3.77 \(\int \sqrt{a+b \cosh ^2(x)} \tanh (x) \, dx\)

Optimal. Leaf size=39 \[ \sqrt{a+b \cosh ^2(x)}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0687702, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3194, 50, 63, 208} \[ \sqrt{a+b \cosh ^2(x)}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cosh[x]^2]*Tanh[x],x]

[Out]

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

Rule 3194

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 \sqrt{a+b \cosh ^2(x)} \tanh (x) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\cosh ^2(x)\right )\\ &=\sqrt{a+b \cosh ^2(x)}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\cosh ^2(x)\right )\\ &=\sqrt{a+b \cosh ^2(x)}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^2(x)}\right )}{b}\\ &=-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right )+\sqrt{a+b \cosh ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0248286, size = 39, normalized size = 1. \[ \sqrt{a+b \cosh ^2(x)}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Cosh[x]^2]*Tanh[x],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 42, normalized size = 1.1 \begin{align*} \sqrt{a+b \left ( \cosh \left ( x \right ) \right ) ^{2}}-\sqrt{a}\ln \left ({\frac{1}{\cosh \left ( x \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \cosh \left ( x \right ) \right ) ^{2}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cosh(x)^2)^(1/2)*tanh(x),x)

[Out]

(a+b*cosh(x)^2)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(a+b*cosh(x)^2)^(1/2))/cosh(x))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (x\right )^{2} + a} \tanh \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x)^2)^(1/2)*tanh(x),x, algorithm="maxima")

[Out]

integrate(sqrt(b*cosh(x)^2 + a)*tanh(x), x)

________________________________________________________________________________________

Fricas [B]  time = 4.66662, size = 1192, normalized size = 30.56 \begin{align*} \left [\frac{\sqrt{a}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \,{\left (4 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + b\right )} \sinh \left (x\right )^{2} - 4 \, \sqrt{2} \sqrt{a} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 4 \,{\left (b \cosh \left (x\right )^{3} +{\left (4 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) + \sqrt{2} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}, \frac{2 \, \sqrt{-a}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}\right ) + \sqrt{2} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x)^2)^(1/2)*tanh(x),x, algorithm="fricas")

[Out]

[1/2*(sqrt(a)*(cosh(x) + sinh(x))*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(4*a + b)*cosh(x)
^2 + 2*(3*b*cosh(x)^2 + 4*a + b)*sinh(x)^2 - 4*sqrt(2)*sqrt(a)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(cos
h(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))*(cosh(x) + sinh(x)) + 4*(b*cosh(x)^3 + (4*a + b)*cosh(x))*sinh(x) + b
)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 +
cosh(x))*sinh(x) + 1)) + sqrt(2)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + s
inh(x)^2)))/(cosh(x) + sinh(x)), 1/2*(2*sqrt(-a)*(cosh(x) + sinh(x))*arctan(1/2*sqrt(2)*sqrt(-a)*sqrt((b*cosh(
x)^2 + b*sinh(x)^2 + 2*a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*cosh(x) + a*sinh(x))) + sqrt(2)*
sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x) + sinh(x))]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x)**2)**(1/2)*tanh(x),x)

[Out]

Integral(sqrt(a + b*cosh(x)**2)*tanh(x), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (x\right )^{2} + a} \tanh \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x)^2)^(1/2)*tanh(x),x, algorithm="giac")

[Out]

integrate(sqrt(b*cosh(x)^2 + a)*tanh(x), x)