3.119 \(\int \frac{A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{2 i B \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}}-\frac{2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 i (A b-a B) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]

[Out]

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

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Rubi [A]  time = 0.204889, antiderivative size = 152, 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 = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 i (A b-a B) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{2 i B \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2),x]

[Out]

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

Rule 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*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] && LtQ[m, -1] && 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 \frac{A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx &=-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 \int \frac{\frac{1}{2} (-a A+b B)-\frac{1}{2} (A b-a B) \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx}{a^2-b^2}\\ &=-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}+\frac{B \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx}{b}+\frac{(A b-a B) \int \sqrt{a+b \cosh (x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}+\frac{\left ((A b-a B) \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{\left (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}{b \sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i (A b-a B) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{2 i B \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}}-\frac{2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}\\ \end{align*}

Mathematica [A]  time = 0.356721, size = 133, normalized size = 0.88 \[ \frac{-2 i B \left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )+2 b \sinh (x) (a B-A b)+2 i (a+b) (a B-A b) \sqrt{\frac{a+b \cosh (x)}{a+b}} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b (a-b) (a+b) \sqrt{a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2),x]

[Out]

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

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Maple [B]  time = 0.197, size = 483, normalized size = 3.2 \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))/(a+b*cosh(x))^(3/2),x)

[Out]

((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)*(2*B/b/(-2*b/(a-b))^(1/2)*((2*cosh(1/2*x)^2*b+a-b)/(a-b))^(1/2)*
(-sinh(1/2*x)^2)^(1/2)/(2*b*sinh(1/2*x)^4+(a+b)*sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),
1/2*((-2*a+2*b)/b)^(1/2))+2*(A*b-B*a)/b/sinh(1/2*x)^2/(2*sinh(1/2*x)^2*b+a+b)/(-2*b/(a-b))^(1/2)/(a^2-b^2)*(2*
b*sinh(1/2*x)^4+(a+b)*sinh(1/2*x)^2)^(1/2)*(-2*(-2*b/(a-b))^(1/2)*b*cosh(1/2*x)*sinh(1/2*x)^2+(2*b/(a-b)*sinh(
1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))*(-sinh(1/2*x)^2
)^(1/2)*a+(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)
/b)^(1/2))*(-sinh(1/2*x)^2)^(1/2)*b-2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/
(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))*(-sinh(1/2*x)^2)^(1/2)*b))/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 \frac{B \cosh \left (x\right ) + A}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x, algorithm="maxima")

[Out]

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B \cosh \left (x\right ) + A\right )} \sqrt{b \cosh \left (x\right ) + a}}{b^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x, algorithm="fricas")

[Out]

integral((B*cosh(x) + A)*sqrt(b*cosh(x) + a)/(b^2*cosh(x)^2 + 2*a*b*cosh(x) + a^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+b*cosh(x))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cosh \left (x\right ) + A}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x, algorithm="giac")

[Out]

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(3/2), x)