Integrand size = 17, antiderivative size = 207 \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {2}{15} (5 A b+3 a B) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2 i \left (20 a A b+3 a^2 B-9 b^2 B\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{15 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) (5 A b+3 a B) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{15 b \sqrt {a+b \sinh (x)}} \]
2/5*B*cosh(x)*(a+b*sinh(x))^(3/2)+2/15*(5*A*b+3*B*a)*cosh(x)*(a+b*sinh(x)) ^(1/2)+2/15*I*(20*A*a*b+3*B*a^2-9*B*b^2)*(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+b*sinh(x))/(a-I*b))^(1/2)-2/15*I*(a^2+b^2)*(5*A* b+3*B*a)*(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)
Time = 0.44 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.95 \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {2 \left (\frac {i \left (b \left (15 a^2 A-5 A b^2-12 a b B\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )+\left (20 a A b+3 a^2 B-9 b^2 B\right ) \left ((a-i b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )\right )\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b}+\cosh (x) (a+b \sinh (x)) (5 A b+6 a B+3 b B \sinh (x))\right )}{15 \sqrt {a+b \sinh (x)}} \]
(2*((I*(b*(15*a^2*A - 5*A*b^2 - 12*a*b*B)*EllipticF[(Pi - (2*I)*x)/4, ((-2 *I)*b)/(a - I*b)] + (20*a*A*b + 3*a^2*B - 9*b^2*B)*((a - I*b)*EllipticE[(P i - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)] - a*EllipticF[(Pi - (2*I)*x)/4, ((-2 *I)*b)/(a - I*b)]))*Sqrt[(a + b*Sinh[x])/(a - I*b)])/b + Cosh[x]*(a + b*Si nh[x])*(5*A*b + 6*a*B + 3*b*B*Sinh[x])))/(15*Sqrt[a + b*Sinh[x]])
Time = 1.03 (sec) , antiderivative size = 213, 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, 3232, 27, 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.
\(\displaystyle \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a-i b \sin (i x))^{3/2} (A-i B \sin (i x))dx\) |
\(\Big \downarrow \) 3232 |
\(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \sinh (x)} (5 a A-3 b B+(5 A b+3 a B) \sinh (x))dx+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \sqrt {a+b \sinh (x)} (5 a A-3 b B+(5 A b+3 a B) \sinh (x))dx+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \int \sqrt {a-i b \sin (i x)} (5 a A-3 b B-i (5 A b+3 a B) \sin (i x))dx\) |
\(\Big \downarrow \) 3232 |
\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {15 A a^2-12 b B a-5 A b^2+\left (3 B a^2+20 A b a-9 b^2 B\right ) \sinh (x)}{2 \sqrt {a+b \sinh (x)}}dx+\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}\right )+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {15 A a^2-12 b B a-5 A b^2+\left (3 B a^2+20 A b a-9 b^2 B\right ) \sinh (x)}{\sqrt {a+b \sinh (x)}}dx+\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}\right )+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \int \frac {15 A a^2-12 b B a-5 A b^2-i \left (3 B a^2+20 A b a-9 b^2 B\right ) \sin (i x)}{\sqrt {a-i b \sin (i x)}}dx\right )\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b-9 b^2 B\right ) \int \sqrt {a+b \sinh (x)}dx}{b}-\frac {\left (a^2+b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sinh (x)}}dx}{b}\right )+\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}\right )+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b-9 b^2 B\right ) \int \sqrt {a-i b \sin (i x)}dx}{b}-\frac {\left (a^2+b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\right )\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b-9 b^2 B\right ) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b-9 b^2 B\right ) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\right )\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (3 a^2 B+20 a A b-9 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}\right )\right )\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (3 a^2 B+20 a A b-9 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (3 a^2 B+20 a A b-9 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}\right )\right )\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (3 a^2 B+20 a A b-9 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}}\right )\right )\) |
(2*B*Cosh[x]*(a + b*Sinh[x])^(3/2))/5 + ((2*(5*A*b + 3*a*B)*Cosh[x]*Sqrt[a + b*Sinh[x]])/3 + (((2*I)*(20*a*A*b + 3*a^2*B - 9*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)*(5*A*b + 3*a*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)/5
3.2.27.3.1 Defintions of rubi rules used
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. 1036 vs. \(2 (231 ) = 462\).
