\(\int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx\) [128]

Optimal result

Integrand size = 17, antiderivative size = 164 \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2 i (3 A b+a B) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) B \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \sqrt {a+b \sinh (x)}} \]

[Out]

2/3*B*cosh(x)*(a+b*sinh(x))^(1/2)+2/3*I*(3*A*b+B*a)*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*Elliptic
E(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*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(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)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=-\frac {2 i B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{3 b \sqrt {a+b \sinh (x)}}+\frac {2 i (a B+3 A b) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)} \]

[In]

Int[Sqrt[a + b*Sinh[x]]*(A + B*Sinh[x]),x]

[Out]

(2*B*Cosh[x]*Sqrt[a + b*Sinh[x]])/3 + (((2*I)/3)*(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)/3)*(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]])

Rule 2732

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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 2832

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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{3} \int \frac {\frac {1}{2} (3 a A-b B)+\frac {1}{2} (3 A b+a B) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx \\ & = \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}-\frac {\left (\left (a^2+b^2\right ) B\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx}{3 b}+\frac {(3 A b+a B) \int \sqrt {a+b \sinh (x)} \, dx}{3 b} \\ & = \frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {\left ((3 A b+a B) \sqrt {a+b \sinh (x)}\right ) \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}} \, dx}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (\left (a^2+b^2\right ) B \sqrt {\frac {a+b \sinh (x)}{a-i b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}} \, dx}{3 b \sqrt {a+b \sinh (x)}} \\ & = \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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\frac {2 b B \cosh (x) (a+b \sinh (x))+2 (i a+b) (3 A b+a B) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}-2 i \left (a^2+b^2\right ) B \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \sqrt {a+b \sinh (x)}} \]

[In]

Integrate[Sqrt[a + b*Sinh[x]]*(A + B*Sinh[x]),x]

[Out]

(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)]*S
qrt[(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]])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (192 ) = 384\).

Time = 3.80 (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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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}+b^{3} \sinh \left (x \right )+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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 {\left (i+\sinh \left (x \right )\right ) b}{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 a \,b^{2} \sinh \left (x \right )^{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 )}}\)	\(897\)

[In]

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

[Out]

-2*A*(I*b-a)*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a))^(1/2)/b*(I*E
llipticF((-(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)*((I+sinh(x))*b/(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*b+I*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh
(x))*b/(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)*((I+sinh(x))*b/(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)*((I+sinh(x))*b
/(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*s
inh(x)^2+b^3*sinh(x)+a*b^2)/b^2/cosh(x)/(a+b*sinh(x))^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.98 \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (2 \, B a^{2} - 3 \, A a b + 3 \, B b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (2 \, B a^{2} - 3 \, A a b + 3 \, B b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 6 \, {\left (\sqrt {2} {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (B a b + 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (B b^{2} \cosh \left (x\right )^{2} + B b^{2} \sinh \left (x\right )^{2} + B b^{2} - 2 \, {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (B b^{2} \cosh \left (x\right ) - B a b - 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{9 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

-1/9*(2*(sqrt(2)*(2*B*a^2 - 3*A*a*b + 3*B*b^2)*cosh(x) + sqrt(2)*(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) + 6*(sqrt(2)*(B*a*b + 3*A*b^2)*cosh(x) + sqrt(2)*(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*sinh(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))

Sympy [F]

\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int \left (A + B \sinh {\left (x \right )}\right ) \sqrt {a + b \sinh {\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral((A + B*sinh(x))*sqrt(a + b*sinh(x)), x)

Maxima [F]

\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} \sqrt {b \sinh \left (x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((B*sinh(x) + A)*sqrt(b*sinh(x) + a), x)

Giac [F]

\[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} \sqrt {b \sinh \left (x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((B*sinh(x) + A)*sqrt(b*sinh(x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,\sqrt {a+b\,\mathrm {sinh}\left (x\right )} \,d x \]

[In]

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

[Out]

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