\(\int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx\) [139]

Optimal result

Integrand size = 17, antiderivative size = 251 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=-\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}+\frac {2 i \left (4 a A b-a^2 B+3 b^2 B\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i (A b-a B) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \]

[Out]

-2/3*(A*b-B*a)*cosh(x)/(a^2+b^2)/(a+b*sinh(x))^(3/2)-2/3*(4*A*a*b-B*a^2+3*B*b^2)*cosh(x)/(a^2+b^2)^2/(a+b*sinh
(x))^(1/2)+2/3*I*(4*A*a*b-B*a^2+3*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*P
i+1/2*I*x),2^(1/2)*(b/(I*a+b))^(1/2))*(a+b*sinh(x))^(1/2)/b/(a^2+b^2)^2/((a+b*sinh(x))/(a-I*b))^(1/2)-2/3*I*(A
*b-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^2+b^2)/(a+b*sinh(x))^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=-\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 i (A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}} \]

[In]

Int[(A + B*Sinh[x])/(a + b*Sinh[x])^(5/2),x]

[Out]

(-2*(A*b - a*B)*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^(3/2)) - (2*(4*a*A*b - a^2*B + 3*b^2*B)*Cosh[x])/(3*(a
^2 + b^2)^2*Sqrt[a + b*Sinh[x]]) + (((2*I)/3)*(4*a*A*b - a^2*B + 3*b^2*B)*EllipticE[Pi/4 - (I/2)*x, (2*b)/(I*a
+ b)]*Sqrt[a + b*Sinh[x]])/(b*(a^2 + b^2)^2*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (((2*I)/3)*(A*b - a*B)*Ellipti
cF[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*(a^2 + b^2)*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)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} (a A+b B)+\frac {1}{2} (A b-a B) \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx}{3 \left (a^2+b^2\right )} \\ & = -\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^2 A-A b^2+4 a b B\right )+\frac {1}{4} \left (4 a A b-a^2 B+3 b^2 B\right ) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx}{3 \left (a^2+b^2\right )^2} \\ & = -\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}-\frac {(A b-a B) \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx}{3 b \left (a^2+b^2\right )}+\frac {\left (4 a A b-a^2 B+3 b^2 B\right ) \int \sqrt {a+b \sinh (x)} \, dx}{3 b \left (a^2+b^2\right )^2} \\ & = -\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}+\frac {\left (\left (4 a A b-a^2 B+3 b^2 B\right ) \sqrt {a+b \sinh (x)}\right ) \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}} \, dx}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left ((A b-a 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 \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \\ & = -\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}+\frac {2 i \left (4 a A b-a^2 B+3 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)}}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i (A b-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}}}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\frac {2 i \left (\left (b \left (3 a^2 A-A b^2+4 a b B\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )+\left (4 a A b-a^2 B+3 b^2 B\right ) \left ((a-i b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )\right )\right ) (a+b \sinh (x)) \sqrt {\frac {a+b \sinh (x)}{a-i b}}+i b \cosh (x) \left (-\left (\left (a^2+b^2\right ) (-A b+a B)\right )-\left (-4 a A b+a^2 B-3 b^2 B\right ) (a+b \sinh (x))\right )\right )}{3 b \left (a^2+b^2\right )^2 (a+b \sinh (x))^{3/2}} \]

[In]

Integrate[(A + B*Sinh[x])/(a + b*Sinh[x])^(5/2),x]

[Out]

(((2*I)/3)*((b*(3*a^2*A - A*b^2 + 4*a*b*B)*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)] + (4*a*A*b - a^2*
B + 3*b^2*B)*((a - I*b)*EllipticE[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)] - a*EllipticF[(Pi - (2*I)*x)/4, ((-2
*I)*b)/(a - I*b)]))*(a + b*Sinh[x])*Sqrt[(a + b*Sinh[x])/(a - I*b)] + I*b*Cosh[x]*(-((a^2 + b^2)*(-(A*b) + a*B
)) - (-4*a*A*b + a^2*B - 3*b^2*B)*(a + b*Sinh[x]))))/(b*(a^2 + b^2)^2*(a + b*Sinh[x])^(3/2))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (275 ) = 550\).

Time = 4.20 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.21

method	result	size
default	\(\frac {\sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}\, \left (\frac {B \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}+\frac {\left (A b -a B \right ) \left (-\frac {2 \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}{3 b \left (a^{2}+b^{2}\right ) \left (\sinh \left (x \right )+\frac {a}{b}\right )^{2}}-\frac {8 b \cosh \left (x \right )^{2} a}{3 \left (a^{2}+b^{2}\right )^{2} \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {2 \left (3 a^{2}-b^{2}\right ) \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 (3 a^{4}+6 a^{2} b^{2}+3 b^{4}\right ) \sqrt {\cosh \left (x \right )^{2} \left (a +b \sinh \left (x \right )\right )}}+\frac {8 a 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 )}{3 \left (a^{2}+b^{2}\right )^{2} \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 )}}\)	\(806\)
parts	\(\text {Expression too large to display}\)	\(1239\)

[In]

int((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

(cosh(x)^2*(a+b*sinh(x)))^(1/2)*(B/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))))+(A*b-B*a)/b*(-2/3/b/(a^2+b^2)*(cosh(x)^2*(a+b*sinh(x)))^(1/2)/(
sinh(x)+a/b)^2-8/3*b*cosh(x)^2/(a^2+b^2)^2*a/(cosh(x)^2*(a+b*sinh(x)))^(1/2)+2*(3*a^2-b^2)/(3*a^4+6*a^2*b^2+3*
b^4)*(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))+8/3*a*b/(a^2+b^2)^
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 = 0.16 (sec) , antiderivative size = 2167, normalized size of antiderivative = 8.63 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x, algorithm="fricas")

[Out]

2/9*((sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*cosh(x)^4 + sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 + 6
*B*a*b^4 - 3*A*b^5)*sinh(x)^4 + 4*sqrt(2)*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*a*b^4)*cosh(x)^3 + 4*(sqr
t(2)*(2*B*a^3*b^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*cosh(x) + sqrt(2)*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 -
3*A*a*b^4))*sinh(x)^3 + 2*sqrt(2)*(4*B*a^5 + 2*A*a^4*b + 10*B*a^3*b^2 - 7*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cos
h(x)^2 + 2*(3*sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*cosh(x)^2 + 6*sqrt(2)*(2*B*a^4*b + A*a^3
*b^2 + 6*B*a^2*b^3 - 3*A*a*b^4)*cosh(x) + sqrt(2)*(4*B*a^5 + 2*A*a^4*b + 10*B*a^3*b^2 - 7*A*a^2*b^3 - 6*B*a*b^
4 + 3*A*b^5))*sinh(x)^2 - 4*sqrt(2)*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*a*b^4)*cosh(x) + 4*(sqrt(2)*(2*
B*a^3*b^2 + A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*cosh(x)^3 + 3*sqrt(2)*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*
a*b^4)*cosh(x)^2 + sqrt(2)*(4*B*a^5 + 2*A*a^4*b + 10*B*a^3*b^2 - 7*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x) -
sqrt(2)*(2*B*a^4*b + A*a^3*b^2 + 6*B*a^2*b^3 - 3*A*a*b^4))*sinh(x) + sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 + 6*B*a*
b^4 - 3*A*b^5))*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*cos
h(x) + 3*b*sinh(x) + 2*a)/b) + 3*(sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x)^4 + sqrt(2)*(B*a^2*b^3 - 4
*A*a*b^4 - 3*B*b^5)*sinh(x)^4 + 4*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4)*cosh(x)^3 + 4*(sqrt(2)*(B*a^2*
b^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x) + sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4))*sinh(x)^3 + 2*sqrt(2)*(2*B
*a^4*b - 8*A*a^3*b^2 - 7*B*a^2*b^3 + 4*A*a*b^4 + 3*B*b^5)*cosh(x)^2 + 2*(3*sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 - 3*
B*b^5)*cosh(x)^2 + 6*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4)*cosh(x) + sqrt(2)*(2*B*a^4*b - 8*A*a^3*b^2
- 7*B*a^2*b^3 + 4*A*a*b^4 + 3*B*b^5))*sinh(x)^2 - 4*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4)*cosh(x) + 4*
(sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x)^3 + 3*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4)*cosh(x)
^2 + sqrt(2)*(2*B*a^4*b - 8*A*a^3*b^2 - 7*B*a^2*b^3 + 4*A*a*b^4 + 3*B*b^5)*cosh(x) - sqrt(2)*(B*a^3*b^2 - 4*A*
a^2*b^3 - 3*B*a*b^4))*sinh(x) + sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5))*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)) + 6*((B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x)^4 + (B*a^2*b^3
- 4*A*a*b^4 - 3*B*b^5)*sinh(x)^4 + (4*B*a^3*b^2 - 13*A*a^2*b^3 - 8*B*a*b^4 - A*b^5)*cosh(x)^3 + (4*B*a^3*b^2
- 13*A*a^2*b^3 - 8*B*a*b^4 - A*b^5 + 4*(B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x))*sinh(x)^3 + (2*B*a^4*b - 8*A
*a^3*b^2 - 7*B*a^2*b^3 + 4*A*a*b^4 + 3*B*b^5)*cosh(x)^2 + (2*B*a^4*b - 8*A*a^3*b^2 - 7*B*a^2*b^3 + 4*A*a*b^4 +
3*B*b^5 + 6*(B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x)^2 + 3*(4*B*a^3*b^2 - 13*A*a^2*b^3 - 8*B*a*b^4 - A*b^5)*
cosh(x))*sinh(x)^2 + (3*A*a^2*b^3 + 4*B*a*b^4 - A*b^5)*cosh(x) + (3*A*a^2*b^3 + 4*B*a*b^4 - A*b^5 + 4*(B*a^2*b
^3 - 4*A*a*b^4 - 3*B*b^5)*cosh(x)^3 + 3*(4*B*a^3*b^2 - 13*A*a^2*b^3 - 8*B*a*b^4 - A*b^5)*cosh(x)^2 + 2*(2*B*a^
4*b - 8*A*a^3*b^2 - 7*B*a^2*b^3 + 4*A*a*b^4 + 3*B*b^5)*cosh(x))*sinh(x))*sqrt(b*sinh(x) + a))/(a^4*b^4 + 2*a^2
*b^6 + b^8 + (a^4*b^4 + 2*a^2*b^6 + b^8)*cosh(x)^4 + (a^4*b^4 + 2*a^2*b^6 + b^8)*sinh(x)^4 + 4*(a^5*b^3 + 2*a^
3*b^5 + a*b^7)*cosh(x)^3 + 4*(a^5*b^3 + 2*a^3*b^5 + a*b^7 + (a^4*b^4 + 2*a^2*b^6 + b^8)*cosh(x))*sinh(x)^3 + 2
*(2*a^6*b^2 + 3*a^4*b^4 - b^8)*cosh(x)^2 + 2*(2*a^6*b^2 + 3*a^4*b^4 - b^8 + 3*(a^4*b^4 + 2*a^2*b^6 + b^8)*cosh
(x)^2 + 6*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*cosh(x))*sinh(x)^2 - 4*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*cosh(x) - 4*(a^5*
b^3 + 2*a^3*b^5 + a*b^7 - (a^4*b^4 + 2*a^2*b^6 + b^8)*cosh(x)^3 - 3*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*cosh(x)^2 -
(2*a^6*b^2 + 3*a^4*b^4 - b^8)*cosh(x))*sinh(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{{\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((B*sinh(x) + A)/(b*sinh(x) + a)^(5/2), x)

Giac [F]

\[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{{\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x, algorithm="giac")

[Out]

integrate((B*sinh(x) + A)/(b*sinh(x) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx=\int \frac {A+B\,\mathrm {sinh}\left (x\right )}{{\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{5/2}} \,d x \]

[In]

int((A + B*sinh(x))/(a + b*sinh(x))^(5/2),x)

[Out]

int((A + B*sinh(x))/(a + b*sinh(x))^(5/2), x)