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

Optimal. Leaf size=138 \[ \frac{2 i B \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{3 b \sqrt{a+b \cosh (x)}}-\frac{2 i (a B+3 A b) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2}{3} B \sinh (x) \sqrt{a+b \cosh (x)} \]

[Out]

(((-2*I)/3)*(3*A*b + a*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a +
b)]) + (((2*I)/3)*(a^2 - b^2)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b
*Cosh[x]]) + (2*B*Sqrt[a + b*Cosh[x]]*Sinh[x])/3

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Rubi [A]  time = 0.206277, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 i B \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \sqrt{a+b \cosh (x)}}-\frac{2 i (a B+3 A b) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2}{3} B \sinh (x) \sqrt{a+b \cosh (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*I)/3)*(3*A*b + a*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a +
b)]) + (((2*I)/3)*(a^2 - b^2)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b
*Cosh[x]]) + (2*B*Sqrt[a + b*Cosh[x]]*Sinh[x])/3

Rule 2753

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2752

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

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

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \sqrt{a+b \cosh (x)} (A+B \cosh (x)) \, dx &=\frac{2}{3} B \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{3} \int \frac{\frac{1}{2} (3 a A+b B)+\frac{1}{2} (3 A b+a B) \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx\\ &=\frac{2}{3} B \sqrt{a+b \cosh (x)} \sinh (x)-\frac{\left (\left (a^2-b^2\right ) B\right ) \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx}{3 b}+\frac{(3 A b+a B) \int \sqrt{a+b \cosh (x)} \, dx}{3 b}\\ &=\frac{2}{3} B \sqrt{a+b \cosh (x)} \sinh (x)+\frac{\left ((3 A b+a B) \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{3 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) B \sqrt{\frac{a+b \cosh (x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}}} \, dx}{3 b \sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i (3 A b+a B) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2 i \left (a^2-b^2\right ) B \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \sqrt{a+b \cosh (x)}}+\frac{2}{3} B \sqrt{a+b \cosh (x)} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.333301, size = 123, normalized size = 0.89 \[ \frac{2 i B \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )-2 i (a+b) (a B+3 A b) \sqrt{\frac{a+b \cosh (x)}{a+b}} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )+2 b B \sinh (x) (a+b \cosh (x))}{3 b \sqrt{a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*(a + b)*(3*A*b + a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticE[(I/2)*x, (2*b)/(a + b)] + (2*I)*(a^2 - b
^2)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)] + 2*b*B*(a + b*Cosh[x])*Sinh[x])/(3*b*Sq
rt[a + b*Cosh[x]])

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Maple [B]  time = 0.094, size = 605, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(4*B*(-2*b/(a-b))^(1/2)*b*cosh(1/2*x)*sinh(1/2*x)^4+(2*B*(-2*b/(a-b))^(1/2)*a+2*B*(-2*b/(a-b))^(1/2)*b)*si
nh(1/2*x)^2*cosh(1/2*x)+3*A*a*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cos
h(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+3*A*b*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1
/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-6*A*(2*b/(a-b)*sinh(1/2*x)^2+(
a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*b+a*
B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),
1/2*(-2*(a-b)/b)^(1/2))+b*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(
1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-2*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x
)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*a)*((2*cosh(1/2*x)^2*b+a-b)*sinh(1
/2*x)^2)^(1/2)/(-2*b/(a-b))^(1/2)/(2*b*sinh(1/2*x)^4+(a+b)*sinh(1/2*x)^2)^(1/2)/sinh(1/2*x)/(2*sinh(1/2*x)^2*b
+a+b)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cosh(x) + A)*sqrt(b*cosh(x) + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B \cosh \left (x\right ) + A\right )} \sqrt{b \cosh \left (x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*cosh(x) + A)*sqrt(b*cosh(x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cosh(x) + A)*sqrt(b*cosh(x) + a), x)