\(\int (a \cosh ^4(x))^{5/2} \, dx\) [134]

Optimal result

Integrand size = 10, antiderivative size = 132 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {63}{256} a^2 x \sqrt {a \cosh ^4(x)} \text {sech}^2(x)+\frac {21}{128} a^2 \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {21}{160} a^2 \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {9}{80} a^2 \cosh ^5(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {63}{256} a^2 \sqrt {a \cosh ^4(x)} \tanh (x) \] Output:

63/256*a^2*x*(a*cosh(x)^4)^(1/2)*sech(x)^2+21/128*a^2*cosh(x)*(a*cosh(x)^4 
)^(1/2)*sinh(x)+21/160*a^2*cosh(x)^3*(a*cosh(x)^4)^(1/2)*sinh(x)+9/80*a^2* 
cosh(x)^5*(a*cosh(x)^4)^(1/2)*sinh(x)+1/10*a^2*cosh(x)^7*(a*cosh(x)^4)^(1/ 
2)*sinh(x)+63/256*a^2*(a*cosh(x)^4)^(1/2)*tanh(x)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.40 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {a \left (a \cosh ^4(x)\right )^{3/2} \text {sech}^6(x) (2520 x+2100 \sinh (2 x)+600 \sinh (4 x)+150 \sinh (6 x)+25 \sinh (8 x)+2 \sinh (10 x))}{10240} \] Input:

Integrate[(a*Cosh[x]^4)^(5/2),x]
 

Output:

(a*(a*Cosh[x]^4)^(3/2)*Sech[x]^6*(2520*x + 2100*Sinh[2*x] + 600*Sinh[4*x] 
+ 150*Sinh[6*x] + 25*Sinh[8*x] + 2*Sinh[10*x]))/10240
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 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.

\(\displaystyle \int \left (a \cosh ^4(x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \sin \left (\frac {\pi }{2}+i x\right )^4\right )^{5/2}dx\)

\(\Big \downarrow \) 3686

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \int \cosh ^{10}(x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \int \sin \left (i x+\frac {\pi }{2}\right )^{10}dx\)

\(\Big \downarrow \) 3115

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {9}{10} \int \cosh ^8(x)dx+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \int \sin \left (i x+\frac {\pi }{2}\right )^8dx\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \int \cosh ^6(x)dx+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \int \sin \left (i x+\frac {\pi }{2}\right )^6dx\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \cosh ^4(x)dx+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \int \sin \left (i x+\frac {\pi }{2}\right )^4dx\right )\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cosh ^2(x)dx+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )\right )\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh (x) \cosh ^3(x)\right )+\frac {1}{6} \sinh (x) \cosh ^5(x)\right )+\frac {1}{8} \sinh (x) \cosh ^7(x)\right )+\frac {1}{10} \sinh (x) \cosh ^9(x)\right )\)

\(\Big \downarrow \) 24

\(\displaystyle a^2 \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \left (\frac {1}{10} \sinh (x) \cosh ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sinh (x) \cosh ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sinh (x) \cosh ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )\right )\right )\right )\right )\)

Input:

Int[(a*Cosh[x]^4)^(5/2),x]
 

Output:

a^2*Sqrt[a*Cosh[x]^4]*Sech[x]^2*((Cosh[x]^9*Sinh[x])/10 + (9*((Cosh[x]^7*S 
inh[x])/8 + (7*((Cosh[x]^5*Sinh[x])/6 + (5*((Cosh[x]^3*Sinh[x])/4 + (3*(x/ 
2 + (Cosh[x]*Sinh[x])/2))/4))/6))/8))/10)
 

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 = 15.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.30

method	result	size
default	\(\frac {a^{\frac {3}{2}} \left (1+\cosh \left (2 x \right )\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \left (8 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}\, \sinh \left (2 x \right )^{4}+50 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}\, \cosh \left (2 x \right ) \sinh \left (2 x \right )^{2}+160 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}\, \sinh \left (2 x \right )^{2}+325 \cosh \left (2 x \right ) \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+640 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+315 \ln \left (\cosh \left (2 x \right ) \sqrt {a}+\sqrt {a \sinh \left (2 x \right )^{2}}\right ) a \right )}{2560 \sinh \left (2 x \right ) \sqrt {\left (1+\cosh \left (2 x \right )\right )^{2} a}}\)	\(171\)
risch	\(\frac {63 a^{2} {\mathrm e}^{2 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}\, x}{256 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {a^{2} {\mathrm e}^{12 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{10240 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {5 a^{2} {\mathrm e}^{10 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{4096 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {15 a^{2} {\mathrm e}^{8 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{2048 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {15 a^{2} {\mathrm e}^{6 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{512 \left ({\mathrm e}^{2 x}+1\right )^{2}}+\frac {105 a^{2} {\mathrm e}^{4 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{1024 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {105 \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}\, a^{2}}{1024 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {15 a^{2} {\mathrm e}^{-2 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{512 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {15 a^{2} {\mathrm e}^{-4 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{2048 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {5 a^{2} {\mathrm e}^{-6 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{4096 \left ({\mathrm e}^{2 x}+1\right )^{2}}-\frac {a^{2} {\mathrm e}^{-8 x} \sqrt {a \left ({\mathrm e}^{2 x}+1\right )^{4} {\mathrm e}^{-4 x}}}{10240 \left ({\mathrm e}^{2 x}+1\right )^{2}}\)	\(362\)

Input:

int((a*cosh(x)^4)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/2560*a^(3/2)*(1+cosh(2*x))*(a*(-1+cosh(2*x))*(1+cosh(2*x)))^(1/2)*(8*(a* 
sinh(2*x)^2)^(1/2)*a^(1/2)*sinh(2*x)^4+50*(a*sinh(2*x)^2)^(1/2)*a^(1/2)*co 
sh(2*x)*sinh(2*x)^2+160*(a*sinh(2*x)^2)^(1/2)*a^(1/2)*sinh(2*x)^2+325*cosh 
(2*x)*(a*sinh(2*x)^2)^(1/2)*a^(1/2)+640*(a*sinh(2*x)^2)^(1/2)*a^(1/2)+315* 
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. 1597 vs. \(2 (108) = 216\).

Time = 0.15 (sec) , antiderivative size = 1597, normalized size of antiderivative = 12.10 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a*cosh(x)^4)^(5/2),x, algorithm="fricas")
 

Output:

1/20480*(40*a^2*cosh(x)*e^(2*x)*sinh(x)^19 + 2*a^2*e^(2*x)*sinh(x)^20 + 5* 
(76*a^2*cosh(x)^2 + 5*a^2)*e^(2*x)*sinh(x)^18 + 30*(76*a^2*cosh(x)^3 + 15* 
a^2*cosh(x))*e^(2*x)*sinh(x)^17 + 15*(646*a^2*cosh(x)^4 + 255*a^2*cosh(x)^ 
2 + 10*a^2)*e^(2*x)*sinh(x)^16 + 48*(646*a^2*cosh(x)^5 + 425*a^2*cosh(x)^3 
 + 50*a^2*cosh(x))*e^(2*x)*sinh(x)^15 + 60*(1292*a^2*cosh(x)^6 + 1275*a^2* 
cosh(x)^4 + 300*a^2*cosh(x)^2 + 10*a^2)*e^(2*x)*sinh(x)^14 + 120*(1292*a^2 
*cosh(x)^7 + 1785*a^2*cosh(x)^5 + 700*a^2*cosh(x)^3 + 70*a^2*cosh(x))*e^(2 
*x)*sinh(x)^13 + 60*(4199*a^2*cosh(x)^8 + 7735*a^2*cosh(x)^6 + 4550*a^2*co 
sh(x)^4 + 910*a^2*cosh(x)^2 + 35*a^2)*e^(2*x)*sinh(x)^12 + 80*(4199*a^2*co 
sh(x)^9 + 9945*a^2*cosh(x)^7 + 8190*a^2*cosh(x)^5 + 2730*a^2*cosh(x)^3 + 3 
15*a^2*cosh(x))*e^(2*x)*sinh(x)^11 + 2*(184756*a^2*cosh(x)^10 + 546975*a^2 
*cosh(x)^8 + 600600*a^2*cosh(x)^6 + 300300*a^2*cosh(x)^4 + 69300*a^2*cosh( 
x)^2 + 2520*a^2*x)*e^(2*x)*sinh(x)^10 + 20*(16796*a^2*cosh(x)^11 + 60775*a 
^2*cosh(x)^9 + 85800*a^2*cosh(x)^7 + 60060*a^2*cosh(x)^5 + 23100*a^2*cosh( 
x)^3 + 2520*a^2*x*cosh(x))*e^(2*x)*sinh(x)^9 + 30*(8398*a^2*cosh(x)^12 + 3 
6465*a^2*cosh(x)^10 + 64350*a^2*cosh(x)^8 + 60060*a^2*cosh(x)^6 + 34650*a^ 
2*cosh(x)^4 + 7560*a^2*x*cosh(x)^2 - 70*a^2)*e^(2*x)*sinh(x)^8 + 240*(646* 
a^2*cosh(x)^13 + 3315*a^2*cosh(x)^11 + 7150*a^2*cosh(x)^9 + 8580*a^2*cosh( 
x)^7 + 6930*a^2*cosh(x)^5 + 2520*a^2*x*cosh(x)^3 - 70*a^2*cosh(x))*e^(2*x) 
*sinh(x)^7 + 60*(1292*a^2*cosh(x)^14 + 7735*a^2*cosh(x)^12 + 20020*a^2*...
 

Sympy [F(-1)]

Timed out. \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\text {Timed out} \] Input:

integrate((a*cosh(x)**4)**(5/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {63}{256} \, a^{\frac {5}{2}} x + \frac {1}{20480} \, {\left (25 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 150 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 600 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 2100 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} - 2100 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} - 600 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} - 150 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} - 25 \, a^{\frac {5}{2}} e^{\left (-18 \, x\right )} - 2 \, a^{\frac {5}{2}} e^{\left (-20 \, x\right )} + 2 \, a^{\frac {5}{2}}\right )} e^{\left (10 \, x\right )} \] Input:

integrate((a*cosh(x)^4)^(5/2),x, algorithm="maxima")
 

Output:

63/256*a^(5/2)*x + 1/20480*(25*a^(5/2)*e^(-2*x) + 150*a^(5/2)*e^(-4*x) + 6 
00*a^(5/2)*e^(-6*x) + 2100*a^(5/2)*e^(-8*x) - 2100*a^(5/2)*e^(-12*x) - 600 
*a^(5/2)*e^(-14*x) - 150*a^(5/2)*e^(-16*x) - 25*a^(5/2)*e^(-18*x) - 2*a^(5 
/2)*e^(-20*x) + 2*a^(5/2))*e^(10*x)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=-\frac {1}{20480} \, {\left ({\left (5754 \, e^{\left (10 \, x\right )} + 2100 \, e^{\left (8 \, x\right )} + 600 \, e^{\left (6 \, x\right )} + 150 \, e^{\left (4 \, x\right )} + 25 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-10 \, x\right )} - 5040 \, x - 2 \, e^{\left (10 \, x\right )} - 25 \, e^{\left (8 \, x\right )} - 150 \, e^{\left (6 \, x\right )} - 600 \, e^{\left (4 \, x\right )} - 2100 \, e^{\left (2 \, x\right )}\right )} a^{\frac {5}{2}} \] Input:

integrate((a*cosh(x)^4)^(5/2),x, algorithm="giac")
 

Output:

-1/20480*((5754*e^(10*x) + 2100*e^(8*x) + 600*e^(6*x) + 150*e^(4*x) + 25*e 
^(2*x) + 2)*e^(-10*x) - 5040*x - 2*e^(10*x) - 25*e^(8*x) - 150*e^(6*x) - 6 
00*e^(4*x) - 2100*e^(2*x))*a^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\int {\left (a\,{\mathrm {cosh}\left (x\right )}^4\right )}^{5/2} \,d x \] Input:

int((a*cosh(x)^4)^(5/2),x)
 

Output:

int((a*cosh(x)^4)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.66 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {\sqrt {a}\, a^{2} \left (2 e^{20 x}+25 e^{18 x}+150 e^{16 x}+600 e^{14 x}+2100 e^{12 x}+5040 e^{10 x} x -2100 e^{8 x}-600 e^{6 x}-150 e^{4 x}-25 e^{2 x}-2\right )}{20480 e^{10 x}} \] Input:

int((a*cosh(x)^4)^(5/2),x)
 

Output:

(sqrt(a)*a**2*(2*e**(20*x) + 25*e**(18*x) + 150*e**(16*x) + 600*e**(14*x) 
+ 2100*e**(12*x) + 5040*e**(10*x)*x - 2100*e**(8*x) - 600*e**(6*x) - 150*e 
**(4*x) - 25*e**(2*x) - 2))/(20480*e**(10*x))