### 3.756 $$\int (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)) \, dx$$

Optimal. Leaf size=24 $x \sqrt{b^2-c^2}+b \sinh (x)+c \cosh (x)$

[Out]

Sqrt[b^2 - c^2]*x + c*Cosh[x] + b*Sinh[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0108072, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {2637, 2638} $x \sqrt{b^2-c^2}+b \sinh (x)+c \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x],x]

[Out]

Sqrt[b^2 - c^2]*x + c*Cosh[x] + b*Sinh[x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0080919, size = 24, normalized size = 1. $x \sqrt{b^2-c^2}+b \sinh (x)+c \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x],x]

[Out]

Sqrt[b^2 - c^2]*x + c*Cosh[x] + b*Sinh[x]

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 23, normalized size = 1. \begin{align*} c\cosh \left ( x \right ) +b\sinh \left ( x \right ) +x\sqrt{{b}^{2}-{c}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2),x)

[Out]

c*cosh(x)+b*sinh(x)+x*(b^2-c^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.00005, size = 30, normalized size = 1.25 \begin{align*} c \cosh \left (x\right ) + b \sinh \left (x\right ) + \sqrt{b^{2} - c^{2}} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*cosh(x) + b*sinh(x) + sqrt(b^2 - c^2)*x

________________________________________________________________________________________

Fricas [B]  time = 2.72436, size = 196, normalized size = 8.17 \begin{align*} \frac{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{b^{2} - c^{2}}{\left (x \cosh \left (x\right ) + x \sinh \left (x\right )\right )} - b + c}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.268791, size = 20, normalized size = 0.83 \begin{align*} b \sinh{\left (x \right )} + c \cosh{\left (x \right )} + x \sqrt{b^{2} - c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*cosh(x)+c*sinh(x)+(b**2-c**2)**(1/2),x)

[Out]

b*sinh(x) + c*cosh(x) + x*sqrt(b**2 - c**2)

________________________________________________________________________________________

Giac [A]  time = 1.14882, size = 49, normalized size = 2.04 \begin{align*} \frac{1}{2} \, c{\left (e^{\left (-x\right )} + e^{x}\right )} - \frac{1}{2} \, b{\left (e^{\left (-x\right )} - e^{x}\right )} + \sqrt{b^{2} - c^{2}} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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