\(\int (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}) \, dx\) [71]

Optimal result

Integrand size = 17, antiderivative size = 20 \[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=-4 \sqrt {\cosh (x)}+\frac {2 x \sinh (x)}{\sqrt {\cosh (x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3396} \[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=\frac {2 x \sinh (x)}{\sqrt {\cosh (x)}}-4 \sqrt {\cosh (x)} \]

[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 3396

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*} \text {integral}& = \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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(20)=40\).

Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=\frac {2 \sinh (x) \left (x-\frac {2 \cosh (x) \sinh (x) \sqrt {\tanh ^2\left (\frac {x}{2}\right )}}{(-1+\cosh (x))^{3/2} \sqrt {1+\cosh (x)}}\right )}{\sqrt {\cosh (x)}} \]

[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]]

Maple [F]

\[\int \left (\frac {x}{\cosh \left (x \right )^{\frac {3}{2}}}+x \sqrt {\cosh \left (x \right )}\right )d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=\int \frac {x \left (\cosh ^{2}{\left (x \right )} + 1\right )}{\cosh ^{\frac {3}{2}}{\left (x \right )}}\, dx \]

[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)

Maxima [F]

\[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=\int { x \sqrt {\cosh \left (x\right )} + \frac {x}{\cosh \left (x\right )^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=\int { x \sqrt {\cosh \left (x\right )} + \frac {x}{\cosh \left (x\right )^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx=-\frac {2\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\,\left (x+2\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}+1} \]

[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)