3.9.61 \(\int (a+b \cosh (c+d x) \sinh (c+d x))^{5/2} \, dx\) [861]

Optimal. Leaf size=301 \[ \frac {2 \sqrt {2} a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}-\frac {i \left (92 a^2-9 b^2\right ) E\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}{60 \sqrt {2} d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {2 i \sqrt {2} a \left (4 a^2+b^2\right ) F\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{15 d \sqrt {2 a+b \sinh (2 c+2 d x)}} \]

[Out]

1/40*b*cosh(2*d*x+2*c)*(2*a+b*sinh(2*d*x+2*c))^(3/2)/d*2^(1/2)+2/15*a*b*cosh(2*d*x+2*c)*2^(1/2)*(2*a+b*sinh(2*
d*x+2*c))^(1/2)/d+1/120*I*(92*a^2-9*b^2)*(sin(I*c+1/4*Pi+I*d*x)^2)^(1/2)/sin(I*c+1/4*Pi+I*d*x)*EllipticE(cos(I
*c+1/4*Pi+I*d*x),2^(1/2)*(b/(2*I*a+b))^(1/2))*(2*a+b*sinh(2*d*x+2*c))^(1/2)/d*2^(1/2)/((2*a+b*sinh(2*d*x+2*c))
/(2*a-I*b))^(1/2)-2/15*I*a*(4*a^2+b^2)*(sin(I*c+1/4*Pi+I*d*x)^2)^(1/2)/sin(I*c+1/4*Pi+I*d*x)*EllipticF(cos(I*c
+1/4*Pi+I*d*x),2^(1/2)*(b/(2*I*a+b))^(1/2))*2^(1/2)*((2*a+b*sinh(2*d*x+2*c))/(2*a-I*b))^(1/2)/d/(2*a+b*sinh(2*
d*x+2*c))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.26, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2745, 2735, 2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {2 i \sqrt {2} a \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{15 d \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {i \left (92 a^2-9 b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{60 \sqrt {2} d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {2 \sqrt {2} a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{15 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(5/2),x]

[Out]

(2*Sqrt[2]*a*b*Cosh[2*c + 2*d*x]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(15*d) + (b*Cosh[2*c + 2*d*x]*(2*a + b*Sinh[
2*c + 2*d*x])^(3/2))/(20*Sqrt[2]*d) - ((I/60)*(92*a^2 - 9*b^2)*EllipticE[((2*I)*c - Pi/2 + (2*I)*d*x)/2, (2*b)
/((2*I)*a + b)]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(Sqrt[2]*d*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)]) + (
((2*I)/15)*Sqrt[2]*a*(4*a^2 + b^2)*EllipticF[((2*I)*c - Pi/2 + (2*I)*d*x)/2, (2*b)/((2*I)*a + b)]*Sqrt[(2*a +
b*Sinh[2*c + 2*d*x])/(2*a - I*b)])/(d*Sqrt[2*a + b*Sinh[2*c + 2*d*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 2735

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
- 1)/(d*n)), x] + Dist[1/n, Int[(a + b*Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c +
d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]

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 2745

Int[((a_) + cos[(c_.) + (d_.)*(x_)]*(b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[(a + b*(Sin[2*c + 2*
d*x]/2))^n, x] /; FreeQ[{a, b, c, d, n}, x]

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*} \int (a+b \cosh (c+d x) \sinh (c+d x))^{5/2} \, dx &=\int \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^{5/2} \, dx\\ &=\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {2}{5} \int \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)} \left (\frac {1}{8} \left (20 a^2-3 b^2\right )+2 a b \sinh (2 c+2 d x)\right ) \, dx\\ &=\frac {2 \sqrt {2} a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {4}{15} \int \frac {\frac {1}{16} a \left (60 a^2-17 b^2\right )+\frac {1}{32} b \left (92 a^2-9 b^2\right ) \sinh (2 c+2 d x)}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac {2 \sqrt {2} a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {1}{60} \left (92 a^2-9 b^2\right ) \int \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)} \, dx-\frac {1}{15} \left (2 a \left (4 a^2+b^2\right )\right ) \int \frac {1}{\sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac {2 \sqrt {2} a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\left (\left (92 a^2-9 b^2\right ) \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}\right ) \int \sqrt {\frac {a}{a-\frac {i b}{2}}+\frac {b \sinh (2 c+2 d x)}{2 \left (a-\frac {i b}{2}\right )}} \, dx}{60 \sqrt {\frac {a+\frac {1}{2} b \sinh (2 c+2 d x)}{a-\frac {i b}{2}}}}-\frac {\left (2 a \left (4 a^2+b^2\right ) \sqrt {\frac {a+\frac {1}{2} b \sinh (2 c+2 d x)}{a-\frac {i b}{2}}}\right ) \int \frac {1}{\sqrt {\frac {a}{a-\frac {i b}{2}}+\frac {b \sinh (2 c+2 d x)}{2 \left (a-\frac {i b}{2}\right )}}} \, dx}{15 \sqrt {a+\frac {1}{2} b \sinh (2 c+2 d x)}}\\ &=\frac {2 \sqrt {2} a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}-\frac {i \left (92 a^2-9 b^2\right ) E\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}{60 \sqrt {2} d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {2 i \sqrt {2} a \left (4 a^2+b^2\right ) F\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{15 d \sqrt {2 a+b \sinh (2 c+2 d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.96, size = 239, normalized size = 0.79 \begin {gather*} \frac {2 \left (184 i a^3+92 a^2 b-18 i a b^2-9 b^3\right ) E\left (\frac {1}{4} (-4 i c+\pi -4 i d x)|-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}-32 i a \left (4 a^2+b^2\right ) F\left (\frac {1}{4} (-4 i c+\pi -4 i d x)|-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}+b \left (88 a^2 \cosh (2 (c+d x))+b (28 a+3 b \sinh (2 (c+d x))) \sinh (4 (c+d x))\right )}{120 d \sqrt {4 a+2 b \sinh (2 (c+d x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(5/2),x]

