3.84 \(\int \frac{1}{\sqrt{a-a \text{sech}(c+d x)}} \, dx\)

Optimal. Leaf size=87 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-a \text{sech}(c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{2} \sqrt{a-a \text{sech}(c+d x)}}\right )}{\sqrt{a} d} \]

[Out]

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

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Rubi [A]  time = 0.079209, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3776, 3774, 203, 3795} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-a \text{sech}(c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{2} \sqrt{a-a \text{sech}(c+d x)}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a - a*Sech[c + d*x]],x]

[Out]

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

Rule 3776

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

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])

Rule 3795

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2/f, Subst[Int[1/(2
*a + x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0
]

Rubi steps

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

Mathematica [A]  time = 2.16382, size = 118, normalized size = 1.36 \[ \frac{\left (e^{c+d x}-1\right ) \left (\sqrt{2} \sinh ^{-1}\left (e^{c+d x}\right )-2 \tanh ^{-1}\left (\frac{e^{c+d x}+1}{\sqrt{2} \sqrt{e^{2 (c+d x)}+1}}\right )+\sqrt{2} \tanh ^{-1}\left (\sqrt{e^{2 (c+d x)}+1}\right )\right )}{\sqrt{2} d \sqrt{e^{2 (c+d x)}+1} \sqrt{a-a \text{sech}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a - a*Sech[c + d*x]],x]

[Out]

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

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Maple [F]  time = 0.161, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a-a{\rm sech} \left (dx+c\right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(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 \frac{1}{\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(1/(a-a*sech(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.81095, size = 2480, normalized size = 28.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*sqrt(a)*log(-(3*cosh(d*x + c)^2 + 2*(3*cosh(d*x + c) + 1)*sinh(d*x + c) + 3*sinh(d*x + c)^2 - 2*s
qrt(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*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) + 1)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)
*sinh(d*x + c) + sinh(d*x + c)^2 + 1))/sqrt(a) + 2*cosh(d*x + c) + 3)/(cosh(d*x + c)^2 + 2*(cosh(d*x + c) - 1)
*sinh(d*x + c) + sinh(d*x + c)^2 - 2*cosh(d*x + c) + 1)) + 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*cosh(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*cosh(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*co
sh(d*x + c) - a)*sinh(d*x + c) + a)/(cosh(d*x + c) + sinh(d*x + c))))/(a*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(a-a*sech(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(-a*sech(c + d*x) + a), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError