3.137 \(\int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx\)

Optimal. Leaf size=15 \[ \frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \]

[Out]

cosh(x)*sinh(x)/(a*cosh(x)^4)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 3767, 8} \[ \frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cosh[x]^4],x]

[Out]

(Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^4]

Rule 8

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

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 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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx &=\frac {\cosh ^2(x) \int \text {sech}^2(x) \, dx}{\sqrt {a \cosh ^4(x)}}\\ &=\frac {\left (i \cosh ^2(x)\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (x))}{\sqrt {a \cosh ^4(x)}}\\ &=\frac {\cosh (x) \sinh (x)}{\sqrt {a \cosh ^4(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cosh[x]^4],x]

[Out]

(Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^4]

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fricas [B]  time = 2.33, size = 116, normalized size = 7.73 \[ -\frac {2 \, \sqrt {a e^{\left (8 \, x\right )} + 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} + 4 \, a e^{\left (2 \, x\right )} + a}}{a \cosh \relax (x)^{2} + {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{2} + {\left (a \cosh \relax (x)^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (a \cosh \relax (x)^{2} + a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \relax (x) e^{\left (4 \, x\right )} + 2 \, a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x) + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^4)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(a*e^(8*x) + 4*a*e^(6*x) + 6*a*e^(4*x) + 4*a*e^(2*x) + a)/(a*cosh(x)^2 + (a*e^(4*x) + 2*a*e^(2*x) + a)*
sinh(x)^2 + (a*cosh(x)^2 + a)*e^(4*x) + 2*(a*cosh(x)^2 + a)*e^(2*x) + 2*(a*cosh(x)*e^(4*x) + 2*a*cosh(x)*e^(2*
x) + a*cosh(x))*sinh(x) + a)

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giac [A]  time = 0.13, size = 13, normalized size = 0.87 \[ -\frac {2}{\sqrt {a} {\left (e^{\left (2 \, x\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^4)^(1/2),x, algorithm="giac")

[Out]

-2/(sqrt(a)*(e^(2*x) + 1))

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maple [B]  time = 0.37, size = 56, normalized size = 3.73 \[ \frac {\sqrt {8}\, \sqrt {2}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}\, \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}}{4 a \sinh \left (2 x \right ) \sqrt {\left (\cosh \left (2 x \right )+1\right )^{2} a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^4)^(1/2),x)

[Out]

1/4*8^(1/2)*2^(1/2)*(a*(-1+cosh(2*x))*(cosh(2*x)+1))^(1/2)/a*(a*sinh(2*x)^2)^(1/2)/sinh(2*x)/((cosh(2*x)+1)^2*
a)^(1/2)

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maxima [A]  time = 0.42, size = 16, normalized size = 1.07 \[ \frac {2}{\sqrt {a} e^{\left (-2 \, x\right )} + \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^4)^(1/2),x, algorithm="maxima")

[Out]

2/(sqrt(a)*e^(-2*x) + sqrt(a))

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mupad [B]  time = 0.06, size = 39, normalized size = 2.60 \[ -\frac {{\mathrm {e}}^{-x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^4)^(1/2),x)

[Out]

-(exp(-x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(a*(exp(-x)/2 + exp(x)/2)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)**4)**(1/2),x)

[Out]

Timed out

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