\(\int (a \cosh (x))^{7/2} \, dx\) [15]

Optimal result

Integrand size = 8, antiderivative size = 65 \[ \int (a \cosh (x))^{7/2} \, dx=-\frac {10 i a^4 \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{21 \sqrt {a \cosh (x)}}+\frac {10}{21} a^3 \sqrt {a \cosh (x)} \sinh (x)+\frac {2}{7} a (a \cosh (x))^{5/2} \sinh (x) \]

[Out]

2/7*a*(a*cosh(x))^(5/2)*sinh(x)-10/21*I*a^4*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))
*cosh(x)^(1/2)/(a*cosh(x))^(1/2)+10/21*a^3*sinh(x)*(a*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2715, 2721, 2720} \[ \int (a \cosh (x))^{7/2} \, dx=-\frac {10 i a^4 \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{21 \sqrt {a \cosh (x)}}+\frac {10}{21} a^3 \sinh (x) \sqrt {a \cosh (x)}+\frac {2}{7} a \sinh (x) (a \cosh (x))^{5/2} \]

[In]

Int[(a*Cosh[x])^(7/2),x]

[Out]

(((-10*I)/21)*a^4*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2])/Sqrt[a*Cosh[x]] + (10*a^3*Sqrt[a*Cosh[x]]*Sinh[x])/21 +
(2*a*(a*Cosh[x])^(5/2)*Sinh[x])/7

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]

Rule 2721

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

Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} a (a \cosh (x))^{5/2} \sinh (x)+\frac {1}{7} \left (5 a^2\right ) \int (a \cosh (x))^{3/2} \, dx \\ & = \frac {10}{21} a^3 \sqrt {a \cosh (x)} \sinh (x)+\frac {2}{7} a (a \cosh (x))^{5/2} \sinh (x)+\frac {1}{21} \left (5 a^4\right ) \int \frac {1}{\sqrt {a \cosh (x)}} \, dx \\ & = \frac {10}{21} a^3 \sqrt {a \cosh (x)} \sinh (x)+\frac {2}{7} a (a \cosh (x))^{5/2} \sinh (x)+\frac {\left (5 a^4 \sqrt {\cosh (x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{21 \sqrt {a \cosh (x)}} \\ & = -\frac {10 i a^4 \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{21 \sqrt {a \cosh (x)}}+\frac {10}{21} a^3 \sqrt {a \cosh (x)} \sinh (x)+\frac {2}{7} a (a \cosh (x))^{5/2} \sinh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82 \[ \int (a \cosh (x))^{7/2} \, dx=\frac {a^3 \sqrt {a \cosh (x)} \left (-20 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+\sqrt {\cosh (x)} (23 \sinh (x)+3 \sinh (3 x))\right )}{42 \sqrt {\cosh (x)}} \]

[In]

Integrate[(a*Cosh[x])^(7/2),x]

[Out]

(a^3*Sqrt[a*Cosh[x]]*((-20*I)*EllipticF[(I/2)*x, 2] + Sqrt[Cosh[x]]*(23*Sinh[x] + 3*Sinh[3*x])))/(42*Sqrt[Cosh
[x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(66)=132\).

Time = 2.74 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.23

method	result	size
default	\(\frac {\sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right )^{2}}\, a^{4} \left (96 \cosh \left (\frac {x}{2}\right )^{9}-240 \cosh \left (\frac {x}{2}\right )^{7}+256 \cosh \left (\frac {x}{2}\right )^{5}+5 \sqrt {2}\, \sqrt {-2 \cosh \left (\frac {x}{2}\right )^{2}+1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-144 \cosh \left (\frac {x}{2}\right )^{3}+32 \cosh \left (\frac {x}{2}\right )\right )}{21 \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right )}}\)	\(145\)

[In]

int((a*cosh(x))^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/21*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)*a^4*(96*cosh(1/2*x)^9-240*cosh(1/2*x)^7+256*cosh(1/2*x)^5+5*2
^(1/2)*(-2*cosh(1/2*x)^2+1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-144*cosh(1
/2*x)^3+32*cosh(1/2*x))/(a*(2*sinh(1/2*x)^4+sinh(1/2*x)^2))^(1/2)/sinh(1/2*x)/(a*(2*cosh(1/2*x)^2-1))^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 256, normalized size of antiderivative = 3.94 \[ \int (a \cosh (x))^{7/2} \, dx=\frac {40 \, {\left (\sqrt {2} a^{3} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} a^{3} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} a^{3} \sinh \left (x\right )^{3}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (3 \, a^{3} \cosh \left (x\right )^{6} + 18 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, a^{3} \sinh \left (x\right )^{6} + 23 \, a^{3} \cosh \left (x\right )^{4} - 23 \, a^{3} \cosh \left (x\right )^{2} + {\left (45 \, a^{3} \cosh \left (x\right )^{2} + 23 \, a^{3}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (15 \, a^{3} \cosh \left (x\right )^{3} + 23 \, a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 3 \, a^{3} + {\left (45 \, a^{3} \cosh \left (x\right )^{4} + 138 \, a^{3} \cosh \left (x\right )^{2} - 23 \, a^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (9 \, a^{3} \cosh \left (x\right )^{5} + 46 \, a^{3} \cosh \left (x\right )^{3} - 23 \, a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}}{84 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \]

[In]

integrate((a*cosh(x))^(7/2),x, algorithm="fricas")

[Out]

1/84*(40*(sqrt(2)*a^3*cosh(x)^3 + 3*sqrt(2)*a^3*cosh(x)^2*sinh(x) + 3*sqrt(2)*a^3*cosh(x)*sinh(x)^2 + sqrt(2)*
a^3*sinh(x)^3)*sqrt(a)*weierstrassPInverse(-4, 0, cosh(x) + sinh(x)) + (3*a^3*cosh(x)^6 + 18*a^3*cosh(x)*sinh(
x)^5 + 3*a^3*sinh(x)^6 + 23*a^3*cosh(x)^4 - 23*a^3*cosh(x)^2 + (45*a^3*cosh(x)^2 + 23*a^3)*sinh(x)^4 + 4*(15*a
^3*cosh(x)^3 + 23*a^3*cosh(x))*sinh(x)^3 - 3*a^3 + (45*a^3*cosh(x)^4 + 138*a^3*cosh(x)^2 - 23*a^3)*sinh(x)^2 +
2*(9*a^3*cosh(x)^5 + 46*a^3*cosh(x)^3 - 23*a^3*cosh(x))*sinh(x))*sqrt(a*cosh(x)))/(cosh(x)^3 + 3*cosh(x)^2*si
nh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)

Sympy [F(-1)]

Timed out. \[ \int (a \cosh (x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((a*cosh(x))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (a \cosh (x))^{7/2} \, dx=\int { \left (a \cosh \left (x\right )\right )^{\frac {7}{2}} \,d x } \]

[In]

integrate((a*cosh(x))^(7/2),x, algorithm="maxima")

[Out]

integrate((a*cosh(x))^(7/2), x)

Giac [F]

\[ \int (a \cosh (x))^{7/2} \, dx=\int { \left (a \cosh \left (x\right )\right )^{\frac {7}{2}} \,d x } \]

[In]

integrate((a*cosh(x))^(7/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x))^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a \cosh (x))^{7/2} \, dx=\int {\left (a\,\mathrm {cosh}\left (x\right )\right )}^{7/2} \,d x \]

[In]

int((a*cosh(x))^(7/2),x)

[Out]

int((a*cosh(x))^(7/2), x)