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

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Rubi [A]  time = 0.0166324, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 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]

Rule 8

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

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*}

Mathematica [A]  time = 0.0048388, size = 15, normalized size = 1. \[ \frac{\sinh (x) \cosh (x)}{\sqrt{a \cosh ^4(x)}} \]

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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Maple [B]  time = 0.085, size = 56, normalized size = 3.7 \begin{align*}{\frac{\sqrt{8}\sqrt{2}}{4\,a\sinh \left ( 2\,x \right ) }\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) }\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}{\frac{1}{\sqrt{a \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{2}}}}} \end{align*}

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

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Maxima [A]  time = 1.72424, size = 22, normalized size = 1.47 \begin{align*} \frac{2}{\sqrt{a} e^{\left (-2 \, x\right )} + \sqrt{a}} \end{align*}

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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Fricas [B]  time = 1.88053, size = 338, normalized size = 22.53 \begin{align*} -\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 \left (x\right )^{2} +{\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} +{\left (a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, 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 (4 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a} \end{align*}

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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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)**4)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16102, size = 18, normalized size = 1.2 \begin{align*} -\frac{2}{\sqrt{a}{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}

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))