3.19 \(\int \sqrt{b \coth ^2(c+d x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tanh (c+d x) \sqrt{b \coth ^2(c+d x)} \log (\sinh (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0196248, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3658, 3475} \[ \frac{\tanh (c+d x) \sqrt{b \coth ^2(c+d x)} \log (\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Coth[c + d*x]^2],x]

[Out]

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

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3475

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

Rubi steps

\begin{align*} \int \sqrt{b \coth ^2(c+d x)} \, dx &=\left (\sqrt{b \coth ^2(c+d x)} \tanh (c+d x)\right ) \int \coth (c+d x) \, dx\\ &=\frac{\sqrt{b \coth ^2(c+d x)} \log (\sinh (c+d x)) \tanh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0452803, size = 39, normalized size = 1.26 \[ \frac{\tanh (c+d x) \sqrt{b \coth ^2(c+d x)} (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Coth[c + d*x]^2],x]

[Out]

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

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Maple [A]  time = 0.036, size = 45, normalized size = 1.5 \begin{align*} -{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) +\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) }{2\,d{\rm coth} \left (dx+c\right )}\sqrt{b \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d*(b*coth(d*x+c)^2)^(1/2)*(ln(coth(d*x+c)-1)+ln(coth(d*x+c)+1))/coth(d*x+c)

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Maxima [A]  time = 1.59022, size = 73, normalized size = 2.35 \begin{align*} -\frac{{\left (d x + c\right )} \sqrt{b}}{d} - \frac{\sqrt{b} \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{\sqrt{b} \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.93362, size = 304, normalized size = 9.81 \begin{align*} -\frac{{\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x -{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \log \left (\frac{2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )} \sqrt{\frac{b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{d e^{\left (2 \, d x + 2 \, c\right )} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \coth ^{2}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*coth(c + d*x)**2), x)

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Giac [A]  time = 1.15947, size = 73, normalized size = 2.35 \begin{align*} -\frac{{\left ({\left (d x + c\right )} \mathrm{sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) \mathrm{sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )\right )} \sqrt{b}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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