### 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[Sqrt[x]]^2

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Rubi [A]  time = 0.0140994, antiderivative size = 8, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0040892, size = 12, normalized size = 1.5 $\frac{1}{2} \cosh \left (2 \sqrt{x}\right )$

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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Maple [A]  time = 0.006, size = 7, normalized size = 0.9 \begin{align*} \left ( \cosh \left ( \sqrt{x} \right ) \right ) ^{2} \end{align*}

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 = 1.17754, size = 8, normalized size = 1. \begin{align*} \cosh \left (\sqrt{x}\right )^{2} \end{align*}

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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Fricas [B]  time = 1.95028, size = 58, normalized size = 7.25 \begin{align*} \frac{1}{2} \, \cosh \left (\sqrt{x}\right )^{2} + \frac{1}{2} \, \sinh \left (\sqrt{x}\right )^{2} \end{align*}

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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Sympy [A]  time = 0.362589, size = 7, normalized size = 0.88 \begin{align*} \cosh ^{2}{\left (\sqrt{x} \right )} \end{align*}

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]

cosh(sqrt(x))**2

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Giac [B]  time = 1.11933, size = 23, normalized size = 2.88 \begin{align*} \frac{1}{4} \, e^{\left (2 \, \sqrt{x}\right )} + \frac{1}{4} \, e^{\left (-2 \, \sqrt{x}\right )} \end{align*}

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))