3.48 \(\int \sqrt {a \text {sech}^4(x)} \, dx\)

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

[Out]

cosh(x)*sinh(x)*(a*sech(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 = {4123, 3767, 8} \[ \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Sech[x]^4],x]

[Out]

Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x]

Rule 8

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

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 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps

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

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

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Sech[x]^4],x]

[Out]

Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x]

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fricas [B]  time = 0.43, size = 81, normalized size = 5.40 \[ -\frac {2 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )}}{2 \, \cosh \relax (x) e^{\left (2 \, x\right )} \sinh \relax (x) + e^{\left (2 \, x\right )} \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*(e^(4*x) + 2*e^(2*x) + 1)*e^(2*x)/(2*cosh(x)*e^(2
*x)*sinh(x) + e^(2*x)*sinh(x)^2 + (cosh(x)^2 + 1)*e^(2*x))

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giac [A]  time = 0.11, size = 13, normalized size = 0.87 \[ -\frac {2 \, \sqrt {a}}{e^{\left (2 \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a)/(e^(2*x) + 1)

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maple [B]  time = 0.22, size = 29, normalized size = 1.93 \[ -2 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {a}}{e^{\left (-2 \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a)/(e^(-2*x) + 1)

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mupad [B]  time = 0.06, size = 71, normalized size = 4.73 \[ -\frac {\sqrt {a}\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (2\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+2\,{\mathrm {e}}^{6\,x}+\frac {{\mathrm {e}}^{8\,x}}{2}+\frac {1}{2}\right )}{\left ({\mathrm {e}}^{2\,x}+1\right )\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^(1/2)*(1/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(2*exp(2*x) + 3*exp(4*x) + 2*exp(6*x) + exp(8*x)/2 + 1/2))/((exp(
2*x) + 1)*(exp(2*x) + 2*exp(4*x) + exp(6*x)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {sech}^{4}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*sech(x)**4), x)

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