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

Optimal. Leaf size=68 \[ \frac{2 \sqrt{x}}{a}-\frac{4 b \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a+b}}\right )}{a d \sqrt{a-b} \sqrt{a+b}} \]

[Out]

(2*Sqrt[x])/a - (4*b*ArcTan[(Sqrt[a - b]*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)

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Rubi [A]  time = 0.0917246, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5436, 3783, 2659, 208} \[ \frac{2 \sqrt{x}}{a}-\frac{4 b \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a+b}}\right )}{a d \sqrt{a-b} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]])),x]

[Out]

(2*Sqrt[x])/a - (4*b*ArcTan[(Sqrt[a - b]*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)

Rule 5436

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 3783

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x} \left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{a+b \text{sech}(c+d x)} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{x}}{a}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{a \cosh (c+d x)}{b}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 \sqrt{x}}{a}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,i \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )}{a d}\\ &=\frac{2 \sqrt{x}}{a}-\frac{4 b \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a+b}}\right )}{a \sqrt{a-b} \sqrt{a+b} d}\\ \end{align*}

Mathematica [A]  time = 0.124481, size = 69, normalized size = 1.01 \[ \frac{2 \left (\frac{2 b \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a^2-b^2}}\right )}{d \sqrt{a^2-b^2}}+\frac{c}{d}+\sqrt{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]])),x]

[Out]

(2*(c/d + Sqrt[x] + (2*b*ArcTan[((-a + b)*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d)))/a

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Maple [A]  time = 0.051, size = 95, normalized size = 1.4 \begin{align*} -2\,{\frac{\ln \left ( \tanh \left ( c/2+1/2\,d\sqrt{x} \right ) -1 \right ) }{ad}}-4\,{\frac{b}{ad\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tanh \left ( c/2+1/2\,d\sqrt{x} \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{\ln \left ( \tanh \left ( c/2+1/2\,d\sqrt{x} \right ) +1 \right ) }{ad}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/a*ln(tanh(1/2*c+1/2*d*x^(1/2))-1)-4/d*b/a/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*c+1/2*d*x^(1/2))/((a+
b)*(a-b))^(1/2))+2/d/a*ln(tanh(1/2*c+1/2*d*x^(1/2))+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.69687, size = 570, normalized size = 8.38 \begin{align*} \left [\frac{2 \,{\left ({\left (a^{2} - b^{2}\right )} d \sqrt{x} - \sqrt{-a^{2} + b^{2}} b \log \left (\frac{a b +{\left (b^{2} + \sqrt{-a^{2} + b^{2}} b\right )} \cosh \left (d \sqrt{x} + c\right ) +{\left (a^{2} - b^{2} - \sqrt{-a^{2} + b^{2}} b\right )} \sinh \left (d \sqrt{x} + c\right ) + \sqrt{-a^{2} + b^{2}} a}{a \cosh \left (d \sqrt{x} + c\right ) + b}\right )\right )}}{{\left (a^{3} - a b^{2}\right )} d}, \frac{2 \,{\left ({\left (a^{2} - b^{2}\right )} d \sqrt{x} + 2 \, \sqrt{a^{2} - b^{2}} b \arctan \left (-\frac{\sqrt{a^{2} - b^{2}} a \cosh \left (d \sqrt{x} + c\right ) + \sqrt{a^{2} - b^{2}} a \sinh \left (d \sqrt{x} + c\right ) + \sqrt{a^{2} - b^{2}} b}{a^{2} - b^{2}}\right )\right )}}{{\left (a^{3} - a b^{2}\right )} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*((a^2 - b^2)*d*sqrt(x) - sqrt(-a^2 + b^2)*b*log((a*b + (b^2 + sqrt(-a^2 + b^2)*b)*cosh(d*sqrt(x) + c) + (a^
2 - b^2 - sqrt(-a^2 + b^2)*b)*sinh(d*sqrt(x) + c) + sqrt(-a^2 + b^2)*a)/(a*cosh(d*sqrt(x) + c) + b)))/((a^3 -
a*b^2)*d), 2*((a^2 - b^2)*d*sqrt(x) + 2*sqrt(a^2 - b^2)*b*arctan(-(sqrt(a^2 - b^2)*a*cosh(d*sqrt(x) + c) + sqr
t(a^2 - b^2)*a*sinh(d*sqrt(x) + c) + sqrt(a^2 - b^2)*b)/(a^2 - b^2)))/((a^3 - a*b^2)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x} \left (a + b \operatorname{sech}{\left (c + d \sqrt{x} \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1479, size = 82, normalized size = 1.21 \begin{align*} -\frac{4 \, b \arctan \left (\frac{a e^{\left (d \sqrt{x} + c\right )} + b}{\sqrt{a^{2} - b^{2}}}\right )}{\sqrt{a^{2} - b^{2}} a d} + \frac{2 \,{\left (d \sqrt{x} + c\right )}}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*b*arctan((a*e^(d*sqrt(x) + c) + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a*d) + 2*(d*sqrt(x) + c)/(a*d)