Integrand size = 17, antiderivative size = 176 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=-\frac {2 (A b-a B) \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{b \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2 i B \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \sqrt {a+b \sinh (x)}} \] Output:
-2*(A*b-B*a)*cosh(x)/(a^2+b^2)/(a+b*sinh(x))^(1/2)+2*I*(A*b-B*a)*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)/((a+b*sinh(x))/(a-I*b))^(1/2)-2*I*B*InverseJacobiAM(-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+b*sinh(x))^( 1/2)
Time = 0.49 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\frac {2 b (-A b+a B) \cosh (x)+\frac {2 i (A b-a B) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) (a+b \sinh (x))}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}+2 i \left (a^2+b^2\right ) B \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \] Input:
Integrate[(A + B*Sinh[x])/(a + b*Sinh[x])^(3/2),x]
Output:
(2*b*(-(A*b) + a*B)*Cosh[x] + ((2*I)*(A*b - a*B)*EllipticE[(Pi - (2*I)*x)/ 4, ((-2*I)*b)/(a - I*b)]*(a + b*Sinh[x]))/Sqrt[(a + b*Sinh[x])/(a - I*b)] + (2*I)*(a^2 + b^2)*B*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)]*Sq rt[(a + b*Sinh[x])/(a - I*b)])/(b*(a^2 + b^2)*Sqrt[a + b*Sinh[x]])
Time = 0.84 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {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.
| \(\displaystyle \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A-i B \sin (i x)}{(a-i b \sin (i x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {2 \int -\frac {a A+b B+(A b-a B) \sinh (x)}{2 \sqrt {a+b \sinh (x)}}dx}{a^2+b^2}-\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a A+b B+(A b-a B) \sinh (x)}{\sqrt {a+b \sinh (x)}}dx}{a^2+b^2}-\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\int \frac {a A+b B-i (A b-a B) \sin (i x)}{\sqrt {a-i b \sin (i x)}}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}}dx}{b}+\frac {(A b-a B) \int \sqrt {a+b \sinh (x)}dx}{b}}{a^2+b^2}-\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {(A b-a B) \int \sqrt {a-i b \sin (i x)}dx}{b}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {(A b-a B) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {(A b-a B) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {2 i (A b-a B) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\frac {2 i B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}}{a^2+b^2}\) |
Input:
Int[(A + B*Sinh[x])/(a + b*Sinh[x])^(3/2),x]
Output:
(-2*(A*b - a*B)*Cosh[x])/((a^2 + b^2)*Sqrt[a + b*Sinh[x]]) + (((2*I)*(A*b - a*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)*B*EllipticF[Pi/4 - ( I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Si nh[x]]))/(a^2 + b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (161 ) = 322\).
Time = 1.29 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.94
| method | result | size |
| default | \(\frac {\sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}\, \left (\frac {2 B \left (\frac {a}{b}-i\right ) \sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{b \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {\left (A b -a B \right ) \left (-\frac {2 b \cosh \left (x \right )^{2}}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 a \left (\frac {a}{b}-i\right ) \sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 b \left (\frac {a}{b}-i\right ) \sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \left (\left (-\frac {a}{b}-i\right ) \operatorname {EllipticE}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )+i \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}\right )}{b}\right )}{\cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(517\) |
| parts | \(\frac {2 A \left (\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}+\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}-\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}-\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}-b^{2} \sinh \left (x \right )^{2}-b^{2}\right )}{b \left (a^{2}+b^{2}\right ) \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}-\frac {2 B \left (i \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} b +i \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^{3}-\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^{3}-\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 \,b^{2}-a \,b^{2} \sinh \left (x \right )^{2}-a \,b^{2}\right )}{b^{2} \left (a^{2}+b^{2}\right ) \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(922\) |
Input:
int((A+B*sinh(x))/(a+b*sinh(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
(cosh(x)^2*(a+b*sinh(x)))^(1/2)*(2*B/b*(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))+(A*b-B*a)/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-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)*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-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)
Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (154) = 308\).
Time = 0.11 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.39 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((A+B*sinh(x))/(a+b*sinh(x))^(3/2),x, algorithm="fricas")
Output:
4/3*(sqrt(1/2)*(2*B*a^2*b + A*a*b^2 + 3*B*b^3 - (2*B*a^2*b + A*a*b^2 + 3*B *b^3)*cosh(x)^2 - (2*B*a^2*b + A*a*b^2 + 3*B*b^3)*sinh(x)^2 - 2*(2*B*a^3 + A*a^2*b + 3*B*a*b^2)*cosh(x) - 2*(2*B*a^3 + A*a^2*b + 3*B*a*b^2 + (2*B*a^ 2*b + A*a*b^2 + 3*B*b^3)*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*b^2 - A*b^3 - (B*a*b^2 - A*b^3)*cosh( x)^2 - (B*a*b^2 - A*b^3)*sinh(x)^2 - 2*(B*a^2*b - A*a*b^2)*cosh(x) - 2*(B* a^2*b - A*a*b^2 + (B*a*b^2 - A*b^3)*cosh(x))*sinh(x))*sqrt(b)*weierstrassZ eta(4/3*(4*a^2 + 3*b^2)/b^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, weierstrassPInve rse(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*a*b^2 - A*b^3)*cosh(x)^2 + (B*a*b^2 - A* b^3)*sinh(x)^2 + (B*a^2*b - A*a*b^2)*cosh(x) + (B*a^2*b - A*a*b^2 + 2*(B*a *b^2 - A*b^3)*cosh(x))*sinh(x))*sqrt(b*sinh(x) + a))/(a^2*b^3 + b^5 - (a^2 *b^3 + b^5)*cosh(x)^2 - (a^2*b^3 + b^5)*sinh(x)^2 - 2*(a^3*b^2 + a*b^4)*co sh(x) - 2*(a^3*b^2 + a*b^4 + (a^2*b^3 + b^5)*cosh(x))*sinh(x))
Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*sinh(x))/(a+b*sinh(x))**(3/2),x)
Output:
Timed out
\[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{{\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*sinh(x))/(a+b*sinh(x))^(3/2),x, algorithm="maxima")
Output:
integrate((B*sinh(x) + A)/(b*sinh(x) + a)^(3/2), x)
\[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{{\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*sinh(x))/(a+b*sinh(x))^(3/2),x, algorithm="giac")
Output:
integrate((B*sinh(x) + A)/(b*sinh(x) + a)^(3/2), x)
Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\int \frac {A+B\,\mathrm {sinh}\left (x\right )}{{\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{3/2}} \,d x \] Input:
int((A + B*sinh(x))/(a + b*sinh(x))^(3/2),x)
Output:
int((A + B*sinh(x))/(a + b*sinh(x))^(3/2), x)
\[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\int \frac {\sqrt {\sinh \left (x \right ) b +a}}{\sinh \left (x \right ) b +a}d x \] Input:
int((A+B*sinh(x))/(a+b*sinh(x))^(3/2),x)
Output:
int(sqrt(sinh(x)*b + a)/(sinh(x)*b + a),x)