3.331 \(\int (\frac{x}{\cosh ^{\frac{7}{2}}(x)}+\frac{3}{5} x \sqrt{\cosh (x)}) \, dx\)

Optimal. Leaf size=47 \[ \frac{4}{15 \cosh ^{\frac{3}{2}}(x)}-\frac{12 \sqrt{\cosh (x)}}{5}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{6 x \sinh (x)}{5 \sqrt{\cosh (x)}} \]

[Out]

4/(15*Cosh[x]^(3/2)) - (12*Sqrt[Cosh[x]])/5 + (2*x*Sinh[x])/(5*Cosh[x]^(5/2)) + (6*x*Sinh[x])/(5*Sqrt[Cosh[x]]
)

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Rubi [A]  time = 0.0708335, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3315} \[ \frac{4}{15 \cosh ^{\frac{3}{2}}(x)}-\frac{12 \sqrt{\cosh (x)}}{5}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{6 x \sinh (x)}{5 \sqrt{\cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Cosh[x]^(7/2) + (3*x*Sqrt[Cosh[x]])/5,x]

[Out]

4/(15*Cosh[x]^(3/2)) - (12*Sqrt[Cosh[x]])/5 + (2*x*Sinh[x])/(5*Cosh[x]^(5/2)) + (6*x*Sinh[x])/(5*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{7}{2}}(x)}+\frac{3}{5} x \sqrt{\cosh (x)}\right ) \, dx &=\frac{3}{5} \int x \sqrt{\cosh (x)} \, dx+\int \frac{x}{\cosh ^{\frac{7}{2}}(x)} \, dx\\ &=\frac{4}{15 \cosh ^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{3}{5} \int \frac{x}{\cosh ^{\frac{3}{2}}(x)} \, dx+\frac{3}{5} \int x \sqrt{\cosh (x)} \, dx\\ &=\frac{4}{15 \cosh ^{\frac{3}{2}}(x)}-\frac{12 \sqrt{\cosh (x)}}{5}+\frac{2 x \sinh (x)}{5 \cosh ^{\frac{5}{2}}(x)}+\frac{6 x \sinh (x)}{5 \sqrt{\cosh (x)}}\\ \end{align*}

Mathematica [A]  time = 0.583576, size = 64, normalized size = 1.36 \[ \frac{1}{5} \sqrt{\cosh (x)} \left (6 x \tanh (x)+\left (2 x \tanh (x)+\frac{4}{3}\right ) \text{sech}^2(x)-\frac{12 \sinh ^2(x)}{\sqrt{\cosh (x)-1} (\cosh (x)+1)^{3/2} \sqrt{\tanh ^2\left (\frac{x}{2}\right )}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/Cosh[x]^(7/2) + (3*x*Sqrt[Cosh[x]])/5,x]

[Out]

(Sqrt[Cosh[x]]*((-12*Sinh[x]^2)/(Sqrt[-1 + Cosh[x]]*(1 + Cosh[x])^(3/2)*Sqrt[Tanh[x/2]^2]) + 6*x*Tanh[x] + Sec
h[x]^2*(4/3 + 2*x*Tanh[x])))/5

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,x}{5}\sqrt{\cosh \left ( x \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/cosh(x)^(7/2)+3/5*x*cosh(x)^(1/2),x)

[Out]

int(x/cosh(x)^(7/2)+3/5*x*cosh(x)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3}{5} \, x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cosh(x)^(7/2)+3/5*x*cosh(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(3/5*x*sqrt(cosh(x)) + x/cosh(x)^(7/2), 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)^(7/2)+3/5*x*cosh(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cosh(x)**(7/2)+3/5*x*cosh(x)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3}{5} \, x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cosh(x)^(7/2)+3/5*x*cosh(x)^(1/2),x, algorithm="giac")

[Out]

integrate(3/5*x*sqrt(cosh(x)) + x/cosh(x)^(7/2), x)