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

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

[Out]

2*a*Sqrt[x] - (2*b*ArcTanh[Cosh[c + d*Sqrt[x]]])/d

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Rubi [A]  time = 0.0252635, 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, 5437, 3770} \[ 2 a \sqrt{x}-\frac{2 b \tanh ^{-1}\left (\cosh \left (c+d \sqrt{x}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*Sqrt[x] - (2*b*ArcTanh[Cosh[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 5437

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[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{csch}\left (c+d \sqrt{x}\right )}{\sqrt{x}} \, dx &=\int \left (\frac{a}{\sqrt{x}}+\frac{b \text{csch}\left (c+d \sqrt{x}\right )}{\sqrt{x}}\right ) \, dx\\ &=2 a \sqrt{x}+b \int \frac{\text{csch}\left (c+d \sqrt{x}\right )}{\sqrt{x}} \, dx\\ &=2 a \sqrt{x}+(2 b) \operatorname{Subst}\left (\int \text{csch}(c+d x) \, dx,x,\sqrt{x}\right )\\ &=2 a \sqrt{x}-\frac{2 b \tanh ^{-1}\left (\cosh \left (c+d \sqrt{x}\right )\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0438647, size = 34, normalized size = 1.31 \[ \frac{2 \left (a \left (c+d \sqrt{x}\right )+b \log \left (\tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*(c + d*Sqrt[x]) + b*Log[Tanh[(c + d*Sqrt[x])/2]]))/d

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Maple [A]  time = 0.02, size = 26, normalized size = 1. \begin{align*} 2\,a\sqrt{x}+2\,{\frac{b\ln \left ( \tanh \left ( c/2+1/2\,d\sqrt{x} \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09464, size = 34, normalized size = 1.31 \begin{align*} 2 \, a \sqrt{x} + \frac{2 \, b \log \left (\tanh \left (\frac{1}{2} \, d \sqrt{x} + \frac{1}{2} \, c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) + 2*b*log(tanh(1/2*d*sqrt(x) + 1/2*c))/d

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Fricas [B]  time = 1.93188, size = 174, normalized size = 6.69 \begin{align*} \frac{2 \,{\left (a d \sqrt{x} - b \log \left (\cosh \left (d \sqrt{x} + c\right ) + \sinh \left (d \sqrt{x} + c\right ) + 1\right ) + b \log \left (\cosh \left (d \sqrt{x} + c\right ) + \sinh \left (d \sqrt{x} + c\right ) - 1\right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a*d*sqrt(x) - b*log(cosh(d*sqrt(x) + c) + sinh(d*sqrt(x) + c) + 1) + b*log(cosh(d*sqrt(x) + c) + sinh(d*sqr
t(x) + c) - 1))/d

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

[Out]

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

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Giac [B]  time = 1.15436, size = 66, normalized size = 2.54 \begin{align*} \frac{2 \,{\left (d \sqrt{x} + c\right )} a}{d} - \frac{2 \, b \log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right )}{d} + \frac{2 \, b \log \left ({\left | e^{\left (d \sqrt{x} + c\right )} - 1 \right |}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(d*sqrt(x) + c)*a/d - 2*b*log(e^(d*sqrt(x) + c) + 1)/d + 2*b*log(abs(e^(d*sqrt(x) + c) - 1))/d