Time = 4.32 (sec) , antiderivative size = 1037, normalized size of antiderivative = 5.01
method | result | size |
default | \(\text {Expression too large to display}\) | \(1037\) |
parts | \(\text {Expression too large to display}\) | \(1489\) |
(cosh(x)^2*(a+b*sinh(x)))^(1/2)*(2*a^2*A*(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))+B*b^2*(2/5/b*sinh(x)*(cosh(x)^2*(a+b*sinh(x)))^(1/2)-8/15* a/b^2*(cosh(x)^2*(a+b*sinh(x)))^(1/2)-4/15*a/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)/(c osh(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*(-3/5+8/15*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))))+(A*b^2+2*B*a*b)*(2/3/b*(cosh(x)^2*(a+b*sinh(x)))^ (1/2)-2/3*(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)*Ellipti cF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))-4/3*a/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)*((-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*sin h(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))))+2*(2*A*a*b+B*a^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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 635, normalized size of antiderivative = 3.07 \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=-\frac {4 \, {\left (\sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b + 18 \, B a b^{2} + 15 \, A b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b + 18 \, B a b^{2} + 15 \, A b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b + 18 \, B a b^{2} + 15 \, A b^{3}\right )} \sinh \left (x\right )^{2}\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 \, {\left (\sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \sinh \left (x\right )^{2}\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 (3 \, B b^{3} \cosh \left (x\right )^{4} + 3 \, B b^{3} \sinh \left (x\right )^{4} - 3 \, B b^{3} + 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right ) + 6 \, B a b^{2} + 5 \, A b^{3}\right )} \sinh \left (x\right )^{3} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, B b^{3} \cosh \left (x\right )^{2} - 6 \, B a^{2} b - 40 \, A a b^{2} + 18 \, B b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right )^{3} + 6 \, B a b^{2} + 5 \, A b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{90 \, {\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}\right )}} \]
-1/90*(4*(sqrt(2)*(6*B*a^3 - 5*A*a^2*b + 18*B*a*b^2 + 15*A*b^3)*cosh(x)^2 + 2*sqrt(2)*(6*B*a^3 - 5*A*a^2*b + 18*B*a*b^2 + 15*A*b^3)*cosh(x)*sinh(x) + sqrt(2)*(6*B*a^3 - 5*A*a^2*b + 18*B*a*b^2 + 15*A*b^3)*sinh(x)^2)*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(2)*(3*B*a^2*b + 20*A* a*b^2 - 9*B*b^3)*cosh(x)^2 + 2*sqrt(2)*(3*B*a^2*b + 20*A*a*b^2 - 9*B*b^3)* cosh(x)*sinh(x) + sqrt(2)*(3*B*a^2*b + 20*A*a*b^2 - 9*B*b^3)*sinh(x)^2)*sq rt(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*(3*B*b^3*cosh(x)^4 + 3*B*b ^3*sinh(x)^4 - 3*B*b^3 + 2*(6*B*a*b^2 + 5*A*b^3)*cosh(x)^3 + 2*(6*B*b^3*co sh(x) + 6*B*a*b^2 + 5*A*b^3)*sinh(x)^3 - 4*(3*B*a^2*b + 20*A*a*b^2 - 9*B*b ^3)*cosh(x)^2 + 2*(9*B*b^3*cosh(x)^2 - 6*B*a^2*b - 40*A*a*b^2 + 18*B*b^3 + 3*(6*B*a*b^2 + 5*A*b^3)*cosh(x))*sinh(x)^2 + 2*(6*B*a*b^2 + 5*A*b^3)*cosh (x) + 2*(6*B*b^3*cosh(x)^3 + 6*B*a*b^2 + 5*A*b^3 + 3*(6*B*a*b^2 + 5*A*b^3) *cosh(x)^2 - 4*(3*B*a^2*b + 20*A*a*b^2 - 9*B*b^3)*cosh(x))*sinh(x))*sqrt(b *sinh(x) + a))/(b^2*cosh(x)^2 + 2*b^2*cosh(x)*sinh(x) + b^2*sinh(x)^2)
\[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (A + B \sinh {\left (x \right )}\right ) \left (a + b \sinh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,{\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \]