3.1031 \(\int \frac {\cosh (\sqrt {x}) \sinh (\sqrt {x})}{\sqrt {x}} \, dx\)

Optimal. Leaf size=8 \[ \sinh ^2\left (\sqrt {x}\right ) \]

[Out]

sinh(x^(1/2))^2

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {5370} \[ \sinh ^2\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[Sqrt[x]]*Sinh[Sqrt[x]])/Sqrt[x],x]

[Out]

Sinh[Sqrt[x]]^2

Rule 5370

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_)]^(p_.), x_Symbol] :> Simp[Sinh[a +
b*x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.50 \[ \frac {1}{2} \cosh \left (2 \sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[Sqrt[x]]*Sinh[Sqrt[x]])/Sqrt[x],x]

[Out]

Cosh[2*Sqrt[x]]/2

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fricas [B]  time = 0.46, size = 17, normalized size = 2.12 \[ \frac {1}{2} \, \cosh \left (\sqrt {x}\right )^{2} + \frac {1}{2} \, \sinh \left (\sqrt {x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cosh(sqrt(x))^2 + 1/2*sinh(sqrt(x))^2

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giac [B]  time = 0.12, size = 17, normalized size = 2.12 \[ \frac {1}{4} \, e^{\left (2 \, \sqrt {x}\right )} + \frac {1}{4} \, e^{\left (-2 \, \sqrt {x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*e^(2*sqrt(x)) + 1/4*e^(-2*sqrt(x))

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maple [A]  time = 0.09, size = 7, normalized size = 0.88 \[ \cosh ^{2}\left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x^(1/2))*sinh(x^(1/2))/x^(1/2),x)

[Out]

cosh(x^(1/2))^2

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maxima [A]  time = 0.31, size = 6, normalized size = 0.75 \[ \cosh \left (\sqrt {x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(sqrt(x))^2

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mupad [B]  time = 1.78, size = 6, normalized size = 0.75 \[ {\mathrm {cosh}\left (\sqrt {x}\right )}^2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x^(1/2))*sinh(x^(1/2)))/x^(1/2),x)

[Out]

cosh(x^(1/2))^2

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sympy [A]  time = 0.27, size = 7, normalized size = 0.88 \[ \sinh ^{2}{\left (\sqrt {x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x**(1/2))*sinh(x**(1/2))/x**(1/2),x)

[Out]

sinh(sqrt(x))**2

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