3.33 \(\int (a \text{sech}^2(x))^{3/2} \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{2} a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right )+\frac{1}{2} a \tanh (x) \sqrt{a \text{sech}^2(x)} \]

[Out]

(a^(3/2)*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a*Sech[x]^2]])/2 + (a*Sqrt[a*Sech[x]^2]*Tanh[x])/2

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Rubi [A]  time = 0.0244867, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4122, 195, 217, 203} \[ \frac{1}{2} a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right )+\frac{1}{2} a \tanh (x) \sqrt{a \text{sech}^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^2)^(3/2),x]

[Out]

(a^(3/2)*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a*Sech[x]^2]])/2 + (a*Sqrt[a*Sech[x]^2]*Tanh[x])/2

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a \text{sech}^2(x)\right )^{3/2} \, dx &=a \operatorname{Subst}\left (\int \sqrt{a-a x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} a \sqrt{a \text{sech}^2(x)} \tanh (x)+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} a \sqrt{a \text{sech}^2(x)} \tanh (x)+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right )\\ &=\frac{1}{2} a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right )+\frac{1}{2} a \sqrt{a \text{sech}^2(x)} \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.0192026, size = 29, normalized size = 0.63 \[ \frac{1}{2} a \sqrt{a \text{sech}^2(x)} \left (\tanh (x)+2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^2)^(3/2),x]

[Out]

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

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Maple [C]  time = 0.066, size = 106, normalized size = 2.3 \begin{align*}{\frac{a \left ({{\rm e}^{2\,x}}-1 \right ) }{{{\rm e}^{2\,x}}+1}\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}+{\frac{i}{2}}a{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ) \sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}\ln \left ({{\rm e}^{x}}+i \right ) -{\frac{i}{2}}a{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ) \sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}\ln \left ({{\rm e}^{x}}-i \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sech(x)^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.83217, size = 53, normalized size = 1.15 \begin{align*} a^{\frac{3}{2}} \arctan \left (e^{x}\right ) + \frac{a^{\frac{3}{2}} e^{\left (3 \, x\right )} - a^{\frac{3}{2}} e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.26341, size = 952, normalized size = 20.7 \begin{align*} \frac{{\left (a \cosh \left (x\right )^{3} +{\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{3} + 3 \,{\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (a \cosh \left (x\right )^{4} +{\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{4} + 4 \,{\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + 2 \,{\left (3 \, a \cosh \left (x\right )^{2} +{\left (3 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} +{\left (a \cosh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \,{\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right ) +{\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - a \cosh \left (x\right ) +{\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} +{\left (3 \, a \cosh \left (x\right )^{2} +{\left (3 \, a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) +{\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^2)^(3/2),x, algorithm="fricas")

[Out]

(a*cosh(x)^3 + (a*e^(2*x) + a)*sinh(x)^3 + 3*(a*cosh(x)*e^(2*x) + a*cosh(x))*sinh(x)^2 + (a*cosh(x)^4 + (a*e^(
2*x) + a)*sinh(x)^4 + 4*(a*cosh(x)*e^(2*x) + a*cosh(x))*sinh(x)^3 + 2*a*cosh(x)^2 + 2*(3*a*cosh(x)^2 + (3*a*co
sh(x)^2 + a)*e^(2*x) + a)*sinh(x)^2 + (a*cosh(x)^4 + 2*a*cosh(x)^2 + a)*e^(2*x) + 4*(a*cosh(x)^3 + a*cosh(x) +
(a*cosh(x)^3 + a*cosh(x))*e^(2*x))*sinh(x) + a)*arctan(cosh(x) + sinh(x)) - a*cosh(x) + (a*cosh(x)^3 - a*cosh
(x))*e^(2*x) + (3*a*cosh(x)^2 + (3*a*cosh(x)^2 - a)*e^(2*x) - a)*sinh(x))*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^
x/(4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*e^x*sinh(x)^2 + 4*(cosh(x)^3 + cosh(x))*e^x*s
inh(x) + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)**2)**(3/2),x)

[Out]

Integral((a*sech(x)**2)**(3/2), x)

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Giac [A]  time = 1.12518, size = 65, normalized size = 1.41 \begin{align*} \frac{1}{4} \,{\left (\pi - \frac{4 \,{\left (e^{\left (-x\right )} - e^{x}\right )}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4} + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/4*(pi - 4*(e^(-x) - e^x)/((e^(-x) - e^x)^2 + 4) + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*a^(3/2)