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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {-8 i a (a+b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b (a+b \cosh (x)) \sinh (x)}{3 \sqrt {a+b \cosh (x)}} \] Input:

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

Output:

((-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]])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3135, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2}dx\)

\(\Big \downarrow \) 3135

\(\displaystyle \frac {2}{3} \int \frac {3 a^2+4 b \cosh (x) a+b^2}{2 \sqrt {a+b \cosh (x)}}dx+\frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3 a^2+4 b \cosh (x) a+b^2}{\sqrt {a+b \cosh (x)}}dx+\frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \int \frac {3 a^2+4 b \sin \left (i x+\frac {\pi }{2}\right ) a+b^2}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{3} \left (4 a \int \sqrt {a+b \cosh (x)}dx-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx\right )+\frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (4 a \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {4 a \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {4 a \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{\sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3135

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] + Simp[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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(465\) vs. \(2(113)=226\).

Time = 6.50 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.76

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

Input:

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

Output:

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*cosh(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/b*(a-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)*E 
llipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-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(cos 
h(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-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/b*(a-b))^(1/2))*a*b)*((2*b*cosh(1/2*x)^2+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*b*sinh(1/2*x)^2+a+b)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (111) = 222\).

Time = 0.12 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.09 \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {4 \, \sqrt {\frac {1}{2}} {\left ({\left (a^{2} + 3 \, b^{2}\right )} \cosh \left (x\right ) + {\left (a^{2} + 3 \, b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 48 \, \sqrt {\frac {1}{2}} {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} - 8 \, a b \cosh \left (x\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) - 4 \, a b\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{9 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )}} \] Input:

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

Output:

1/9*(4*sqrt(1/2)*((a^2 + 3*b^2)*cosh(x) + (a^2 + 3*b^2)*sinh(x))*sqrt(b)*w 
eierstrassPInverse(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) - 48*sqrt(1/2)*(a*b*cosh(x) + 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, 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)) + 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))*s 
qrt(b*cosh(x) + a))/(b*cosh(x) + b*sinh(x))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(cosh(x)*b + a),x)*a + int(sqrt(cosh(x)*b + a)*cosh(x),x)*b