3.58 \(\int \frac{(a+b \text{csch}(c+d \sqrt{x}))^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=47 \[ 2 a^2 \sqrt{x}-\frac{4 a b \tanh ^{-1}\left (\cosh \left (c+d \sqrt{x}\right )\right )}{d}-\frac{2 b^2 \coth \left (c+d \sqrt{x}\right )}{d} \]

[Out]

2*a^2*Sqrt[x] - (4*a*b*ArcTanh[Cosh[c + d*Sqrt[x]]])/d - (2*b^2*Coth[c + d*Sqrt[x]])/d

________________________________________________________________________________________

Rubi [A]  time = 0.0604685, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {5437, 3773, 3770, 3767, 8} \[ 2 a^2 \sqrt{x}-\frac{4 a b \tanh ^{-1}\left (\cosh \left (c+d \sqrt{x}\right )\right )}{d}-\frac{2 b^2 \coth \left (c+d \sqrt{x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*Sqrt[x]])^2/Sqrt[x],x]

[Out]

2*a^2*Sqrt[x] - (4*a*b*ArcTanh[Cosh[c + d*Sqrt[x]]])/d - (2*b^2*Coth[c + d*Sqrt[x]])/d

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 3773

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

Rule 3770

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.26818, size = 75, normalized size = 1.6 \[ -\frac{-2 a \left (a c+a d \sqrt{x}+2 b \log \left (\tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )\right )+b^2 \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )+b^2 \coth \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Csch[c + d*Sqrt[x]])^2/Sqrt[x],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 44, normalized size = 0.9 \begin{align*} 2\,{\frac{{a}^{2} \left ( c+d\sqrt{x} \right ) -4\,ba{\it Artanh} \left ({{\rm e}^{c+d\sqrt{x}}} \right ) -{b}^{2}{\rm coth} \left (c+d\sqrt{x}\right )}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(a^2*(c+d*x^(1/2))-4*b*a*arctanh(exp(c+d*x^(1/2)))-b^2*coth(c+d*x^(1/2)))

________________________________________________________________________________________

Maxima [A]  time = 1.1064, size = 69, normalized size = 1.47 \begin{align*} 2 \, a^{2} \sqrt{x} + \frac{4 \, a b \log \left (\tanh \left (\frac{1}{2} \, d \sqrt{x} + \frac{1}{2} \, c\right )\right )}{d} + \frac{4 \, b^{2}}{d{\left (e^{\left (-2 \, d \sqrt{x} - 2 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^2*sqrt(x) + 4*a*b*log(tanh(1/2*d*sqrt(x) + 1/2*c))/d + 4*b^2/(d*(e^(-2*d*sqrt(x) - 2*c) - 1))

________________________________________________________________________________________

Fricas [B]  time = 1.9765, size = 801, normalized size = 17.04 \begin{align*} \frac{2 \,{\left (a^{2} d \sqrt{x} \cosh \left (d \sqrt{x} + c\right )^{2} + 2 \, a^{2} d \sqrt{x} \cosh \left (d \sqrt{x} + c\right ) \sinh \left (d \sqrt{x} + c\right ) + a^{2} d \sqrt{x} \sinh \left (d \sqrt{x} + c\right )^{2} - a^{2} d \sqrt{x} - 2 \, b^{2} - 2 \,{\left (a b \cosh \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \cosh \left (d \sqrt{x} + c\right ) \sinh \left (d \sqrt{x} + c\right ) + a b \sinh \left (d \sqrt{x} + c\right )^{2} - a b\right )} \log \left (\cosh \left (d \sqrt{x} + c\right ) + \sinh \left (d \sqrt{x} + c\right ) + 1\right ) + 2 \,{\left (a b \cosh \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \cosh \left (d \sqrt{x} + c\right ) \sinh \left (d \sqrt{x} + c\right ) + a b \sinh \left (d \sqrt{x} + c\right )^{2} - a b\right )} \log \left (\cosh \left (d \sqrt{x} + c\right ) + \sinh \left (d \sqrt{x} + c\right ) - 1\right )\right )}}{d \cosh \left (d \sqrt{x} + c\right )^{2} + 2 \, d \cosh \left (d \sqrt{x} + c\right ) \sinh \left (d \sqrt{x} + c\right ) + d \sinh \left (d \sqrt{x} + c\right )^{2} - d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a^2*d*sqrt(x)*cosh(d*sqrt(x) + c)^2 + 2*a^2*d*sqrt(x)*cosh(d*sqrt(x) + c)*sinh(d*sqrt(x) + c) + a^2*d*sqrt(
x)*sinh(d*sqrt(x) + c)^2 - a^2*d*sqrt(x) - 2*b^2 - 2*(a*b*cosh(d*sqrt(x) + c)^2 + 2*a*b*cosh(d*sqrt(x) + c)*si
nh(d*sqrt(x) + c) + a*b*sinh(d*sqrt(x) + c)^2 - a*b)*log(cosh(d*sqrt(x) + c) + sinh(d*sqrt(x) + c) + 1) + 2*(a
*b*cosh(d*sqrt(x) + c)^2 + 2*a*b*cosh(d*sqrt(x) + c)*sinh(d*sqrt(x) + c) + a*b*sinh(d*sqrt(x) + c)^2 - a*b)*lo
g(cosh(d*sqrt(x) + c) + sinh(d*sqrt(x) + c) - 1))/(d*cosh(d*sqrt(x) + c)^2 + 2*d*cosh(d*sqrt(x) + c)*sinh(d*sq
rt(x) + c) + d*sinh(d*sqrt(x) + c)^2 - d)

________________________________________________________________________________________

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

[Out]

Integral((a + b*csch(c + d*sqrt(x)))**2/sqrt(x), x)

________________________________________________________________________________________

Giac [A]  time = 1.22822, size = 103, normalized size = 2.19 \begin{align*} \frac{2 \,{\left (d \sqrt{x} + c\right )} a^{2}}{d} - \frac{4 \, a b \log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right )}{d} + \frac{4 \, a b \log \left ({\left | e^{\left (d \sqrt{x} + c\right )} - 1 \right |}\right )}{d} - \frac{4 \, b^{2}}{d{\left (e^{\left (2 \, d \sqrt{x} + 2 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(d*sqrt(x) + c)*a^2/d - 4*a*b*log(e^(d*sqrt(x) + c) + 1)/d + 4*a*b*log(abs(e^(d*sqrt(x) + c) - 1))/d - 4*b^2
/(d*(e^(2*d*sqrt(x) + 2*c) - 1))