)^2 + 1)*tan(x) + 1)/(tan(x)^2 + 1)) - 1/2*log((tan(x)^2 - sqrt(tan(x)^2 + 1)*ta
n(x) + 1)/(tan(x)^2 + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\tan ^{2}{\left (x \right )} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(tan(x)**2 + 1), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (3) = 6\).
time = 0.46, size = 16, normalized size = 5.33 \begin {gather*} -\log \left (\sqrt {\tan \left (x\right )^{2} + 1} - \tan \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(tan(x)^2 + 1) - tan(x))

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Mupad [B]
time = 0.06, size = 3, normalized size = 1.00 \begin {gather*} \mathrm {asinh}\left (\mathrm {tan}\left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((tan(x)^2 + 1)^(1/2),x)

[Out]

asinh(tan(x))

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