3.83 \(\int \sqrt{a-a \text{sech}(c+d x)} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-a \text{sech}(c+d x)}}\right )}{d} \]

[Out]

(2*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a - a*Sech[c + d*x]]])/d

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Rubi [A]  time = 0.0232803, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3774, 203} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-a \text{sech}(c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Sech[c + d*x]],x]

[Out]

(2*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a - a*Sech[c + d*x]]])/d

Rule 3774

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

Rule 203

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

Rubi steps

\begin{align*} \int \sqrt{a-a \text{sech}(c+d x)} \, dx &=-\frac{(2 i a) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\frac{i a \tanh (c+d x)}{\sqrt{a-a \text{sech}(c+d x)}}\right )}{d}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-a \text{sech}(c+d x)}}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 2.26553, size = 70, normalized size = 1.84 \[ \frac{\sqrt{e^{2 (c+d x)}+1} \sqrt{a-a \text{sech}(c+d x)} \left (\sinh ^{-1}\left (e^{c+d x}\right )+\tanh ^{-1}\left (\sqrt{e^{2 (c+d x)}+1}\right )\right )}{d \left (e^{c+d x}-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Sech[c + d*x]],x]

[Out]

(Sqrt[1 + E^(2*(c + d*x))]*(ArcSinh[E^(c + d*x)] + ArcTanh[Sqrt[1 + E^(2*(c + d*x))]])*Sqrt[a - a*Sech[c + d*x
]])/(d*(-1 + E^(c + d*x)))

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a-a{\rm sech} \left (dx+c\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*sech(d*x+c))^(1/2),x)

[Out]

int((a-a*sech(d*x+c))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a \operatorname{sech}\left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*sech(d*x + c) + a), x)

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Fricas [B]  time = 2.5674, size = 1808, normalized size = 47.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(a)*log((a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 + 3*a*cosh(d*x + c)^3 + (4*a*cosh(d*x + c) + 3*a)*sinh
(d*x + c)^3 + 5*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 + 9*a*cosh(d*x + c) + 5*a)*sinh(d*x + c)^2 + (cosh(d*
x + c)^5 + (5*cosh(d*x + c) + 3)*sinh(d*x + c)^4 + sinh(d*x + c)^5 + 3*cosh(d*x + c)^4 + (10*cosh(d*x + c)^2 +
12*cosh(d*x + c) + 5)*sinh(d*x + c)^3 + 5*cosh(d*x + c)^3 + (10*cosh(d*x + c)^3 + 18*cosh(d*x + c)^2 + 15*cos
h(d*x + c) + 7)*sinh(d*x + c)^2 + 7*cosh(d*x + c)^2 + (5*cosh(d*x + c)^4 + 12*cosh(d*x + c)^3 + 15*cosh(d*x +
c)^2 + 14*cosh(d*x + c) + 4)*sinh(d*x + c) + 4*cosh(d*x + c) + 4)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x
+ c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) + 4*a*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 + 9*a*cosh(d*x + c)^2 +
10*a*cosh(d*x + c) + 4*a)*sinh(d*x + c) + 4*a)/(cosh(d*x + c)^3 + 3*cosh(d*x + c)^2*sinh(d*x + c) + 3*cosh(d*
x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3)) + sqrt(a)*log(-(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + (cosh(d*x +
c)^3 + (3*cosh(d*x + c) - 1)*sinh(d*x + c)^2 + sinh(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 - 2*cos
h(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) - 1)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x
+ c) + sinh(d*x + c)^2 + 1)) - a*cosh(d*x + c) + (2*a*cosh(d*x + c) - a)*sinh(d*x + c) + a)/(cosh(d*x + c) + s
inh(d*x + c))))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- a \operatorname{sech}{\left (c + d x \right )} + a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-a*sech(c + d*x) + a), x)

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Giac [B]  time = 1.18494, size = 136, normalized size = 3.58 \begin{align*} -\frac{\frac{2 \, a \arctan \left (-\frac{\sqrt{a} e^{\left (d x + c\right )} - \sqrt{a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt{-a}}\right ) \mathrm{sgn}\left (e^{\left (d x + c\right )} - 1\right )}{\sqrt{-a}} + \sqrt{a} \log \left ({\left | -\sqrt{a} e^{\left (d x + c\right )} + \sqrt{a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) \mathrm{sgn}\left (e^{\left (d x + c\right )} - 1\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*arctan(-(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))/sqrt(-a))*sgn(e^(d*x + c) - 1)/sqrt(-a) + sq
rt(a)*log(abs(-sqrt(a)*e^(d*x + c) + sqrt(a*e^(2*d*x + 2*c) + a)))*sgn(e^(d*x + c) - 1))/d