\(\int (a \sinh ^3(x))^{5/2} \, dx\) [146]

Optimal result

Integrand size = 10, antiderivative size = 135 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=-\frac {26}{77} a^2 \coth (x) \sqrt {a \sinh ^3(x)}+\frac {26}{77} i a^2 \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}+\frac {78}{385} a^2 \cosh (x) \sinh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^3(x)} \] Output:

-26/77*a^2*coth(x)*(a*sinh(x)^3)^(1/2)-26/77*I*a^2*csch(x)^2*InverseJacobi 
AM(-1/4*Pi+1/2*I*x,2^(1/2))*(I*sinh(x))^(1/2)*(a*sinh(x)^3)^(1/2)+78/385*a 
^2*cosh(x)*sinh(x)*(a*sinh(x)^3)^(1/2)-26/165*a^2*cosh(x)*sinh(x)^3*(a*sin 
h(x)^3)^(1/2)+2/15*a^2*cosh(x)*sinh(x)^5*(a*sinh(x)^3)^(1/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\frac {a^2 \text {csch}(x) \left (-15465 \cosh (x)+3657 \cosh (3 x)-749 \cosh (5 x)+77 \cosh (7 x)-\frac {12480 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )}{\sqrt {i \sinh (x)}}\right ) \sqrt {a \sinh ^3(x)}}{36960} \] Input:

Integrate[(a*Sinh[x]^3)^(5/2),x]
 

Output:

(a^2*Csch[x]*(-15465*Cosh[x] + 3657*Cosh[3*x] - 749*Cosh[5*x] + 77*Cosh[7* 
x] - (12480*EllipticF[(Pi - (2*I)*x)/4, 2])/Sqrt[I*Sinh[x]])*Sqrt[a*Sinh[x 
]^3])/36960
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3121, 3042, 3120}

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.

\(\displaystyle \int \left (a \sinh ^3(x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (i a \sin (i x)^3\right )^{5/2}dx\)

\(\Big \downarrow \) 3686

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \int \sinh ^{\frac {15}{2}}(x)dx}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \int (-i \sin (i x))^{15/2}dx}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \int \sinh ^{\frac {11}{2}}(x)dx\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \int (-i \sin (i x))^{11/2}dx\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \int \sinh ^{\frac {7}{2}}(x)dx\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \int (-i \sin (i x))^{7/2}dx\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \left (\frac {2}{7} \sinh ^{\frac {5}{2}}(x) \cosh (x)-\frac {5}{7} \int \sinh ^{\frac {3}{2}}(x)dx\right )\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \left (\frac {2}{7} \sinh ^{\frac {5}{2}}(x) \cosh (x)-\frac {5}{7} \int (-i \sin (i x))^{3/2}dx\right )\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \left (\frac {2}{7} \sinh ^{\frac {5}{2}}(x) \cosh (x)-\frac {5}{7} \left (\frac {2}{3} \sqrt {\sinh (x)} \cosh (x)-\frac {1}{3} \int \frac {1}{\sqrt {\sinh (x)}}dx\right )\right )\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \left (\frac {2}{7} \sinh ^{\frac {5}{2}}(x) \cosh (x)-\frac {5}{7} \left (\frac {2}{3} \sqrt {\sinh (x)} \cosh (x)-\frac {1}{3} \int \frac {1}{\sqrt {-i \sin (i x)}}dx\right )\right )\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \left (\frac {2}{7} \sinh ^{\frac {5}{2}}(x) \cosh (x)-\frac {5}{7} \left (\frac {2}{3} \sqrt {\sinh (x)} \cosh (x)-\frac {\sqrt {i \sinh (x)} \int \frac {1}{\sqrt {i \sinh (x)}}dx}{3 \sqrt {\sinh (x)}}\right )\right )\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \left (\frac {2}{7} \sinh ^{\frac {5}{2}}(x) \cosh (x)-\frac {5}{7} \left (\frac {2}{3} \sqrt {\sinh (x)} \cosh (x)-\frac {\sqrt {i \sinh (x)} \int \frac {1}{\sqrt {\sin (i x)}}dx}{3 \sqrt {\sinh (x)}}\right )\right )\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {a^2 \sqrt {a \sinh ^3(x)} \left (\frac {2}{15} \sinh ^{\frac {13}{2}}(x) \cosh (x)-\frac {13}{15} \left (\frac {2}{11} \sinh ^{\frac {9}{2}}(x) \cosh (x)-\frac {9}{11} \left (\frac {2}{7} \sinh ^{\frac {5}{2}}(x) \cosh (x)-\frac {5}{7} \left (\frac {2}{3} \sqrt {\sinh (x)} \cosh (x)-\frac {2 i \sqrt {i \sinh (x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right )}{3 \sqrt {\sinh (x)}}\right )\right )\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

Input:

Int[(a*Sinh[x]^3)^(5/2),x]
 

Output:

(a^2*Sqrt[a*Sinh[x]^3]*((2*Cosh[x]*Sinh[x]^(13/2))/15 - (13*((2*Cosh[x]*Si 
nh[x]^(9/2))/11 - (9*((-5*((((-2*I)/3)*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I 
*Sinh[x]])/Sqrt[Sinh[x]] + (2*Cosh[x]*Sqrt[Sinh[x]])/3))/7 + (2*Cosh[x]*Si 
nh[x]^(5/2))/7))/11))/15))/Sinh[x]^(3/2)
 

Defintions of rubi rules used

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 3120

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 3121

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

\[\int \left (a \sinh \left (x \right )^{3}\right )^{\frac {5}{2}}d x\]

Input:

int((a*sinh(x)^3)^(5/2),x)
 

Output:

int((a*sinh(x)^3)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (105) = 210\).

Time = 0.11 (sec) , antiderivative size = 802, normalized size of antiderivative = 5.94 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a*sinh(x)^3)^(5/2),x, algorithm="fricas")
 

Output:

1/73920*(49920*sqrt(1/2)*(a^2*cosh(x)^7 + 7*a^2*cosh(x)^6*sinh(x) + 21*a^2 
*cosh(x)^5*sinh(x)^2 + 35*a^2*cosh(x)^4*sinh(x)^3 + 35*a^2*cosh(x)^3*sinh( 
x)^4 + 21*a^2*cosh(x)^2*sinh(x)^5 + 7*a^2*cosh(x)*sinh(x)^6 + a^2*sinh(x)^ 
7)*sqrt(a)*weierstrassPInverse(4, 0, cosh(x) + sinh(x)) + (77*a^2*cosh(x)^ 
14 + 1078*a^2*cosh(x)*sinh(x)^13 + 77*a^2*sinh(x)^14 - 749*a^2*cosh(x)^12 
+ 7*(1001*a^2*cosh(x)^2 - 107*a^2)*sinh(x)^12 + 3657*a^2*cosh(x)^10 + 28*( 
1001*a^2*cosh(x)^3 - 321*a^2*cosh(x))*sinh(x)^11 + (77077*a^2*cosh(x)^4 - 
49434*a^2*cosh(x)^2 + 3657*a^2)*sinh(x)^10 - 15465*a^2*cosh(x)^8 + 2*(7707 
7*a^2*cosh(x)^5 - 82390*a^2*cosh(x)^3 + 18285*a^2*cosh(x))*sinh(x)^9 + 3*( 
77077*a^2*cosh(x)^6 - 123585*a^2*cosh(x)^4 + 54855*a^2*cosh(x)^2 - 5155*a^ 
2)*sinh(x)^8 - 15465*a^2*cosh(x)^6 + 24*(11011*a^2*cosh(x)^7 - 24717*a^2*c 
osh(x)^5 + 18285*a^2*cosh(x)^3 - 5155*a^2*cosh(x))*sinh(x)^7 + 3*(77077*a^ 
2*cosh(x)^8 - 230692*a^2*cosh(x)^6 + 255990*a^2*cosh(x)^4 - 144340*a^2*cos 
h(x)^2 - 5155*a^2)*sinh(x)^6 + 3657*a^2*cosh(x)^4 + 2*(77077*a^2*cosh(x)^9 
 - 296604*a^2*cosh(x)^7 + 460782*a^2*cosh(x)^5 - 433020*a^2*cosh(x)^3 - 46 
395*a^2*cosh(x))*sinh(x)^5 + (77077*a^2*cosh(x)^10 - 370755*a^2*cosh(x)^8 
+ 767970*a^2*cosh(x)^6 - 1082550*a^2*cosh(x)^4 - 231975*a^2*cosh(x)^2 + 36 
57*a^2)*sinh(x)^4 - 749*a^2*cosh(x)^2 + 4*(7007*a^2*cosh(x)^11 - 41195*a^2 
*cosh(x)^9 + 109710*a^2*cosh(x)^7 - 216510*a^2*cosh(x)^5 - 77325*a^2*cosh( 
x)^3 + 3657*a^2*cosh(x))*sinh(x)^3 + (7007*a^2*cosh(x)^12 - 49434*a^2*c...
 

Sympy [F]

\[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int \left (a \sinh ^{3}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((a*sinh(x)**3)**(5/2),x)
 

Output:

Integral((a*sinh(x)**3)**(5/2), x)
 

Maxima [F]

\[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int { \left (a \sinh \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \] Input:

integrate((a*sinh(x)^3)^(5/2),x, algorithm="maxima")
 

Output:

integrate((a*sinh(x)^3)^(5/2), x)
 

Giac [F]

\[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int { \left (a \sinh \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \] Input:

integrate((a*sinh(x)^3)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((a*sinh(x)^3)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\int {\left (a\,{\mathrm {sinh}\left (x\right )}^3\right )}^{5/2} \,d x \] Input:

int((a*sinh(x)^3)^(5/2),x)
 

Output:

int((a*sinh(x)^3)^(5/2), x)
 

Reduce [F]

\[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\sinh \left (x \right )}\, \sinh \left (x \right )^{7}d x \right ) a^{2} \] Input:

int((a*sinh(x)^3)^(5/2),x)
 

Output:

sqrt(a)*int(sqrt(sinh(x))*sinh(x)**7,x)*a**2