3.92 \(\int \sqrt{a-a \cosh (x)} (A+B \cosh (x)) \, dx\)

Optimal. Leaf size=44 \[ \frac{2}{3} B \sinh (x) \sqrt{a-a \cosh (x)}-\frac{2 a (3 A-B) \sinh (x)}{3 \sqrt{a-a \cosh (x)}} \]

[Out]

(-2*a*(3*A - B)*Sinh[x])/(3*Sqrt[a - a*Cosh[x]]) + (2*B*Sqrt[a - a*Cosh[x]]*Sinh[x])/3

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Rubi [A]  time = 0.0576204, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2751, 2646} \[ \frac{2}{3} B \sinh (x) \sqrt{a-a \cosh (x)}-\frac{2 a (3 A-B) \sinh (x)}{3 \sqrt{a-a \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Cosh[x]]*(A + B*Cosh[x]),x]

[Out]

(-2*a*(3*A - B)*Sinh[x])/(3*Sqrt[a - a*Cosh[x]]) + (2*B*Sqrt[a - a*Cosh[x]]*Sinh[x])/3

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \sqrt{a-a \cosh (x)} (A+B \cosh (x)) \, dx &=\frac{2}{3} B \sqrt{a-a \cosh (x)} \sinh (x)-\frac{1}{3} (-3 A+B) \int \sqrt{a-a \cosh (x)} \, dx\\ &=-\frac{2 a (3 A-B) \sinh (x)}{3 \sqrt{a-a \cosh (x)}}+\frac{2}{3} B \sqrt{a-a \cosh (x)} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0506901, size = 32, normalized size = 0.73 \[ \frac{2}{3} \coth \left (\frac{x}{2}\right ) \sqrt{a-a \cosh (x)} (3 A+B \cosh (x)-2 B) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Cosh[x]]*(A + B*Cosh[x]),x]

[Out]

(2*Sqrt[a - a*Cosh[x]]*(3*A - 2*B + B*Cosh[x])*Coth[x/2])/3

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Maple [A]  time = 0.052, size = 39, normalized size = 0.9 \begin{align*} -{\frac{4\,a}{3}\sinh \left ({\frac{x}{2}} \right ) \cosh \left ({\frac{x}{2}} \right ) \left ( 2\,B \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+3\,A-3\,B \right ){\frac{1}{\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*cosh(x))^(1/2)*(A+B*cosh(x)),x)

[Out]

-4/3*sinh(1/2*x)*a*cosh(1/2*x)*(2*B*cosh(1/2*x)^2+3*A-3*B)/(-2*sinh(1/2*x)^2*a)^(1/2)

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Maxima [B]  time = 1.6375, size = 147, normalized size = 3.34 \begin{align*} -{\left (\frac{\sqrt{2} \sqrt{a} e^{\left (-x\right )}}{\sqrt{-e^{\left (-x\right )}}} + \frac{\sqrt{2} \sqrt{a}}{\sqrt{-e^{\left (-x\right )}}}\right )} A + \frac{1}{6} \,{\left (\frac{{\left (3 \, \sqrt{2} \sqrt{a} e^{\left (-x\right )} - \sqrt{2} \sqrt{a}\right )} e^{x}}{\sqrt{-e^{\left (-x\right )}}} + \frac{3 \, \sqrt{2} \sqrt{a} e^{\left (-x\right )} - \sqrt{2} \sqrt{a} e^{\left (-2 \, x\right )}}{\sqrt{-e^{\left (-x\right )}}}\right )} B \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(x))^(1/2)*(A+B*cosh(x)),x, algorithm="maxima")

[Out]

-(sqrt(2)*sqrt(a)*e^(-x)/sqrt(-e^(-x)) + sqrt(2)*sqrt(a)/sqrt(-e^(-x)))*A + 1/6*((3*sqrt(2)*sqrt(a)*e^(-x) - s
qrt(2)*sqrt(a))*e^x/sqrt(-e^(-x)) + (3*sqrt(2)*sqrt(a)*e^(-x) - sqrt(2)*sqrt(a)*e^(-2*x))/sqrt(-e^(-x)))*B

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Fricas [B]  time = 2.14688, size = 319, normalized size = 7.25 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (B \cosh \left (x\right )^{3} + B \sinh \left (x\right )^{3} + 3 \,{\left (2 \, A - B\right )} \cosh \left (x\right )^{2} + 3 \,{\left (B \cosh \left (x\right ) + 2 \, A - B\right )} \sinh \left (x\right )^{2} + 3 \,{\left (2 \, A - B\right )} \cosh \left (x\right ) + 3 \,{\left (B \cosh \left (x\right )^{2} + 2 \,{\left (2 \, A - B\right )} \cosh \left (x\right ) + 2 \, A - B\right )} \sinh \left (x\right ) + B\right )} \sqrt{-\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{3 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(x))^(1/2)*(A+B*cosh(x)),x, algorithm="fricas")

[Out]

1/3*sqrt(1/2)*(B*cosh(x)^3 + B*sinh(x)^3 + 3*(2*A - B)*cosh(x)^2 + 3*(B*cosh(x) + 2*A - B)*sinh(x)^2 + 3*(2*A
- B)*cosh(x) + 3*(B*cosh(x)^2 + 2*(2*A - B)*cosh(x) + 2*A - B)*sinh(x) + B)*sqrt(-a/(cosh(x) + sinh(x)))/(cosh
(x) + sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- a \left (\cosh{\left (x \right )} - 1\right )} \left (A + B \cosh{\left (x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(x))**(1/2)*(A+B*cosh(x)),x)

[Out]

Integral(sqrt(-a*(cosh(x) - 1))*(A + B*cosh(x)), x)

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Giac [B]  time = 1.23306, size = 166, normalized size = 3.77 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{-a e^{x}} B a e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + 6 \, \sqrt{-a e^{x}} A a \mathrm{sgn}\left (-e^{x} + 1\right ) - 3 \, \sqrt{-a e^{x}} B a \mathrm{sgn}\left (-e^{x} + 1\right ) - \frac{{\left (6 \, A a^{3} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{3} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + B a^{3} \mathrm{sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-x\right )}}{\sqrt{-a e^{x}} a}\right )}}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(x))^(1/2)*(A+B*cosh(x)),x, algorithm="giac")

[Out]

-1/6*sqrt(2)*(sqrt(-a*e^x)*B*a*e^x*sgn(-e^x + 1) + 6*sqrt(-a*e^x)*A*a*sgn(-e^x + 1) - 3*sqrt(-a*e^x)*B*a*sgn(-
e^x + 1) - (6*A*a^3*e^x*sgn(-e^x + 1) - 3*B*a^3*e^x*sgn(-e^x + 1) + B*a^3*sgn(-e^x + 1))*e^(-x)/(sqrt(-a*e^x)*
a))/a