3.23 \(\int \sqrt {1+\tanh (4 x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (4 x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]

[Out]

1/4*arctanh(1/2*(1+tanh(4*x))^(1/2)*2^(1/2))*2^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3480, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (4 x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Tanh[4*x]],x]

[Out]

ArcTanh[Sqrt[1 + Tanh[4*x]]/Sqrt[2]]/(2*Sqrt[2])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \sqrt {1+\tanh (4 x)} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (4 x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (4 x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (4 x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Tanh[4*x]],x]

[Out]

ArcTanh[Sqrt[1 + Tanh[4*x]]/Sqrt[2]]/(2*Sqrt[2])

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fricas [B]  time = 0.41, size = 68, normalized size = 2.62 \[ \frac {1}{8} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (4 \, x\right )}{\cosh \left (4 \, x\right ) - \sinh \left (4 \, x\right )}} {\left (\cosh \left (4 \, x\right ) + \sinh \left (4 \, x\right )\right )} - 2 \, \cosh \left (4 \, x\right )^{2} - 4 \, \cosh \left (4 \, x\right ) \sinh \left (4 \, x\right ) - 2 \, \sinh \left (4 \, x\right )^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(4*x))^(1/2),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log(-2*sqrt(2)*sqrt(cosh(4*x)/(cosh(4*x) - sinh(4*x)))*(cosh(4*x) + sinh(4*x)) - 2*cosh(4*x)^2 - 4
*cosh(4*x)*sinh(4*x) - 2*sinh(4*x)^2 - 1)

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giac [A]  time = 0.22, size = 21, normalized size = 0.81 \[ -\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {e^{\left (8 \, x\right )} + 1} - e^{\left (4 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(4*x))^(1/2),x, algorithm="giac")

[Out]

-1/4*sqrt(2)*log(sqrt(e^(8*x) + 1) - e^(4*x))

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maple [A]  time = 0.05, size = 20, normalized size = 0.77 \[ \frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {\tanh \left (4 x \right )+1}\, \sqrt {2}}{2}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+tanh(4*x))^(1/2),x)

[Out]

1/4*arctanh(1/2*(1+tanh(4*x))^(1/2)*2^(1/2))*2^(1/2)

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maxima [B]  time = 0.98, size = 43, normalized size = 1.65 \[ -\frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sqrt {2}}{\sqrt {e^{\left (-8 \, x\right )} + 1}}}{\sqrt {2} + \frac {\sqrt {2}}{\sqrt {e^{\left (-8 \, x\right )} + 1}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(4*x))^(1/2),x, algorithm="maxima")

[Out]

-1/8*sqrt(2)*log(-(sqrt(2) - sqrt(2)/sqrt(e^(-8*x) + 1))/(sqrt(2) + sqrt(2)/sqrt(e^(-8*x) + 1)))

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mupad [B]  time = 0.16, size = 19, normalized size = 0.73 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (4\,x\right )+1}}{2}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((tanh(4*x) + 1)^(1/2),x)

[Out]

(2^(1/2)*atanh((2^(1/2)*(tanh(4*x) + 1)^(1/2))/2))/4

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tanh {\left (4 x \right )} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(4*x))**(1/2),x)

[Out]

Integral(sqrt(tanh(4*x) + 1), x)

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