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]

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

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.36, size = 46, normalized size = 2.30 \[ \frac {2 \sinh (x) \left (x-\frac {2 \sinh (x) \cosh (x) \sqrt {\tanh ^2\left (\frac {x}{2}\right )}}{(\cosh (x)-1)^{3/2} \sqrt {\cosh (x)+1}}\right )}{\sqrt {\cosh (x)}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cosh \relax (x)} + \frac {x}{\cosh \relax (x)^{\frac {3}{2}}}\,{d x} \]

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)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cosh \relax (x )^{\frac {3}{2}}}+x \left (\sqrt {\cosh }\relax (x )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cosh \relax (x)} + \frac {x}{\cosh \relax (x)^{\frac {3}{2}}}\,{d x} \]

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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mupad [B]  time = 0.15, size = 39, normalized size = 1.95 \[ -\frac {2\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\,\left (x+2\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\cosh ^{2}{\relax (x )} + 1\right )}{\cosh ^{\frac {3}{2}}{\relax (x )}}\, dx \]

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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