\(\int \cosh ^{\frac {3}{2}}(a+b x) \, dx\) [9]

Optimal result

Integrand size = 10, antiderivative size = 46 \[ \int \cosh ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b}+\frac {2 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{3 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2715, 2720} \[ \int \cosh ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b} \]

[In]

Int[Cosh[a + b*x]^(3/2),x]

[Out]

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

Rule 2715

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

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.76 \[ \int \cosh ^{\frac {3}{2}}(a+b x) \, dx=\frac {\sinh (2 (a+b x))+2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right ) \sqrt {1+\cosh (2 (a+b x))+\sinh (2 (a+b x))}}{3 b \sqrt {\cosh (a+b x)}} \]

[In]

Integrate[Cosh[a + b*x]^(3/2),x]

[Out]

(Sinh[2*(a + b*x)] + 2*Hypergeometric2F1[1/4, 1/2, 5/4, -Cosh[2*(a + b*x)] - Sinh[2*(a + b*x)]]*Sqrt[1 + Cosh[
2*(a + b*x)] + Sinh[2*(a + b*x)]])/(3*b*Sqrt[Cosh[a + b*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(66)=132\).

Time = 0.46 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.78

method	result	size
default	\(\frac {2 \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (4 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}-6 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+\sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, b}\)	\(174\)

[In]

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

[Out]

2/3*((-1+2*cosh(1/2*b*x+1/2*a)^2)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(4*cosh(1/2*b*x+1/2*a)^5-6*cosh(1/2*b*x+1/2*a)^
3+(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))+2*c
osh(1/2*b*x+1/2*a))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/sinh(1/2*b*x+1/2*a)/(-1+2*cosh(1/2*b
*x+1/2*a)^2)^(1/2)/b

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.22 \[ \int \cosh ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \, {\left (\sqrt {2} \cosh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \sqrt {\cosh \left (b x + a\right )}}{3 \, {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \]

[In]

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

[Out]

1/3*(2*(sqrt(2)*cosh(b*x + a) + sqrt(2)*sinh(b*x + a))*weierstrassPInverse(-4, 0, cosh(b*x + a) + sinh(b*x + a
)) + (cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*sqrt(cosh(b*x + a)))/(b*cosh(b*x
+ a) + b*sinh(b*x + a))

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(a + b*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(cosh(b*x + a)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(cosh(b*x + a)^(3/2), x)

Mupad [F(-1)]

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

[In]

int(cosh(a + b*x)^(3/2),x)

[Out]

int(cosh(a + b*x)^(3/2), x)