3.120 \(\int \frac{A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx\)

Optimal. Leaf size=231 \[ \frac{2 i (A b-a B) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 \sinh (x) \left (a^2 (-B)+4 a A b-3 b^2 B\right )}{3 \left (a^2-b^2\right )^2 \sqrt{a+b \cosh (x)}}-\frac{2 \sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac{2 i \left (a^2 (-B)+4 a A b-3 b^2 B\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]

[Out]

(((-2*I)/3)*(4*a*A*b - a^2*B - 3*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*(a^2 - b^2)^
2*Sqrt[(a + b*Cosh[x])/(a + b)]) + (((2*I)/3)*(A*b - a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*
b)/(a + b)])/(b*(a^2 - b^2)*Sqrt[a + b*Cosh[x]]) - (2*(A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2
)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sinh[x])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cosh[x]])

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Rubi [A]  time = 0.349833, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{2 \sinh (x) \left (a^2 (-B)+4 a A b-3 b^2 B\right )}{3 \left (a^2-b^2\right )^2 \sqrt{a+b \cosh (x)}}-\frac{2 \sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac{2 i (A b-a B) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 i \left (a^2 (-B)+4 a A b-3 b^2 B\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(a + b*Cosh[x])^(5/2),x]

[Out]

(((-2*I)/3)*(4*a*A*b - a^2*B - 3*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*(a^2 - b^2)^
2*Sqrt[(a + b*Cosh[x])/(a + b)]) + (((2*I)/3)*(A*b - a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*
b)/(a + b)])/(b*(a^2 - b^2)*Sqrt[a + b*Cosh[x]]) - (2*(A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2
)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sinh[x])/(3*(a^2 - b^2)^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))^{5/2}} \, dx &=-\frac{2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac{2 \int \frac{-\frac{3}{2} (a A-b B)+\frac{1}{2} (A b-a B) \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac{2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt{a+b \cosh (x)}}+\frac{4 \int \frac{\frac{1}{4} \left (3 a^2 A+A b^2-4 a b B\right )+\frac{1}{4} \left (4 a A b-a^2 B-3 b^2 B\right ) \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=-\frac{2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt{a+b \cosh (x)}}-\frac{(A b-a B) \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx}{3 b \left (a^2-b^2\right )}+\frac{\left (4 a A b-a^2 B-3 b^2 B\right ) \int \sqrt{a+b \cosh (x)} \, dx}{3 b \left (a^2-b^2\right )^2}\\ &=-\frac{2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt{a+b \cosh (x)}}+\frac{\left (\left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{3 b \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{\left ((A b-a 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 \left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i \left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2 i (A b-a B) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac{2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt{a+b \cosh (x)}}\\ \end{align*}

Mathematica [A]  time = 0.841302, size = 172, normalized size = 0.74 \[ \frac{2 \left (\frac{\sinh (x) \left (b \cosh (x) \left (a^2 B-4 a A b+3 b^2 B\right )-5 a^2 A b+2 a^3 B+2 a b^2 B+A b^3\right )}{\left (a^2-b^2\right )^2}+\frac{i \left (\frac{a+b \cosh (x)}{a+b}\right )^{3/2} \left (\left (a^2 B-4 a A b+3 b^2 B\right ) E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )-(a-b) (a B-A b) \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )\right )}{b (a-b)^2}\right )}{3 (a+b \cosh (x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/(a + b*Cosh[x])^(5/2),x]

[Out]

(2*((I*((a + b*Cosh[x])/(a + b))^(3/2)*((-4*a*A*b + a^2*B + 3*b^2*B)*EllipticE[(I/2)*x, (2*b)/(a + b)] - (a -
b)*(-(A*b) + a*B)*EllipticF[(I/2)*x, (2*b)/(a + b)]))/((a - b)^2*b) + ((-5*a^2*A*b + A*b^3 + 2*a^3*B + 2*a*b^2
*B + b*(-4*a*A*b + a^2*B + 3*b^2*B)*Cosh[x])*Sinh[x])/(a^2 - b^2)^2))/(3*(a + b*Cosh[x])^(3/2))

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Maple [B]  time = 0.346, size = 797, normalized size = 3.5 \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))^(5/2),x)

[Out]

((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)*(-2*B/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)+2*(A*b-B*a)/b*(-1/6/b/(a-b)/(a+b
)*cosh(1/2*x)*(2*b*sinh(1/2*x)^4+(a+b)*sinh(1/2*x)^2)^(1/2)/(cosh(1/2*x)^2+1/2*(a-b)/b)^2-8/3*b*sinh(1/2*x)^2/
(a-b)^2/(a+b)^2*cosh(1/2*x)*a/((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)+(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b
^3)/(-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)*s
inh(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))-16/3*a*b/(a+b)^2/(a-b)^
2*(-a+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)/(2*a-2*b)*(EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-Ellipti
cE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(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 \frac{B \cosh \left (x\right ) + A}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac{5}{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))^(5/2),x, algorithm="maxima")

[Out]

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(5/2), 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^{3} \cosh \left (x\right )^{3} + 3 \, a b^{2} \cosh \left (x\right )^{2} + 3 \, a^{2} b \cosh \left (x\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+b*cosh(x))^(5/2),x, algorithm="fricas")

[Out]

integral((B*cosh(x) + A)*sqrt(b*cosh(x) + a)/(b^3*cosh(x)^3 + 3*a*b^2*cosh(x)^2 + 3*a^2*b*cosh(x) + a^3), 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))**(5/2),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{5}{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))^(5/2),x, algorithm="giac")

[Out]

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(5/2), x)