\(\int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx\) [138]

Optimal result

Integrand size = 17, antiderivative size = 176 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=-\frac {2 (A b-a B) \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{b \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2 i B \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \sqrt {a+b \sinh (x)}} \]

[Out]

-2*(A*b-B*a)*cosh(x)/(a^2+b^2)/(a+b*sinh(x))^(1/2)+2*I*(A*b-B*a)*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*
I*x)*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*(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 = 0.15 (sec) , antiderivative size = 176, 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 = {2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=-\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2 i B \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}} \]

[In]

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

[Out]

(-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*(a^2 + b^2)*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + ((2*I)*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 2833

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] + Dist[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]

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

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\frac {2 b (-A b+a B) \cosh (x)+\frac {2 i (A b-a B) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) (a+b \sinh (x))}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}+2 i \left (a^2+b^2\right ) B \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \]

[In]

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

[Out]

(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)]*
Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*(a^2 + b^2)*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. 516 vs. \(2 (210 ) = 420\).

Time = 3.63 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.94

method	result	size
default	\(\frac {\sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}\, \left (\frac {2 B \left (\frac {a}{b}-i\right ) \sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{b \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {\left (A b -a B \right ) \left (-\frac {2 b \cosh \left (x \right )^{2}}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 a \left (\frac {a}{b}-i\right ) \sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 b \left (\frac {a}{b}-i\right ) \sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \left (\left (-\frac {a}{b}-i\right ) \operatorname {EllipticE}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )+i \operatorname {EllipticF}\left (\sqrt {\frac {-a -b \sinh \left (x \right )}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}\right )}{b}\right )}{\cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\)	\(517\)
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 {\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}+\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^{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 {\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}-\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^{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 {\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}-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 )}}\)	\(922\)

[In]

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

[Out]

(cosh(x)^2*(a+b*sinh(x)))^(1/2)*(2*B/b*(a/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)/(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)*(a/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)/(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)*(a/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)/(cosh(x)^2*(a+b*sinh(x)))^(1/2
)*((-a/b-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 [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.60 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=-\frac {2 \, {\left ({\left (\sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} + 3 \, B b^{3}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} + 3 \, B b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (2 \, B a^{3} + A a^{2} b + 3 \, B a b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (\sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} + 3 \, B b^{3}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (2 \, B a^{3} + A a^{2} b + 3 \, B a b^{2}\right )}\right )} \sinh \left (x\right ) - \sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} + 3 \, B b^{3}\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 \, {\left (\sqrt {2} {\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (B a b^{2} - A b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (B a^{2} b - A a b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (\sqrt {2} {\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (B a^{2} b - A a b^{2}\right )}\right )} \sinh \left (x\right ) - \sqrt {2} {\left (B a b^{2} - A b^{3}\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 ) + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right )^{2} + {\left (B a b^{2} - A b^{3}\right )} \sinh \left (x\right )^{2} + {\left (B a^{2} b - A a b^{2}\right )} \cosh \left (x\right ) + {\left (B a^{2} b - A a b^{2} + 2 \, {\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}\right )}}{3 \, {\left (a^{2} b^{3} + b^{5} - {\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{2} - {\left (a^{2} b^{3} + b^{5}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{3} b^{2} + a b^{4} + {\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

-2/3*((sqrt(2)*(2*B*a^2*b + A*a*b^2 + 3*B*b^3)*cosh(x)^2 + sqrt(2)*(2*B*a^2*b + A*a*b^2 + 3*B*b^3)*sinh(x)^2 +
2*sqrt(2)*(2*B*a^3 + A*a^2*b + 3*B*a*b^2)*cosh(x) + 2*(sqrt(2)*(2*B*a^2*b + A*a*b^2 + 3*B*b^3)*cosh(x) + sqrt
(2)*(2*B*a^3 + A*a^2*b + 3*B*a*b^2))*sinh(x) - sqrt(2)*(2*B*a^2*b + A*a*b^2 + 3*B*b^3))*sqrt(b)*weierstrassPIn
verse(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
(2)*(B*a*b^2 - A*b^3)*cosh(x)^2 + sqrt(2)*(B*a*b^2 - A*b^3)*sinh(x)^2 + 2*sqrt(2)*(B*a^2*b - A*a*b^2)*cosh(x)
+ 2*(sqrt(2)*(B*a*b^2 - A*b^3)*cosh(x) + sqrt(2)*(B*a^2*b - A*a*b^2))*sinh(x) - sqrt(2)*(B*a*b^2 - A*b^3))*sqr
t(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)) + 6*((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)*cos
h(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)*cosh(x) - 2*(a^3*b^2 + a*b^4 + (a^2*b^3 + b^5)*cosh(x))*sinh(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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