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

Optimal. Leaf size=26 \[ 2 a \sqrt{x}+\frac{2 b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )}{d} \]

[Out]

2*a*Sqrt[x] + (2*b*ArcTan[Sinh[c + d*Sqrt[x]]])/d

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Rubi [A]  time = 0.0231044, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {14, 5436, 3770} \[ 2 a \sqrt{x}+\frac{2 b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*Sqrt[x] + (2*b*ArcTan[Sinh[c + d*Sqrt[x]]])/d

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0295161, size = 30, normalized size = 1.15 \[ \frac{2 \left (a \left (c+d \sqrt{x}\right )+b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*(c + d*Sqrt[x]) + b*ArcTan[Sinh[c + d*Sqrt[x]]]))/d

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Maple [A]  time = 0.017, size = 23, normalized size = 0.9 \begin{align*} 2\,{\frac{b\arctan \left ( \sinh \left ( c+d\sqrt{x} \right ) \right ) }{d}}+2\,a\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*arctan(sinh(c+d*x^(1/2)))/d+2*a*x^(1/2)

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Maxima [A]  time = 1.1875, size = 30, normalized size = 1.15 \begin{align*} 2 \, a \sqrt{x} + \frac{2 \, b \arctan \left (\sinh \left (d \sqrt{x} + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) + 2*b*arctan(sinh(d*sqrt(x) + c))/d

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Fricas [A]  time = 2.25631, size = 101, normalized size = 3.88 \begin{align*} \frac{2 \,{\left (a d \sqrt{x} + 2 \, b \arctan \left (\cosh \left (d \sqrt{x} + c\right ) + \sinh \left (d \sqrt{x} + c\right )\right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a*d*sqrt(x) + 2*b*arctan(cosh(d*sqrt(x) + c) + sinh(d*sqrt(x) + c)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14721, size = 39, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (d \sqrt{x} + c\right )} a}{d} + \frac{4 \, b \arctan \left (e^{\left (d \sqrt{x} + c\right )}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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