\(\int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx\) [139]

Optimal result

Integrand size = 17, antiderivative size = 251 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=-\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}+\frac {2 i \left (4 a A b-a^2 B+3 b^2 B\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i (A b-a B) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \] Output:

-2/3*(A*b-B*a)*cosh(x)/(a^2+b^2)/(a+b*sinh(x))^(3/2)-2/3*(4*A*a*b-B*a^2+3* 
B*b^2)*cosh(x)/(a^2+b^2)^2/(a+b*sinh(x))^(1/2)+2/3*I*(4*A*a*b-B*a^2+3*B*b^ 
2)*EllipticE(cos(1/4*Pi+1/2*I*x),2^(1/2)*(b/(I*a+b))^(1/2))*(a+b*sinh(x))^ 
(1/2)/b/(a^2+b^2)^2/((a+b*sinh(x))/(a-I*b))^(1/2)+2/3*I*(A*b-B*a)*InverseJ 
acobiAM(-1/4*Pi+1/2*I*x,2^(1/2)*(b/(I*a+b))^(1/2))*((a+b*sinh(x))/(a-I*b)) 
^(1/2)/b/(a^2+b^2)/(a+b*sinh(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\frac {2 i \left (\left (b \left (3 a^2 A-A b^2+4 a b B\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )+\left (4 a A b-a^2 B+3 b^2 B\right ) \left ((a-i b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )\right )\right ) (a+b \sinh (x)) \sqrt {\frac {a+b \sinh (x)}{a-i b}}+i b \cosh (x) \left (-\left (\left (a^2+b^2\right ) (-A b+a B)\right )-\left (-4 a A b+a^2 B-3 b^2 B\right ) (a+b \sinh (x))\right )\right )}{3 b \left (a^2+b^2\right )^2 (a+b \sinh (x))^{3/2}} \] Input:

Integrate[(A + B*Sinh[x])/(a + b*Sinh[x])^(5/2),x]
 

Output:

(((2*I)/3)*((b*(3*a^2*A - A*b^2 + 4*a*b*B)*EllipticF[(Pi - (2*I)*x)/4, ((- 
2*I)*b)/(a - I*b)] + (4*a*A*b - a^2*B + 3*b^2*B)*((a - I*b)*EllipticE[(Pi 
- (2*I)*x)/4, ((-2*I)*b)/(a - I*b)] - a*EllipticF[(Pi - (2*I)*x)/4, ((-2*I 
)*b)/(a - I*b)]))*(a + b*Sinh[x])*Sqrt[(a + b*Sinh[x])/(a - I*b)] + I*b*Co 
sh[x]*(-((a^2 + b^2)*(-(A*b) + a*B)) - (-4*a*A*b + a^2*B - 3*b^2*B)*(a + b 
*Sinh[x]))))/(b*(a^2 + b^2)^2*(a + b*Sinh[x])^(3/2))
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {3042, 3233, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

\(\displaystyle \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A-i B \sin (i x)}{(a-i b \sin (i x))^{5/2}}dx\)

\(\Big \downarrow \) 3233

\(\displaystyle -\frac {2 \int -\frac {3 (a A+b B)-(A b-a B) \sinh (x)}{2 (a+b \sinh (x))^{3/2}}dx}{3 \left (a^2+b^2\right )}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 (a A+b B)-(A b-a B) \sinh (x)}{(a+b \sinh (x))^{3/2}}dx}{3 \left (a^2+b^2\right )}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {\int \frac {3 (a A+b B)+i (A b-a B) \sin (i x)}{(a-i b \sin (i x))^{3/2}}dx}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {2 \int -\frac {3 A a^2+4 b B a-A b^2+\left (-B a^2+4 A b a+3 b^2 B\right ) \sinh (x)}{2 \sqrt {a+b \sinh (x)}}dx}{a^2+b^2}-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}}{3 \left (a^2+b^2\right )}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 A a^2+4 b B a-A b^2+\left (-B a^2+4 A b a+3 b^2 B\right ) \sinh (x)}{\sqrt {a+b \sinh (x)}}dx}{a^2+b^2}-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}}{3 \left (a^2+b^2\right )}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\int \frac {3 A a^2+4 b B a-A b^2-i \left (-B a^2+4 A b a+3 b^2 B\right ) \sin (i x)}{\sqrt {a-i b \sin (i x)}}dx}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\frac {\left (a^2 (-B)+4 a A b+3 b^2 B\right ) \int \sqrt {a+b \sinh (x)}dx}{b}-\frac {\left (a^2+b^2\right ) (A b-a B) \int \frac {1}{\sqrt {a+b \sinh (x)}}dx}{b}}{a^2+b^2}-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}}{3 \left (a^2+b^2\right )}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {\left (a^2 (-B)+4 a A b+3 b^2 B\right ) \int \sqrt {a-i b \sin (i x)}dx}{b}-\frac {\left (a^2+b^2\right ) (A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {\left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {\left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}+\frac {-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) (A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\)

Input:

Int[(A + B*Sinh[x])/(a + b*Sinh[x])^(5/2),x]
 

Output:

(-2*(A*b - a*B)*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^(3/2)) + ((-2*(4*a 
*A*b - a^2*B + 3*b^2*B)*Cosh[x])/((a^2 + b^2)*Sqrt[a + b*Sinh[x]]) + (((2* 
I)*(4*a*A*b - a^2*B + 3*b^2*B)*EllipticE[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]* 
Sqrt[a + b*Sinh[x]])/(b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - ((2*I)*(a^2 + b 
^2)*(A*b - a*B)*EllipticF[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a + b*Sin 
h[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]]))/(a^2 + b^2))/(3*(a^2 + b^2))
 

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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/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 3231

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

rule 3233

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (226 ) = 452\).

Time = 1.94 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.21

method	result	size
default	\(\frac {\sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}\, \left (\frac {B \left (-\frac {2 b \cosh \left (x \right )^{2}}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 a \left (\frac {a}{b}-i\right ) \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 )}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 b \left (\frac {a}{b}-i\right ) \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}}\, \left (\left (-\frac {a}{b}-i\right ) \operatorname {EllipticE}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )+i \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}\right )}{b}+\frac {\left (A b -a B \right ) \left (-\frac {2 \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}{3 b \left (a^{2}+b^{2}\right ) \left (\sinh \left (x \right )+\frac {a}{b}\right )^{2}}-\frac {8 b \cosh \left (x \right )^{2} a}{3 \left (a^{2}+b^{2}\right )^{2} \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 \left (3 a^{2}-b^{2}\right ) \left (\frac {a}{b}-i\right ) \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 )}{\left (3 a^{4}+6 a^{2} b^{2}+3 b^{4}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {8 a b \left (\frac {a}{b}-i\right ) \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}}\, \left (\left (-\frac {a}{b}-i\right ) \operatorname {EllipticE}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )+i \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )\right )}{3 \left (a^{2}+b^{2}\right )^{2} \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}\right )}{b}\right )}{\cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\)	\(806\)
parts	\(\text {Expression too large to display}\)	\(1239\)

Input:

int((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

(cosh(x)^2*(a+b*sinh(x)))^(1/2)*(B/b*(-2*b*cosh(x)^2/(a^2+b^2)/(cosh(x)^2* 
(a+b*sinh(x)))^(1/2)+2*a/(a^2+b^2)*(1/b*a-I)*((-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)/(cosh(x)^2*( 
a+b*sinh(x)))^(1/2)*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b 
+a))^(1/2))+2*b/(a^2+b^2)*(1/b*a-I)*((-a-b*sinh(x))/(I*b-a))^(1/2)*((I-sin 
h(x))*b/(I*b+a))^(1/2)*(b*(I+sinh(x))/(I*b-a))^(1/2)/(cosh(x)^2*(a+b*sinh( 
x)))^(1/2)*((-1/b*a-I)*EllipticE(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/( 
I*b+a))^(1/2))+I*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a) 
)^(1/2))))+(A*b-B*a)/b*(-2/3/b/(a^2+b^2)*(cosh(x)^2*(a+b*sinh(x)))^(1/2)/( 
sinh(x)+1/b*a)^2-8/3*b*cosh(x)^2/(a^2+b^2)^2*a/(cosh(x)^2*(a+b*sinh(x)))^( 
1/2)+2*(3*a^2-b^2)/(3*a^4+6*a^2*b^2+3*b^4)*(1/b*a-I)*((-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)/(cos 
h(x)^2*(a+b*sinh(x)))^(1/2)*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I 
*b)/(I*b+a))^(1/2))+8/3*a*b/(a^2+b^2)^2*(1/b*a-I)*((-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)/(cosh(x 
)^2*(a+b*sinh(x)))^(1/2)*((-1/b*a-I)*EllipticE(((-a-b*sinh(x))/(I*b-a))^(1 
/2),((a-I*b)/(I*b+a))^(1/2))+I*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),(( 
a-I*b)/(I*b+a))^(1/2)))))/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. 2069 vs. \(2 (218) = 436\).

Time = 0.16 (sec) , antiderivative size = 2069, normalized size of antiderivative = 8.24 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x, algorithm="fricas")
 

Output:

4/9*(sqrt(1/2)*(2*B*a^3*b^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5 + (2*B*a^3*b 
^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*cosh(x)^4 + (2*B*a^3*b^2 + A*a^2*b^3 
 + 6*B*a*b^4 - 3*A*b^5)*sinh(x)^4 + 4*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 
 - 3*A*a*b^4)*cosh(x)^3 + 4*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*a*b 
^4 + (2*B*a^3*b^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*cosh(x))*sinh(x)^3 + 
2*(4*B*a^5 + 2*A*a^4*b + 10*B*a^3*b^2 - 7*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5) 
*cosh(x)^2 + 2*(4*B*a^5 + 2*A*a^4*b + 10*B*a^3*b^2 - 7*A*a^2*b^3 - 6*B*a*b 
^4 + 3*A*b^5 + 3*(2*B*a^3*b^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*cosh(x)^2 
 + 6*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*a*b^4)*cosh(x))*sinh(x)^2 
- 4*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*a*b^4)*cosh(x) - 4*(2*B*a^4 
*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*a*b^4 - (2*B*a^3*b^2 + A*a^2*b^3 + 6*B* 
a*b^4 - 3*A*b^5)*cosh(x)^3 - 3*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A* 
a*b^4)*cosh(x)^2 - (4*B*a^5 + 2*A*a^4*b + 10*B*a^3*b^2 - 7*A*a^2*b^3 - 6*B 
*a*b^4 + 3*A*b^5)*cosh(x))*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*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5 + (B*a^2*b^3 - 4 
*A*a*b^4 - 3*B*b^5)*cosh(x)^4 + (B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*sinh(x)^ 
4 + 4*(B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4)*cosh(x)^3 + 4*(B*a^3*b^2 - 4*A 
*a^2*b^3 - 3*B*a*b^4 + (B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x))*sinh(x)^ 
3 + 2*(2*B*a^4*b - 8*A*a^3*b^2 - 7*B*a^2*b^3 + 4*A*a*b^4 + 3*B*b^5)*cos...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x, algorithm="maxima")
 

Output:

integrate((B*sinh(x) + A)/(b*sinh(x) + a)^(5/2), x)
 

Giac [F]

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

integrate((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x, algorithm="giac")
 

Output:

integrate((B*sinh(x) + A)/(b*sinh(x) + a)^(5/2), x)
 

Mupad [F(-1)]

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

int((A + B*sinh(x))/(a + b*sinh(x))^(5/2),x)
 

Output:

int((A + B*sinh(x))/(a + b*sinh(x))^(5/2), x)
 

Reduce [F]

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

int((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x)
 

Output:

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