3.125 \(\int \frac{1}{\sqrt{a \cosh ^2(x)}} \, dx\)

Optimal. Leaf size=16 \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{a \cosh ^2(x)}} \]

[Out]

(ArcTan[Sinh[x]]*Cosh[x])/Sqrt[a*Cosh[x]^2]

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Rubi [A]  time = 0.0149485, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3770} \[ \frac{\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt{a \cosh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cosh[x]^2],x]

[Out]

(ArcTan[Sinh[x]]*Cosh[x])/Sqrt[a*Cosh[x]^2]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[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 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{1}{\sqrt{a \cosh ^2(x)}} \, dx &=\frac{\cosh (x) \int \text{sech}(x) \, dx}{\sqrt{a \cosh ^2(x)}}\\ &=\frac{\tan ^{-1}(\sinh (x)) \cosh (x)}{\sqrt{a \cosh ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0065387, size = 21, normalized size = 1.31 \[ \frac{2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a \cosh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cosh[x]^2],x]

[Out]

(2*ArcTan[Tanh[x/2]]*Cosh[x])/Sqrt[a*Cosh[x]^2]

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Maple [B]  time = 0.044, size = 55, normalized size = 3.4 \begin{align*} -{\frac{\cosh \left ( x \right ) }{\sinh \left ( x \right ) }\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{-a}\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}-a}{\cosh \left ( x \right ) }} \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.69339, size = 11, normalized size = 0.69 \begin{align*} \frac{2 \, \arctan \left (e^{x}\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(e^x)/sqrt(a)

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Fricas [B]  time = 2.18351, size = 552, normalized size = 34.5 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (\frac{a \cosh \left (x\right )^{2} - 2 \, \sqrt{a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a}{\left (\cosh \left (x\right ) e^{x} + e^{x} \sinh \left (x\right )\right )} \sqrt{-a} e^{\left (-x\right )} +{\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} +{\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \,{\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a}{{\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right )}{a}, \frac{2 \, \sqrt{a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a e^{\left (2 \, x\right )} + a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-sqrt(-a)*log((a*cosh(x)^2 - 2*sqrt(a*e^(4*x) + 2*a*e^(2*x) + a)*(cosh(x)*e^x + e^x*sinh(x))*sqrt(-a)*e^(-x)
+ (a*e^(2*x) + a)*sinh(x)^2 + (a*cosh(x)^2 - a)*e^(2*x) + 2*(a*cosh(x)*e^(2*x) + a*cosh(x))*sinh(x) - a)/((e^(
2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) + cosh(x))*sinh(x) + 1))/a, 2*s
qrt(a*e^(4*x) + 2*a*e^(2*x) + a)*arctan(cosh(x) + sinh(x))/(a*e^(2*x) + a)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)**2)**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError