\(\int (a \sinh ^3(x))^{3/2} \, dx\) [147]

Optimal result

Integrand size = 10, antiderivative size = 83 \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=-\frac {14}{45} a \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {14 i a \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \sinh ^3(x)}}{15 \sqrt {i \sinh (x)}}+\frac {2}{9} a \cosh (x) \sinh ^2(x) \sqrt {a \sinh ^3(x)} \] Output:

-14/45*a*cosh(x)*(a*sinh(x)^3)^(1/2)+14/15*I*a*csch(x)*EllipticE(cos(1/4*P 
i+1/2*I*x),2^(1/2))*(a*sinh(x)^3)^(1/2)/(I*sinh(x))^(1/2)+2/9*a*cosh(x)*si 
nh(x)^2*(a*sinh(x)^3)^(1/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.69 \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=\frac {1}{180} a \text {csch}(x) \sqrt {a \sinh ^3(x)} \left (168 \text {csch}(x) E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}-38 \sinh (2 x)+5 \sinh (4 x)\right ) \] Input:

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

Output:

(a*Csch[x]*Sqrt[a*Sinh[x]^3]*(168*Csch[x]*EllipticE[(Pi - (2*I)*x)/4, 2]*S 
qrt[I*Sinh[x]] - 38*Sinh[2*x] + 5*Sinh[4*x]))/180
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3121, 3042, 3119}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3686

\(\displaystyle \frac {a \sqrt {a \sinh ^3(x)} \int \sinh ^{\frac {9}{2}}(x)dx}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \sinh ^3(x)} \int (-i \sin (i x))^{9/2}dx}{\sinh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

\(\displaystyle \frac {a \sqrt {a \sinh ^3(x)} \left (\frac {2}{9} \sinh ^{\frac {7}{2}}(x) \cosh (x)-\frac {7}{9} \left (\frac {2}{5} \sinh ^{\frac {3}{2}}(x) \cosh (x)-\frac {6 i \sqrt {\sinh (x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{5 \sqrt {i \sinh (x)}}\right )\right )}{\sinh ^{\frac {3}{2}}(x)}\)

Input:

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

Output:

(a*Sqrt[a*Sinh[x]^3]*((2*Cosh[x]*Sinh[x]^(7/2))/9 - (7*((((-6*I)/5)*Ellipt 
icE[Pi/4 - (I/2)*x, 2]*Sqrt[Sinh[x]])/Sqrt[I*Sinh[x]] + (2*Cosh[x]*Sinh[x] 
^(3/2))/5))/9))/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 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (62) = 124\).

Time = 0.10 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.67 \[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=-\frac {672 \, \sqrt {\frac {1}{2}} {\left (a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (5 \, a \cosh \left (x\right )^{8} + 40 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + 5 \, a \sinh \left (x\right )^{8} - 38 \, a \cosh \left (x\right )^{6} + 2 \, {\left (70 \, a \cosh \left (x\right )^{2} - 19 \, a\right )} \sinh \left (x\right )^{6} + 4 \, {\left (70 \, a \cosh \left (x\right )^{3} - 57 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{4} + 2 \, {\left (175 \, a \cosh \left (x\right )^{4} - 285 \, a \cosh \left (x\right )^{2} - 168 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (35 \, a \cosh \left (x\right )^{5} - 95 \, a \cosh \left (x\right )^{3} - 168 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 38 \, a \cosh \left (x\right )^{2} + 2 \, {\left (70 \, a \cosh \left (x\right )^{6} - 285 \, a \cosh \left (x\right )^{4} - 1008 \, a \cosh \left (x\right )^{2} + 19 \, a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (10 \, a \cosh \left (x\right )^{7} - 57 \, a \cosh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{3} + 19 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {a \sinh \left (x\right )}}{360 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \] Input:

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

Output:

-1/360*(672*sqrt(1/2)*(a*cosh(x)^4 + 4*a*cosh(x)^3*sinh(x) + 6*a*cosh(x)^2 
*sinh(x)^2 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4)*sqrt(a)*weierstrassZeta( 
4, 0, weierstrassPInverse(4, 0, cosh(x) + sinh(x))) - (5*a*cosh(x)^8 + 40* 
a*cosh(x)*sinh(x)^7 + 5*a*sinh(x)^8 - 38*a*cosh(x)^6 + 2*(70*a*cosh(x)^2 - 
 19*a)*sinh(x)^6 + 4*(70*a*cosh(x)^3 - 57*a*cosh(x))*sinh(x)^5 - 336*a*cos 
h(x)^4 + 2*(175*a*cosh(x)^4 - 285*a*cosh(x)^2 - 168*a)*sinh(x)^4 + 8*(35*a 
*cosh(x)^5 - 95*a*cosh(x)^3 - 168*a*cosh(x))*sinh(x)^3 + 38*a*cosh(x)^2 + 
2*(70*a*cosh(x)^6 - 285*a*cosh(x)^4 - 1008*a*cosh(x)^2 + 19*a)*sinh(x)^2 + 
 4*(10*a*cosh(x)^7 - 57*a*cosh(x)^5 - 336*a*cosh(x)^3 + 19*a*cosh(x))*sinh 
(x) - 5*a)*sqrt(a*sinh(x)))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2 
*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a \sinh ^3(x)\right )^{3/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\sinh \left (x \right )}\, \sinh \left (x \right )^{4}d x \right ) a \] Input:

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

Output:

sqrt(a)*int(sqrt(sinh(x))*sinh(x)**4,x)*a