3.8.56 \(\int (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)) \, dx\) [756]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {2717, 2718} \begin {gather*} x \sqrt {b^2-c^2}+b \sinh (x)+c \cosh (x) \end {gather*}

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 2717

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

Rule 2718

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*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \sqrt {b^2-c^2} x+c \cosh (x)+b \sinh (x) \end {gather*}

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]

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Maple [A]
time = 0.46, size = 23, normalized size = 0.96

method	result	size
default	\(c \cosh \left (x \right )+b \sinh \left (x \right )+x \sqrt {b^{2}-c^{2}}\)	\(23\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 22, normalized size = 0.92 \begin {gather*} c \cosh \left (x\right ) + b \sinh \left (x\right ) + \sqrt {b^{2} - c^{2}} x \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
time = 0.36, size = 61, normalized size = 2.54 \begin {gather*} \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 {gather*}

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))

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Sympy [A]
time = 0.05, size = 20, normalized size = 0.83 \begin {gather*} b \sinh {\left (x \right )} + c \cosh {\left (x \right )} + x \sqrt {b^{2} - c^{2}} \end {gather*}

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)

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Giac [A]
time = 0.42, size = 36, normalized size = 1.50 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.06, size = 22, normalized size = 0.92 \begin {gather*} x\,\sqrt {b^2-c^2}+c\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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