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

Optimal. Leaf size=181 \[ \frac{2 i \left (a^2-b^2\right ) (3 a B+5 A b) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{15 b \sqrt{a+b \cosh (x)}}-\frac{2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{15 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2}{15} \sinh (x) (3 a B+5 A b) \sqrt{a+b \cosh (x)}+\frac{2}{5} B \sinh (x) (a+b \cosh (x))^{3/2} \]

[Out]

(((-2*I)/15)*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[(a
+ b*Cosh[x])/(a + b)]) + (((2*I)/15)*(a^2 - b^2)*(5*A*b + 3*a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)
*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]]) + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Cosh[x]]*Sinh[x])/15 + (2*B*(a + b
*Cosh[x])^(3/2)*Sinh[x])/5

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Rubi [A]  time = 0.322156, antiderivative size = 181, 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 = {2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 i \left (a^2-b^2\right ) (3 a B+5 A b) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{15 b \sqrt{a+b \cosh (x)}}-\frac{2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{15 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2}{15} \sinh (x) (3 a B+5 A b) \sqrt{a+b \cosh (x)}+\frac{2}{5} B \sinh (x) (a+b \cosh (x))^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*I)/15)*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[(a
+ b*Cosh[x])/(a + b)]) + (((2*I)/15)*(a^2 - b^2)*(5*A*b + 3*a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)
*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]]) + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Cosh[x]]*Sinh[x])/15 + (2*B*(a + b
*Cosh[x])^(3/2)*Sinh[x])/5

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 (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx &=\frac{2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)+\frac{2}{5} \int \sqrt{a+b \cosh (x)} \left (\frac{1}{2} (5 a A+3 b B)+\frac{1}{2} (5 A b+3 a B) \cosh (x)\right ) \, dx\\ &=\frac{2}{15} (5 A b+3 a B) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)+\frac{4}{15} \int \frac{\frac{1}{4} \left (15 a^2 A+5 A b^2+12 a b B\right )+\frac{1}{4} \left (20 a A b+3 a^2 B+9 b^2 B\right ) \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx\\ &=\frac{2}{15} (5 A b+3 a B) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)-\frac{\left (\left (a^2-b^2\right ) (5 A b+3 a B)\right ) \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx}{15 b}+\frac{\left (20 a A b+3 a^2 B+9 b^2 B\right ) \int \sqrt{a+b \cosh (x)} \, dx}{15 b}\\ &=\frac{2}{15} (5 A b+3 a B) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)+\frac{\left (\left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{15 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) (5 A b+3 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}{15 b \sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i \left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{15 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2 i \left (a^2-b^2\right ) (5 A b+3 a B) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{15 b \sqrt{a+b \cosh (x)}}+\frac{2}{15} (5 A b+3 a B) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.63913, size = 124, normalized size = 0.69 \[ \frac{2}{15} \sqrt{a+b \cosh (x)} \left (\sinh (x) (6 a B+5 A b+3 b B \cosh (x))-\frac{i \left (\left (3 a^2 B+20 a A b+9 b^2 B\right ) E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )-(a-b) (3 a B+5 A b) \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )\right )}{b \sqrt{\frac{a+b \cosh (x)}{a+b}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*Cosh[x]]*(((-I)*((20*a*A*b + 3*a^2*B + 9*b^2*B)*EllipticE[(I/2)*x, (2*b)/(a + b)] - (a - b)*(5*A
*b + 3*a*B)*EllipticF[(I/2)*x, (2*b)/(a + b)]))/(b*Sqrt[(a + b*Cosh[x])/(a + b)]) + (5*A*b + 6*a*B + 3*b*B*Cos
h[x])*Sinh[x]))/15

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Maple [B]  time = 0.105, size = 973, normalized size = 5.4 \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))^(3/2)*(A+B*cosh(x)),x)

[Out]

2/15*(24*B*(-2*b/(a-b))^(1/2)*b^2*cosh(1/2*x)*sinh(1/2*x)^6+(20*A*(-2*b/(a-b))^(1/2)*b^2+36*B*(-2*b/(a-b))^(1/
2)*a*b+24*B*(-2*b/(a-b))^(1/2)*b^2)*sinh(1/2*x)^4*cosh(1/2*x)+(10*A*(-2*b/(a-b))^(1/2)*a*b+10*A*(-2*b/(a-b))^(
1/2)*b^2+12*B*(-2*b/(a-b))^(1/2)*a^2+18*B*(-2*b/(a-b))^(1/2)*a*b+6*B*(-2*b/(a-b))^(1/2)*b^2)*sinh(1/2*x)^2*cos
h(1/2*x)+15*A*a^2*(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))+20*A*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))+5*A*b^2*(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))-40*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))*a*b+3*a^2*B*(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))+12*b*B*a*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sin
h(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))+9*B*b^2*(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*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^2-18*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*Ellip
ticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*b^2)*((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 )}{\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))^(3/2)*(A+B*cosh(x)),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 ({\left (B b \cosh \left (x\right )^{2} + A a +{\left (B a + A b\right )} \cosh \left (x\right )\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))^(3/2)*(A+B*cosh(x)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cosh \left (x\right ) + A\right )}{\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))^(3/2)*(A+B*cosh(x)),x, algorithm="giac")

[Out]

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