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

3.9.61.1 Optimal result

Integrand size = 20, antiderivative size = 301 \[ \int (a+b \cosh (c+d x) \sinh (c+d x))^{5/2} \, dx=\frac {2 \sqrt {2} a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}-\frac {i \left (92 a^2-9 b^2\right ) E\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}{60 \sqrt {2} d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {2 i \sqrt {2} a \left (4 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right ),\frac {2 b}{2 i a+b}\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{15 d \sqrt {2 a+b \sinh (2 c+2 d x)}} \]

output

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*co 
sh(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)
 

3.9.61.2 Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79 \[ \int (a+b \cosh (c+d x) \sinh (c+d x))^{5/2} \, dx=\frac {2 \left (184 i a^3+92 a^2 b-18 i a b^2-9 b^3\right ) E\left (\frac {1}{4} (-4 i c+\pi -4 i d x)|-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}-32 i a \left (4 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-4 i c+\pi -4 i d x),-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}+b \left (88 a^2 \cosh (2 (c+d x))+b (28 a+3 b \sinh (2 (c+d x))) \sinh (4 (c+d x))\right )}{120 d \sqrt {4 a+2 b \sinh (2 (c+d x))}} \]

input

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

output

(2*((184*I)*a^3 + 92*a^2*b - (18*I)*a*b^2 - 9*b^3)*EllipticE[((-4*I)*c + P 
i - (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)]])
 

3.9.61.3 Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.99, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.850, Rules used = {3042, 3145, 3042, 3135, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (a-i b \sin (i c+i d x) \cos (i c+i d x))^{5/2}dx\)

\(\Big \downarrow \) 3145

\(\displaystyle \int \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^{5/2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a-\frac {1}{2} i b \sin (2 i c+2 i d x)\right )^{5/2}dx\)

\(\Big \downarrow \) 3135

\(\displaystyle \frac {2}{5} \int \frac {\sqrt {2 a+b \sinh (2 c+2 d x)} \left (20 a^2+16 b \sinh (2 c+2 d x) a-3 b^2\right )}{8 \sqrt {2}}dx+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {2 a+b \sinh (2 c+2 d x)} \left (20 a^2+16 b \sinh (2 c+2 d x) a-3 b^2\right )dx}{20 \sqrt {2}}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\int \sqrt {2 a-i b \sin (2 i c+2 i d x)} \left (20 a^2-16 i b \sin (2 i c+2 i d x) a-3 b^2\right )dx}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {2}{3} \int \frac {2 a \left (60 a^2-17 b^2\right )+b \left (92 a^2-9 b^2\right ) \sinh (2 c+2 d x)}{2 \sqrt {2 a+b \sinh (2 c+2 d x)}}dx+\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}}{20 \sqrt {2}}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \int \frac {2 a \left (60 a^2-17 b^2\right )+b \left (92 a^2-9 b^2\right ) \sinh (2 c+2 d x)}{\sqrt {2 a+b \sinh (2 c+2 d x)}}dx+\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}}{20 \sqrt {2}}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \int \frac {2 a \left (60 a^2-17 b^2\right )-i b \left (92 a^2-9 b^2\right ) \sin (2 i c+2 i d x)}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {1}{3} \left (\left (92 a^2-9 b^2\right ) \int \sqrt {2 a+b \sinh (2 c+2 d x)}dx-16 a \left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a+b \sinh (2 c+2 d x)}}dx\right )+\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}}{20 \sqrt {2}}+\frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \left (\left (92 a^2-9 b^2\right ) \int \sqrt {2 a-i b \sin (2 i c+2 i d x)}dx-16 a \left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx\right )}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \left (\frac {\left (92 a^2-9 b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)} \int \sqrt {\frac {2 a}{2 a-i b}+\frac {b \sinh (2 c+2 d x)}{2 a-i b}}dx}{\sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}-16 a \left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx\right )}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \left (\frac {\left (92 a^2-9 b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)} \int \sqrt {\frac {2 a}{2 a-i b}-\frac {i b \sin (2 i c+2 i d x)}{2 a-i b}}dx}{\sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}-16 a \left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx\right )}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \left (-16 a \left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx-\frac {i \left (92 a^2-9 b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}\right )}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \left (-\frac {16 a \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \int \frac {1}{\sqrt {\frac {2 a}{2 a-i b}+\frac {b \sinh (2 c+2 d x)}{2 a-i b}}}dx}{\sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {i \left (92 a^2-9 b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}\right )}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \left (-\frac {16 a \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \int \frac {1}{\sqrt {\frac {2 a}{2 a-i b}-\frac {i b \sin (2 i c+2 i d x)}{2 a-i b}}}dx}{\sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {i \left (92 a^2-9 b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}\right )}{20 \sqrt {2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt {2} d}+\frac {\frac {16 a b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d}+\frac {1}{3} \left (\frac {16 i a \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right ),\frac {2 b}{2 i a+b}\right )}{d \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {i \left (92 a^2-9 b^2\right ) \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}\right )}{20 \sqrt {2}}\)

input

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

output

(b*Cosh[2*c + 2*d*x]*(2*a + b*Sinh[2*c + 2*d*x])^(3/2))/(20*Sqrt[2]*d) + ( 
(16*a*b*Cosh[2*c + 2*d*x]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(3*d) + (((-I)* 
(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]])/(d*Sqrt[(2*a + b*Sinh[2*c + 2*d*x]) 
/(2*a - I*b)]) + ((16*I)*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]]))/3)/(20*Sqrt[2])
 

3.9.61.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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 3134

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]
 

rule 3135

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] + Simp[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 3140

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]
 

rule 3142

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]
 

rule 3145

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 3231

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]
 

rule 3232

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]
 

3.9.61.4 Maple [B] (verified)

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 = 6.30 (sec) , antiderivative size = 1260, normalized size of antiderivative = 4.19

method	result	size
default	\(\text {Expression too large to display}\)	\(1260\)

input

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

output

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+24 
0*(-(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*sin 
h(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*si 
nh(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))^...
 

3.9.61.5 Fricas [F]

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

input

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

output

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)
 

3.9.61.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.9.61.7 Maxima [F]

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

input

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

output

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

3.9.61.8 Giac [F(-2)]

Exception generated. \[ \int (a+b \cosh (c+d x) \sinh (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

3.9.61.9 Mupad [F(-1)]

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

input

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

output

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