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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 1.32, size = 186, normalized size = 11.62 \[ \left [-\frac {\sqrt {-a} \log \left (\frac {a \cosh \relax (x)^{2} - 2 \, \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} {\left (\cosh \relax (x) e^{x} + e^{x} \sinh \relax (x)\right )} \sqrt {-a} e^{\left (-x\right )} + {\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{2} + {\left (a \cosh \relax (x)^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x) - a}{{\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) + 1}\right )}{a}, \frac {2 \, \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right )}{a e^{\left (2 \, x\right )} + a}\right ] \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.22, size = 55, normalized size = 3.44 \[ -\frac {\cosh \relax (x ) \sqrt {a \left (\sinh ^{2}\relax (x )\right )}\, \ln \left (\frac {2 \sqrt {-a}\, \sqrt {a \left (\sinh ^{2}\relax (x )\right )}-2 a}{\cosh \relax (x )}\right )}{\sqrt {-a}\, \sinh \relax (x ) \sqrt {a \left (\cosh ^{2}\relax (x )\right )}} \]

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 = 0.46, size = 8, normalized size = 0.50 \[ \frac {2 \, \arctan \left (e^{x}\right )}{\sqrt {a}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {a\,{\mathrm {cosh}\relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*cosh(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh ^{2}{\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*cosh(x)**2), x)

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