3.329 \(\int (\frac{x}{\cosh ^{\frac{3}{2}}(x)}+x \sqrt{\cosh (x)}) \, dx\)

Optimal. Leaf size=20 \[ \frac{2 x \sinh (x)}{\sqrt{\cosh (x)}}-4 \sqrt{\cosh (x)} \]

[Out]

-4*Sqrt[Cosh[x]] + (2*x*Sinh[x])/Sqrt[Cosh[x]]

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Rubi [A]  time = 0.0498668, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3315} \[ \frac{2 x \sinh (x)}{\sqrt{\cosh (x)}}-4 \sqrt{\cosh (x)} \]

Antiderivative was successfully verified.

[In]

Int[x/Cosh[x]^(3/2) + x*Sqrt[Cosh[x]],x]

[Out]

-4*Sqrt[Cosh[x]] + (2*x*Sinh[x])/Sqrt[Cosh[x]]

Rule 3315

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)*Cos[e + f*x]*(b*Si
n[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2),
x], x] - Simp[(d*(b*Sin[e + f*x])^(n + 2))/(b^2*f^2*(n + 1)*(n + 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[
n, -1] && NeQ[n, -2]

Rubi steps

\begin{align*} \int \left (\frac{x}{\cosh ^{\frac{3}{2}}(x)}+x \sqrt{\cosh (x)}\right ) \, dx &=\int \frac{x}{\cosh ^{\frac{3}{2}}(x)} \, dx+\int x \sqrt{\cosh (x)} \, dx\\ &=-4 \sqrt{\cosh (x)}+\frac{2 x \sinh (x)}{\sqrt{\cosh (x)}}\\ \end{align*}

Mathematica [B]  time = 0.346142, size = 46, normalized size = 2.3 \[ \frac{2 \sinh (x) \left (x-\frac{2 \sinh (x) \cosh (x) \sqrt{\tanh ^2\left (\frac{x}{2}\right )}}{(\cosh (x)-1)^{3/2} \sqrt{\cosh (x)+1}}\right )}{\sqrt{\cosh (x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/Cosh[x]^(3/2) + x*Sqrt[Cosh[x]],x]

[Out]

(2*Sinh[x]*(x - (2*Cosh[x]*Sinh[x]*Sqrt[Tanh[x/2]^2])/((-1 + Cosh[x])^(3/2)*Sqrt[1 + Cosh[x]])))/Sqrt[Cosh[x]]

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}+x\sqrt{\cosh \left ( x \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/cosh(x)^(3/2)+x*cosh(x)^(1/2),x)

[Out]

int(x/cosh(x)^(3/2)+x*cosh(x)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cosh(x)^(3/2)+x*cosh(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*sqrt(cosh(x)) + x/cosh(x)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cosh(x)^(3/2)+x*cosh(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (\cosh ^{2}{\left (x \right )} + 1\right )}{\cosh ^{\frac{3}{2}}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cosh(x)**(3/2)+x*cosh(x)**(1/2),x)

[Out]

Integral(x*(cosh(x)**2 + 1)/cosh(x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cosh(x)^(3/2)+x*cosh(x)^(1/2),x, algorithm="giac")

[Out]

integrate(x*sqrt(cosh(x)) + x/cosh(x)^(3/2), x)