\(\int \frac {1}{(a+b \sinh (x))^{3/2}} \, dx\) [109]

Optimal result

Integrand size = 10, antiderivative size = 94 \[ \int \frac {1}{(a+b \sinh (x))^{3/2}} \, dx=-\frac {2 b \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{\left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}} \] Output:

-2*b*cosh(x)/(a^2+b^2)/(a+b*sinh(x))^(1/2)+2*I*EllipticE(cos(1/4*Pi+1/2*I* 
x),2^(1/2)*(b/(I*a+b))^(1/2))*(a+b*sinh(x))^(1/2)/(a^2+b^2)/((a+b*sinh(x)) 
/(a-I*b))^(1/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b \sinh (x))^{3/2}} \, dx=\frac {-2 b \cosh (x)+2 (i a+b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \] Input:

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

Output:

(-2*b*Cosh[x] + 2*(I*a + b)*EllipticE[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I* 
b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/((a^2 + b^2)*Sqrt[a + b*Sinh[x]])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 3143, 27, 3042, 3134, 3042, 3132}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3143

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {a+b \sinh (x)}dx}{a^2+b^2}-\frac {2 b \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\int \sqrt {a-i b \sin (i x)}dx}{a^2+b^2}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {2 b \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{\left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{\left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {2 b \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

Input:

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

Output:

(-2*b*Cosh[x])/((a^2 + b^2)*Sqrt[a + b*Sinh[x]]) + ((2*I)*EllipticE[Pi/4 - 
 (I/2)*x, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/((a^2 + b^2)*Sqrt[(a + b*S 
inh[x])/(a - I*b)])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3143

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos 
[c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp 
[1/((n + 1)*(a^2 - b^2))   Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) 
- b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (86 ) = 172\).

Time = 0.34 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.85

method	result	size
default	\(\frac {2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2}+2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{2}-2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2}-2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{2}-2 b^{2} \sinh \left (x \right )^{2}-2 b^{2}}{\left (a^{2}+b^{2}\right ) b \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\)	\(456\)

Input:

int(1/(a+b*sinh(x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2*((-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*(b*(I+sinh 
(x))/(I*b-a))^(1/2)*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I* 
b+a))^(1/2))*a^2+(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1 
/2)*(b*(I+sinh(x))/(I*b-a))^(1/2)*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2) 
,(-(I*b-a)/(I*b+a))^(1/2))*b^2-(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x)) 
*b/(I*b+a))^(1/2)*(b*(I+sinh(x))/(I*b-a))^(1/2)*EllipticE((-(a+b*sinh(x))/ 
(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^2-(-(a+b*sinh(x))/(I*b-a))^(1/2 
)*((I-sinh(x))*b/(I*b+a))^(1/2)*(b*(I+sinh(x))/(I*b-a))^(1/2)*EllipticE((- 
(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*b^2-b^2*sinh(x)^2-b 
^2)/(a^2+b^2)/b/cosh(x)/(a+b*sinh(x))^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (82) = 164\).

Time = 0.10 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.00 \[ \int \frac {1}{(a+b \sinh (x))^{3/2}} \, dx=-\frac {4 \, {\left (\sqrt {\frac {1}{2}} {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) - a b + 2 \, {\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + a b \cosh \left (x\right ) + {\left (2 \, b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}\right )}}{3 \, {\left (a^{2} b^{2} + b^{4} - {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - {\left (a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{3} b + a b^{3} + {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \] Input:

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

Output:

-4/3*(sqrt(1/2)*(a*b*cosh(x)^2 + a*b*sinh(x)^2 + 2*a^2*cosh(x) - a*b + 2*( 
a*b*cosh(x) + a^2)*sinh(x))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 + 3*b^2 
)/b^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/ 
b) - 3*sqrt(1/2)*(b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) - b^2 + 2* 
(b^2*cosh(x) + a*b)*sinh(x))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 + 3*b^2)/b 
^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 + 3*b^2)/b 
^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b)) 
 - 3*(b^2*cosh(x)^2 + b^2*sinh(x)^2 + a*b*cosh(x) + (2*b^2*cosh(x) + a*b)* 
sinh(x))*sqrt(b*sinh(x) + a))/(a^2*b^2 + b^4 - (a^2*b^2 + b^4)*cosh(x)^2 - 
 (a^2*b^2 + b^4)*sinh(x)^2 - 2*(a^3*b + a*b^3)*cosh(x) - 2*(a^3*b + a*b^3 
+ (a^2*b^2 + b^4)*cosh(x))*sinh(x))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sinh(x)*b + a)/(sinh(x)**2*b**2 + 2*sinh(x)*a*b + a**2),x)