Integrand size = 17, antiderivative size = 164 \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2 i (3 A b+a B) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) B \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \sqrt {a+b \sinh (x)}} \] Output:
2/3*B*cosh(x)*(a+b*sinh(x))^(1/2)+2/3*I*(3*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+b*sinh(x))/(a -I*b))^(1/2)+2/3*I*(a^2+b^2)*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 = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\frac {2 b B \cosh (x) (a+b \sinh (x))+2 (i a+b) (3 A b+a B) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}-2 i \left (a^2+b^2\right ) B \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \sqrt {a+b \sinh (x)}} \] Input:
Integrate[Sqrt[a + b*Sinh[x]]*(A + B*Sinh[x]),x]
Output:
(2*b*B*Cosh[x]*(a + b*Sinh[x]) + 2*(I*a + b)*(3*A*b + a*B)*EllipticE[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)]*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)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*b*Sqrt[a + b*Sinh[x]])
Time = 1.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3042, 3232, 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.
| \(\displaystyle \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a-i b \sin (i x)} (A-i B \sin (i x))dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2}{3} \int \frac {3 a A-b B+(3 A b+a B) \sinh (x)}{2 \sqrt {a+b \sinh (x)}}dx+\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {3 a A-b B+(3 A b+a B) \sinh (x)}{\sqrt {a+b \sinh (x)}}dx+\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \int \frac {3 a A-b B-i (3 A b+a B) \sin (i x)}{\sqrt {a-i b \sin (i x)}}dx\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{3} \left (\frac {(a B+3 A b) \int \sqrt {a+b \sinh (x)}dx}{b}-\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}}dx}{b}\right )+\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {(a B+3 A b) \int \sqrt {a-i b \sin (i x)}dx}{b}-\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {(a B+3 A b) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {(a B+3 A b) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i (a B+3 A b) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {B \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i (a B+3 A b) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i (a B+3 A b) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i (a B+3 A b) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}}\right )\) |
Input:
Int[Sqrt[a + b*Sinh[x]]*(A + B*Sinh[x]),x]
Output:
(2*B*Cosh[x]*Sqrt[a + b*Sinh[x]])/3 + (((2*I)*(3*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*Sinh[x]]))/3
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[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*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] && GtQ[m, 0] && IntegerQ[2*m]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (143 ) = 286\).
Time = 1.66 (sec) , antiderivative size = 731, normalized size of antiderivative = 4.46
| method | result | size |
| parts | \(-\frac {2 A \left (i b -a \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 (i \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b -i \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b +\operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) 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 \right )}{b \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}+b^{3} \sinh \left (x \right )^{3}+a \,b^{2} \sinh \left (x \right )^{2}+\sinh \left (x \right ) b^{3}+a \,b^{2}\right )}{3 b^{2} \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(731\) |
| default | \(\frac {\frac {2 i B \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}{3}+\frac {2 i B \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}}{3}+2 A \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 +2 A \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}-2 A \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} b -2 A \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^{3}-\frac {2 B \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}}{3}-\frac {2 B \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}}{3}+\frac {2 B \,b^{3} \sinh \left (x \right )^{3}}{3}+\frac {2 B \sinh \left (x \right )^{2} a \,b^{2}}{3}+\frac {2 B \,b^{3} \sinh \left (x \right )}{3}+\frac {2 B a \,b^{2}}{3}}{b^{2} \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(897\) |
Input:
int((a+b*sinh(x))^(1/2)*(A+B*sinh(x)),x,method=_RETURNVERBOSE)
Output:
-2*A*(I*b-a)*(-(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)/b*(I*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2 ),(-(I*b-a)/(I*b+a))^(1/2))*b-I*EllipticE((-(a+b*sinh(x))/(I*b-a))^(1/2),( -(I*b-a)/(I*b+a))^(1/2))*b+EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b -a)/(I*b+a))^(1/2))*a-EllipticE((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/( I*b+a))^(1/2))*a)/cosh(x)/(a+b*sinh(x))^(1/2)+2/3*B*(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)*El lipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^2*b+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)*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a) )^(1/2))*b^3-(-(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^3-(-(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*b^2+b^3*sinh(x)^3+a*b^2*sinh(x)^2+s inh(x)*b^3+a*b^2)/b^2/cosh(x)/(a+b*sinh(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (140) = 280\).
Time = 0.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.94 \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=-\frac {4 \, \sqrt {\frac {1}{2}} {\left ({\left (2 \, B a^{2} - 3 \, A a b + 3 \, B b^{2}\right )} \cosh \left (x\right ) + {\left (2 \, B a^{2} - 3 \, A a b + 3 \, B b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 12 \, \sqrt {\frac {1}{2}} {\left ({\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + {\left (B a b + 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (B b^{2} \cosh \left (x\right )^{2} + B b^{2} \sinh \left (x\right )^{2} + B b^{2} - 2 \, {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (B b^{2} \cosh \left (x\right ) - B a b - 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{9 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )}} \] Input:
integrate((a+b*sinh(x))^(1/2)*(A+B*sinh(x)),x, algorithm="fricas")
Output:
-1/9*(4*sqrt(1/2)*((2*B*a^2 - 3*A*a*b + 3*B*b^2)*cosh(x) + (2*B*a^2 - 3*A* a*b + 3*B*b^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) + 12*sqrt(1/2)*((B*a*b + 3*A*b^2)*cosh(x) + (B*a*b + 3*A*b^2)*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*b^2*cosh(x)^2 + B*b^2*sin h(x)^2 + B*b^2 - 2*(B*a*b + 3*A*b^2)*cosh(x) + 2*(B*b^2*cosh(x) - B*a*b - 3*A*b^2)*sinh(x))*sqrt(b*sinh(x) + a))/(b^2*cosh(x) + b^2*sinh(x))
\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int \left (A + B \sinh {\left (x \right )}\right ) \sqrt {a + b \sinh {\left (x \right )}}\, dx \] Input:
integrate((a+b*sinh(x))**(1/2)*(A+B*sinh(x)),x)
Output:
Integral((A + B*sinh(x))*sqrt(a + b*sinh(x)), x)
\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} \sqrt {b \sinh \left (x\right ) + a} \,d x } \] Input:
integrate((a+b*sinh(x))^(1/2)*(A+B*sinh(x)),x, algorithm="maxima")
Output:
integrate((B*sinh(x) + A)*sqrt(b*sinh(x) + a), x)
\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} \sqrt {b \sinh \left (x\right ) + a} \,d x } \] Input:
integrate((a+b*sinh(x))^(1/2)*(A+B*sinh(x)),x, algorithm="giac")
Output:
integrate((B*sinh(x) + A)*sqrt(b*sinh(x) + a), x)
Timed out. \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,\sqrt {a+b\,\mathrm {sinh}\left (x\right )} \,d x \] Input:
int((A + B*sinh(x))*(a + b*sinh(x))^(1/2),x)
Output:
int((A + B*sinh(x))*(a + b*sinh(x))^(1/2), x)
\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\left (\int \sqrt {\sinh \left (x \right ) b +a}d x \right ) a +\left (\int \sqrt {\sinh \left (x \right ) b +a}\, \sinh \left (x \right )d x \right ) b \] Input:
int((a+b*sinh(x))^(1/2)*(A+B*sinh(x)),x)
Output:
int(sqrt(sinh(x)*b + a),x)*a + int(sqrt(sinh(x)*b + a)*sinh(x),x)*b