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

Optimal. Leaf size=40 \[ \frac{2 a (3 A+B) \sinh (x)}{3 \sqrt{a \cosh (x)+a}}+\frac{2}{3} B \sinh (x) \sqrt{a \cosh (x)+a} \]

[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 = 0.0524759, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2751, 2646} \[ \frac{2 a (3 A+B) \sinh (x)}{3 \sqrt{a \cosh (x)+a}}+\frac{2}{3} B \sinh (x) \sqrt{a \cosh (x)+a} \]

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 = 0.038382, size = 31, normalized size = 0.78 \[ \frac{2}{3} \tanh \left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} (3 A+B \cosh (x)+2 B) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a*(1 + Cosh[x])]*(3*A + 2*B + B*Cosh[x])*Tanh[x/2])/3

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Maple [A]  time = 0.042, size = 39, normalized size = 1. \begin{align*}{\frac{2\,a\sqrt{2}}{3}\cosh \left ({\frac{x}{2}} \right ) \sinh \left ({\frac{x}{2}} \right ) \left ( 2\,B \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+3\,A+B \right ){\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{2}}}}} \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]

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

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Maxima [B]  time = 1.6292, size = 122, normalized size = 3.05 \begin{align*}{\left (\sqrt{2} \sqrt{a} e^{\left (\frac{1}{2} \, x\right )} - \sqrt{2} \sqrt{a} e^{\left (-\frac{1}{2} \, x\right )}\right )} A + \frac{1}{6} \,{\left ({\left (\sqrt{2} \sqrt{a} e^{\left (-x\right )} + 3 \, \sqrt{2} \sqrt{a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac{5}{2} \, x\right )} -{\left (3 \, \sqrt{2} \sqrt{a} e^{\left (-x\right )} + \sqrt{2} \sqrt{a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac{1}{2} \, 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^(1/2*x) - sqrt(2)*sqrt(a)*e^(-1/2*x))*A + 1/6*((sqrt(2)*sqrt(a)*e^(-x) + 3*sqrt(2)*sqrt(a)*
e^(-2*x))*e^(5/2*x) - (3*sqrt(2)*sqrt(a)*e^(-x) + sqrt(2)*sqrt(a)*e^(-2*x))*e^(1/2*x))*B

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Fricas [B]  time = 2.17391, size = 317, normalized size = 7.92 \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 = 1.11449, size = 93, normalized size = 2.32 \begin{align*} \frac{\sqrt{2}{\left (B a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} + 6 \, A a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} + 3 \, B a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - \frac{{\left (6 \, A a^{3} e^{x} + 3 \, B a^{3} e^{x} + B a^{3}\right )} e^{\left (-\frac{3}{2} \, x\right )}}{a^{\frac{3}{2}}}\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)*(B*a^(3/2)*e^(3/2*x) + 6*A*a^(3/2)*e^(1/2*x) + 3*B*a^(3/2)*e^(1/2*x) - (6*A*a^3*e^x + 3*B*a^3*e^x
+ B*a^3)*e^(-3/2*x)/a^(3/2))/a