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]

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

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Rubi [A]  time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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]

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)]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [B]  time = 0.49, size = 100, normalized size = 2.50 \[ \frac {\sqrt {\frac {1}{2}} {\left (B \cosh \relax (x)^{3} + B \sinh \relax (x)^{3} + 3 \, {\left (2 \, A + B\right )} \cosh \relax (x)^{2} + 3 \, {\left (B \cosh \relax (x) + 2 \, A + B\right )} \sinh \relax (x)^{2} - 3 \, {\left (2 \, A + B\right )} \cosh \relax (x) + 3 \, {\left (B \cosh \relax (x)^{2} + 2 \, {\left (2 \, A + B\right )} \cosh \relax (x) - 2 \, A - B\right )} \sinh \relax (x) - B\right )} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{3 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

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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giac [B]  time = 0.15, size = 71, normalized size = 1.78 \[ -\frac {1}{6} \, \sqrt {2} {\left (\frac {{\left (6 \, A a^{2} e^{x} + 3 \, B a^{2} e^{x} + B a^{2}\right )} e^{\left (-\frac {3}{2} \, x\right )}}{a^{\frac {3}{2}}} - \frac {B a^{\frac {7}{2}} e^{\left (\frac {3}{2} \, x\right )} + 6 \, A a^{\frac {7}{2}} e^{\left (\frac {1}{2} \, x\right )} + 3 \, B a^{\frac {7}{2}} e^{\left (\frac {1}{2} \, x\right )}}{a^{3}}\right )} \]

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)*((6*A*a^2*e^x + 3*B*a^2*e^x + B*a^2)*e^(-3/2*x)/a^(3/2) - (B*a^(7/2)*e^(3/2*x) + 6*A*a^(7/2)*e^(1
/2*x) + 3*B*a^(7/2)*e^(1/2*x))/a^3)

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maple [A]  time = 0.23, size = 39, normalized size = 0.98 \[ \frac {2 \cosh \left (\frac {x}{2}\right ) a \sinh \left (\frac {x}{2}\right ) \left (2 B \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+3 A +B \right ) \sqrt {2}}{3 \sqrt {a \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )}} \]

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 = 0.46, size = 90, normalized size = 2.25 \[ {\left (\sqrt {2} \sqrt {a} e^{\left (\frac {1}{2} \, x\right )} - \sqrt {2} \sqrt {a} e^{\left (-\frac {1}{2} \, x\right )}\right )} A + \frac {1}{6} \, {\left ({\left (\sqrt {2} \sqrt {a} e^{\left (-x\right )} + 3 \, \sqrt {2} \sqrt {a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac {5}{2} \, x\right )} - {\left (3 \, \sqrt {2} \sqrt {a} e^{\left (-x\right )} + \sqrt {2} \sqrt {a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac {1}{2} \, x\right )}\right )} B \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \left (A+B\,\mathrm {cosh}\relax (x)\right )\,\sqrt {a+a\,\mathrm {cosh}\relax (x)} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\cosh {\relax (x )} + 1\right )} \left (A + B \cosh {\relax (x )}\right )\, dx \]

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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