\(\int (a+b \cosh (x))^{3/2} \, dx\) [80]

Optimal result

Integrand size = 10, antiderivative size = 124 \[ \int (a+b \cosh (x))^{3/2} \, dx=-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \sqrt {a+b \cosh (x)}}+\frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x) \]

[Out]

2/3*b*sinh(x)*(a+b*cosh(x))^(1/2)-8/3*I*a*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2)*(b
/(a+b))^(1/2))*(a+b*cosh(x))^(1/2)/((a+b*cosh(x))/(a+b))^(1/2)+2/3*I*(a^2-b^2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*
x)*EllipticF(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cosh(x))/(a+b))^(1/2)/(a+b*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2735, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \sqrt {a+b \cosh (x)}}+\frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

[In]

Int[(a + b*Cosh[x])^(3/2),x]

[Out]

(((-8*I)/3)*a*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/Sqrt[(a + b*Cosh[x])/(a + b)] + (((2*I)/3
)*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/Sqrt[a + b*Cosh[x]] + (2*b*Sqrt
[a + b*Cosh[x]]*Sinh[x])/3

Rule 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2735

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

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{3} \int \frac {\frac {1}{2} \left (3 a^2+b^2\right )+2 a b \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx \\ & = \frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {1}{3} (4 a) \int \sqrt {a+b \cosh (x)} \, dx+\frac {1}{3} \left (-a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx \\ & = \frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {\left (4 a \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {\left (\left (-a^2+b^2\right ) \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 \sqrt {a+b \cosh (x)}} \\ & = -\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \sqrt {a+b \cosh (x)}}+\frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {-8 i a (a+b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b (a+b \cosh (x)) \sinh (x)}{3 \sqrt {a+b \cosh (x)}} \]

[In]

Integrate[(a + b*Cosh[x])^(3/2),x]

[Out]

((-8*I)*a*(a + b)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticE[(I/2)*x, (2*b)/(a + b)] + (2*I)*(a^2 - b^2)*Sqrt[(a
+ b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)] + 2*b*(a + b*Cosh[x])*Sinh[x])/(3*Sqrt[a + b*Cosh[x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(465\) vs. \(2(144)=288\).

Time = 1.43 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.76

method	result	size
default	\(\frac {2 \left (4 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{4} b^{2}+2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} a b +2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b^{2}+3 a^{2} \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+4 a b \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+b^{2} \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-8 \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) a b \right ) \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{3 \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\)	\(466\)

[In]

int((a+b*cosh(x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(4*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^4*b^2+2*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^2*a*b+2*c
osh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^2*b^2+3*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))+4*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^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))-8*(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)*((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)/(-2*b/(a-b))^(1
/2)/(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)/sinh(1/2*x)/(2*sinh(1/2*x)^2*b+a+b)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.14 \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (a^{2} + 3 \, b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a^{2} + 3 \, b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 24 \, {\left (\sqrt {2} a b \cosh \left (x\right ) + \sqrt {2} a b \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} - 8 \, a b \cosh \left (x\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) - 4 \, a b\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{9 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

1/9*(2*(sqrt(2)*(a^2 + 3*b^2)*cosh(x) + sqrt(2)*(a^2 + 3*b^2)*sinh(x))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2
- 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b) - 24*(sqrt(2)*a*b*cosh(x)
+ sqrt(2)*a*b*sinh(x))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstra
ssPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b)) + 3
*(b^2*cosh(x)^2 + b^2*sinh(x)^2 - 8*a*b*cosh(x) - b^2 + 2*(b^2*cosh(x) - 4*a*b)*sinh(x))*sqrt(b*cosh(x) + a))/
(b*cosh(x) + b*sinh(x))

Sympy [F]

\[ \int (a+b \cosh (x))^{3/2} \, dx=\int \left (a + b \cosh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*cosh(x))**(3/2), x)

Maxima [F]

\[ \int (a+b \cosh (x))^{3/2} \, dx=\int { {\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((b*cosh(x) + a)^(3/2), x)

Giac [F]

\[ \int (a+b \cosh (x))^{3/2} \, dx=\int { {\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((b*cosh(x) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \cosh (x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \]

[In]

int((a + b*cosh(x))^(3/2),x)

[Out]

int((a + b*cosh(x))^(3/2), x)