[Out]

(2*((184*I)*a^3 + 92*a^2*b - (18*I)*a*b^2 - 9*b^3)*EllipticE[((-4*I)*c + Pi - (4*I)*d*x)/4, ((-2*I)*b)/(2*a -
I*b)]*Sqrt[(2*a + b*Sinh[2*(c + d*x)])/(2*a - I*b)] - (32*I)*a*(4*a^2 + b^2)*EllipticF[((-4*I)*c + Pi - (4*I)*
d*x)/4, ((-2*I)*b)/(2*a - I*b)]*Sqrt[(2*a + b*Sinh[2*(c + d*x)])/(2*a - I*b)] + b*(88*a^2*Cosh[2*(c + d*x)] +
b*(28*a + 3*b*Sinh[2*(c + d*x)])*Sinh[4*(c + d*x)]))/(120*d*Sqrt[4*a + 2*b*Sinh[2*(c + d*x)]])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1259 vs. \(2 (331 ) = 662\).
time = 3.89, size = 1260, normalized size = 4.19

method	result	size
default	\(\frac {64 i \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticF \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{3} b +16 i \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticF \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a \,b^{3}+240 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticF \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{4}+24 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticF \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{2} b^{2}-9 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticF \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) b^{4}-368 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticE \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{4}-56 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticE \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{2} b^{2}+9 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {\left (\sinh \left (2 d x +2 c \right )+i\right ) b}{i b -2 a}}\, \EllipticE \left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) b^{4}+3 b^{4} \left (\sinh ^{4}\left (2 d x +2 c \right )\right )+28 a \,b^{3} \left (\sinh ^{3}\left (2 d x +2 c \right )\right )+44 a^{2} b^{2} \left (\sinh ^{2}\left (2 d x +2 c \right )\right )+3 b^{4} \left (\sinh ^{2}\left (2 d x +2 c \right )\right )+28 a \,b^{3} \sinh \left (2 d x +2 c \right )+44 a^{2} b^{2}}{60 b \cosh \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sinh \left (2 d x +2 c \right )}\, d}\)	\(1260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/60*(64*I*(-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*
c)+I)*b/(I*b-2*a))^(1/2)*EllipticF((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*a^
3*b+16*I*(-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*c)
+I)*b/(I*b-2*a))^(1/2)*EllipticF((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*a*b^
3+240*(-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*c)+I)
*b/(I*b-2*a))^(1/2)*EllipticF((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*a^4+24*
(-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*c)+I)*b/(I*
b-2*a))^(1/2)*EllipticF((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*a^2*b^2-9*(-(
2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*c)+I)*b/(I*b-2
*a))^(1/2)*EllipticF((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*b^4-368*(-(2*a+b
*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*c)+I)*b/(I*b-2*a))^
(1/2)*EllipticE((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*a^4-56*(-(2*a+b*sinh(
2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*c)+I)*b/(I*b-2*a))^(1/2)*
EllipticE((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*a^2*b^2+9*(-(2*a+b*sinh(2*d
*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((sinh(2*d*x+2*c)+I)*b/(I*b-2*a))^(1/2)*Ell
ipticE((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*b^4+3*b^4*sinh(2*d*x+2*c)^4+28
*a*b^3*sinh(2*d*x+2*c)^3+44*a^2*b^2*sinh(2*d*x+2*c)^2+3*b^4*sinh(2*d*x+2*c)^2+28*a*b^3*sinh(2*d*x+2*c)+44*a^2*
b^2)/b/cosh(2*d*x+2*c)/(4*a+2*b*sinh(2*d*x+2*c))^(1/2)/d

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(d*x + c)*sinh(d*x + c) + a)^(5/2), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a^2)*sqrt(b*cosh(d*x + c)*
sinh(d*x + c) + a), x)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cosh(c + d*x)*sinh(c + d*x))^(5/2),x)

[Out]

int((a + b*cosh(c + d*x)*sinh(c + d*x))^(5/2), x)

________________________________________________________________________________________