\(\int (a \cosh ^4(x))^{3/2} \, dx\) [135]

Optimal result

Integrand size = 10, antiderivative size = 78 \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=\frac {5}{16} a x \sqrt {a \cosh ^4(x)} \text {sech}^2(x)+\frac {5}{24} a \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{6} a \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {5}{16} a \sqrt {a \cosh ^4(x)} \tanh (x) \] Output:

5/16*a*x*(a*cosh(x)^4)^(1/2)*sech(x)^2+5/24*a*cosh(x)*(a*cosh(x)^4)^(1/2)* 
sinh(x)+1/6*a*cosh(x)^3*(a*cosh(x)^4)^(1/2)*sinh(x)+5/16*a*(a*cosh(x)^4)^( 
1/2)*tanh(x)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.49 \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=\frac {1}{192} \left (a \cosh ^4(x)\right )^{3/2} \text {sech}^6(x) (60 x+45 \sinh (2 x)+9 \sinh (4 x)+\sinh (6 x)) \] Input:

Integrate[(a*Cosh[x]^4)^(3/2),x]
 

Output:

((a*Cosh[x]^4)^(3/2)*Sech[x]^6*(60*x + 45*Sinh[2*x] + 9*Sinh[4*x] + Sinh[6 
*x]))/192
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.77, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3115, 24}

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.

\(\displaystyle \int \left (a \cosh ^4(x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \sin \left (\frac {\pi }{2}+i x\right )^4\right )^{3/2}dx\)

\(\Big \downarrow \) 3686

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \int \cosh ^6(x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \int \sin \left (i x+\frac {\pi }{2}\right )^6dx\)

\(\Big \downarrow \) 3115

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {5}{6} \int \cosh ^4(x)dx+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \int \sin \left (i x+\frac {\pi }{2}\right )^4dx\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {5}{6} \left (\frac {3}{4} \int \cosh ^2(x)dx+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )\)

\(\Big \downarrow \) 24

\(\displaystyle a \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\right )\right )\)

Input:

Int[(a*Cosh[x]^4)^(3/2),x]
 

Output:

a*Sqrt[a*Cosh[x]^4]*Sech[x]^2*((Cosh[x]^5*Sinh[x])/6 + (5*((Cosh[x]^3*Sinh 
[x])/4 + (3*(x/2 + (Cosh[x]*Sinh[x])/2))/4))/6)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 3042

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

rule 3115

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] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3686

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff 
= FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ 
n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p]))   Int[ActivateTrig[u]*(Si 
n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] 
 && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / 
; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.60

method	result	size
default	\(\frac {\sqrt {a}\, \left (1+\cosh \left (2 x \right )\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \left (2 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}\, \sinh \left (2 x \right )^{2}+9 \cosh \left (2 x \right ) \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+24 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+15 \ln \left (\cosh \left (2 x \right ) \sqrt {a}+\sqrt {a \sinh \left (2 x \right )^{2}}\right ) a \right )}{96 \sinh \left (2 x \right ) \sqrt {\left (1+\cosh \left (2 x \right )\right )^{2} a}}\)	\(125\)
risch	\(\frac {5 a \,{\mathrm e}^{2 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}\, x}{16 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {a \,{\mathrm e}^{8 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{384 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {3 a \,{\mathrm e}^{6 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{128 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {15 a \,{\mathrm e}^{4 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{128 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {15 \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}\, a}{128 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {3 a \,{\mathrm e}^{-2 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{128 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {a \,{\mathrm e}^{-4 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{384 \left ({\mathrm e}^{2 x}+1\right )^{2}}\)	\(216\)

Input:

int((a*cosh(x)^4)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/96*a^(1/2)*(1+cosh(2*x))*(a*(-1+cosh(2*x))*(1+cosh(2*x)))^(1/2)*(2*(a*si 
nh(2*x)^2)^(1/2)*a^(1/2)*sinh(2*x)^2+9*cosh(2*x)*(a*sinh(2*x)^2)^(1/2)*a^( 
1/2)+24*(a*sinh(2*x)^2)^(1/2)*a^(1/2)+15*ln(cosh(2*x)*a^(1/2)+(a*sinh(2*x) 
^2)^(1/2))*a)/sinh(2*x)/((1+cosh(2*x))^2*a)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (62) = 124\).

Time = 0.16 (sec) , antiderivative size = 659, normalized size of antiderivative = 8.45 \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((a*cosh(x)^4)^(3/2),x, algorithm="fricas")
 

Output:

1/384*(12*a*cosh(x)*e^(2*x)*sinh(x)^11 + a*e^(2*x)*sinh(x)^12 + 3*(22*a*co 
sh(x)^2 + 3*a)*e^(2*x)*sinh(x)^10 + 10*(22*a*cosh(x)^3 + 9*a*cosh(x))*e^(2 
*x)*sinh(x)^9 + 45*(11*a*cosh(x)^4 + 9*a*cosh(x)^2 + a)*e^(2*x)*sinh(x)^8 
+ 72*(11*a*cosh(x)^5 + 15*a*cosh(x)^3 + 5*a*cosh(x))*e^(2*x)*sinh(x)^7 + 6 
*(154*a*cosh(x)^6 + 315*a*cosh(x)^4 + 210*a*cosh(x)^2 + 20*a*x)*e^(2*x)*si 
nh(x)^6 + 36*(22*a*cosh(x)^7 + 63*a*cosh(x)^5 + 70*a*cosh(x)^3 + 20*a*x*co 
sh(x))*e^(2*x)*sinh(x)^5 + 45*(11*a*cosh(x)^8 + 42*a*cosh(x)^6 + 70*a*cosh 
(x)^4 + 40*a*x*cosh(x)^2 - a)*e^(2*x)*sinh(x)^4 + 20*(11*a*cosh(x)^9 + 54* 
a*cosh(x)^7 + 126*a*cosh(x)^5 + 120*a*x*cosh(x)^3 - 9*a*cosh(x))*e^(2*x)*s 
inh(x)^3 + 3*(22*a*cosh(x)^10 + 135*a*cosh(x)^8 + 420*a*cosh(x)^6 + 600*a* 
x*cosh(x)^4 - 90*a*cosh(x)^2 - 3*a)*e^(2*x)*sinh(x)^2 + 6*(2*a*cosh(x)^11 
+ 15*a*cosh(x)^9 + 60*a*cosh(x)^7 + 120*a*x*cosh(x)^5 - 30*a*cosh(x)^3 - 3 
*a*cosh(x))*e^(2*x)*sinh(x) + (a*cosh(x)^12 + 9*a*cosh(x)^10 + 45*a*cosh(x 
)^8 + 120*a*x*cosh(x)^6 - 45*a*cosh(x)^4 - 9*a*cosh(x)^2 - a)*e^(2*x))*sqr 
t(a*e^(8*x) + 4*a*e^(6*x) + 6*a*e^(4*x) + 4*a*e^(2*x) + a)*e^(-2*x)/(cosh( 
x)^6*e^(4*x) + 2*cosh(x)^6*e^(2*x) + (e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^6 + 
 cosh(x)^6 + 6*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^5 + 
 15*(cosh(x)^2*e^(4*x) + 2*cosh(x)^2*e^(2*x) + cosh(x)^2)*sinh(x)^4 + 20*( 
cosh(x)^3*e^(4*x) + 2*cosh(x)^3*e^(2*x) + cosh(x)^3)*sinh(x)^3 + 15*(cosh( 
x)^4*e^(4*x) + 2*cosh(x)^4*e^(2*x) + cosh(x)^4)*sinh(x)^2 + 6*(cosh(x)^...
 

Sympy [F(-1)]

Timed out. \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=\text {Timed out} \] Input:

integrate((a*cosh(x)**4)**(3/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=\frac {5}{16} \, a^{\frac {3}{2}} x + \frac {1}{384} \, {\left (9 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 45 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} - 45 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} - 9 \, a^{\frac {3}{2}} e^{\left (-10 \, x\right )} - a^{\frac {3}{2}} e^{\left (-12 \, x\right )} + a^{\frac {3}{2}}\right )} e^{\left (6 \, x\right )} \] Input:

integrate((a*cosh(x)^4)^(3/2),x, algorithm="maxima")
 

Output:

5/16*a^(3/2)*x + 1/384*(9*a^(3/2)*e^(-2*x) + 45*a^(3/2)*e^(-4*x) - 45*a^(3 
/2)*e^(-8*x) - 9*a^(3/2)*e^(-10*x) - a^(3/2)*e^(-12*x) + a^(3/2))*e^(6*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.67 \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=-\frac {1}{384} \, {\left ({\left (110 \, e^{\left (6 \, x\right )} + 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-6 \, x\right )} - 120 \, x - e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 45 \, e^{\left (2 \, x\right )}\right )} a^{\frac {3}{2}} \] Input:

integrate((a*cosh(x)^4)^(3/2),x, algorithm="giac")
 

Output:

-1/384*((110*e^(6*x) + 45*e^(4*x) + 9*e^(2*x) + 1)*e^(-6*x) - 120*x - e^(6 
*x) - 9*e^(4*x) - 45*e^(2*x))*a^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {cosh}\left (x\right )}^4\right )}^{3/2} \,d x \] Input:

int((a*cosh(x)^4)^(3/2),x)
 

Output:

int((a*cosh(x)^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int \left (a \cosh ^4(x)\right )^{3/2} \, dx=\frac {\sqrt {a}\, a \left (e^{12 x}+9 e^{10 x}+45 e^{8 x}+120 e^{6 x} x -45 e^{4 x}-9 e^{2 x}-1\right )}{384 e^{6 x}} \] Input:

int((a*cosh(x)^4)^(3/2),x)
 

Output:

(sqrt(a)*a*(e**(12*x) + 9*e**(10*x) + 45*e**(8*x) + 120*e**(6*x)*x - 45*e* 
*(4*x) - 9*e**(2*x) - 1))/(384*e**(6*x))