\(\int \frac {1}{(a \sinh ^3(x))^{3/2}} \, dx\) [150]

Optimal result

Integrand size = 10, antiderivative size = 87 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {10 i \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)} \sinh (x)}{21 a \sqrt {a \sinh ^3(x)}} \] Output:

10/21*cosh(x)/a/(a*sinh(x)^3)^(1/2)-2/7*coth(x)*csch(x)/a/(a*sinh(x)^3)^(1 
/2)-10/21*I*InverseJacobiAM(-1/4*Pi+1/2*I*x,2^(1/2))*(I*sinh(x))^(1/2)*sin 
h(x)/a/(a*sinh(x)^3)^(1/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx=\frac {2 \left (5 \cosh (x)-3 \coth (x) \text {csch}(x)+5 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right ) (i \sinh (x))^{3/2}\right )}{21 a \sqrt {a \sinh ^3(x)}} \] Input:

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

Output:

(2*(5*Cosh[x] - 3*Coth[x]*Csch[x] + 5*EllipticF[(Pi - (2*I)*x)/4, 2]*(I*Si 
nh[x])^(3/2)))/(21*a*Sqrt[a*Sinh[x]^3])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, 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, 3116, 3042, 3116, 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.

\(\displaystyle \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (i a \sin (i x)^3\right )^{3/2}}dx\)

\(\Big \downarrow \) 3686

\(\displaystyle \frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {9}{2}}(x)}dx}{a \sqrt {a \sinh ^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{(-i \sin (i x))^{9/2}}dx}{a \sqrt {a \sinh ^3(x)}}\)

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 3042

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

rule 3116

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

\[\int \frac {1}{\left (a \sinh \left (x \right )^{3}\right )^{\frac {3}{2}}}d x\]

Input:

int(1/(a*sinh(x)^3)^(3/2),x)
 

Output:

int(1/(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. 561 vs. \(2 (65) = 130\).

Time = 0.10 (sec) , antiderivative size = 561, normalized size of antiderivative = 6.45 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(a*sinh(x)^3)^(3/2),x, algorithm="fricas")
 

Output:

4/21*(5*sqrt(1/2)*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh 
(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 
 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh 
(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^ 
4 + 9*cosh(x)^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)^5 
+ 3*cosh(x)^3 - cosh(x))*sinh(x) + 1)*sqrt(a)*weierstrassPInverse(4, 0, co 
sh(x) + sinh(x)) + (5*cosh(x)^7 + 35*cosh(x)*sinh(x)^6 + 5*sinh(x)^7 + (10 
5*cosh(x)^2 - 17)*sinh(x)^5 - 17*cosh(x)^5 + 5*(35*cosh(x)^3 - 17*cosh(x)) 
*sinh(x)^4 + (175*cosh(x)^4 - 170*cosh(x)^2 - 17)*sinh(x)^3 - 17*cosh(x)^3 
 + (105*cosh(x)^5 - 170*cosh(x)^3 - 51*cosh(x))*sinh(x)^2 + (35*cosh(x)^6 
- 85*cosh(x)^4 - 51*cosh(x)^2 + 5)*sinh(x) + 5*cosh(x))*sqrt(a*sinh(x)))/( 
a^2*cosh(x)^8 + 8*a^2*cosh(x)*sinh(x)^7 + a^2*sinh(x)^8 - 4*a^2*cosh(x)^6 
+ 4*(7*a^2*cosh(x)^2 - a^2)*sinh(x)^6 + 6*a^2*cosh(x)^4 + 8*(7*a^2*cosh(x) 
^3 - 3*a^2*cosh(x))*sinh(x)^5 + 2*(35*a^2*cosh(x)^4 - 30*a^2*cosh(x)^2 + 3 
*a^2)*sinh(x)^4 - 4*a^2*cosh(x)^2 + 8*(7*a^2*cosh(x)^5 - 10*a^2*cosh(x)^3 
+ 3*a^2*cosh(x))*sinh(x)^3 + 4*(7*a^2*cosh(x)^6 - 15*a^2*cosh(x)^4 + 9*a^2 
*cosh(x)^2 - a^2)*sinh(x)^2 + a^2 + 8*(a^2*cosh(x)^7 - 3*a^2*cosh(x)^5 + 3 
*a^2*cosh(x)^3 - a^2*cosh(x))*sinh(x))
 

Sympy [F]

\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \sinh ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(a*sinh(x)**3)**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \sinh \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a*sinh(x)^3)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \sinh \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a*sinh(x)^3)^(3/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {sinh}\left (x\right )}^3\right )}^{3/2}} \,d x \] Input:

int(1/(a*sinh(x)^3)^(3/2),x)
 

Output:

int(1/(a*sinh(x)^3)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sinh \left (x \right )}}{\sinh \left (x \right )^{5}}d x \right )}{a^{2}} \] Input:

int(1/(a*sinh(x)^3)^(3/2),x)
 

Output:

(sqrt(a)*int(sqrt(sinh(x))/sinh(x)**5,x))/a**2