3.9.62 \(\int (a+b \cosh (c+d x) \sinh (c+d x))^{3/2} \, dx\) [862]

3.9.62.1 Optimal result

Integrand size = 20, antiderivative size = 248 \[ \int (a+b \cosh (c+d x) \sinh (c+d x))^{3/2} \, dx=\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}-\frac {2 i \sqrt {2} a E\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}{3 d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac {i \left (4 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right ),\frac {2 b}{2 i a+b}\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{6 \sqrt {2} d \sqrt {2 a+b \sinh (2 c+2 d x)}} \]

output

1/12*b*cosh(2*d*x+2*c)*(2*a+b*sinh(2*d*x+2*c))^(1/2)/d*2^(1/2)+2/3*I*a*(si 
n(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*P 
i+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))^(1/2 
)/d/((2*a+b*sinh(2*d*x+2*c))/(2*a-I*b))^(1/2)-1/12*I*(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*a+b*sinh(2*d*x+2*c))/(2*a-I*b))^(1/2)/ 
d*2^(1/2)/(2*a+b*sinh(2*d*x+2*c))^(1/2)
 

3.9.62.2 Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.81 \[ \int (a+b \cosh (c+d x) \sinh (c+d x))^{3/2} \, dx=\frac {16 a (2 i a+b) E\left (\frac {1}{4} (-4 i c+\pi -4 i d x)|-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}-2 i \left (4 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-4 i c+\pi -4 i d x),-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}+b (4 a \cosh (2 (c+d x))+b \sinh (4 (c+d x)))}{12 d \sqrt {4 a+2 b \sinh (2 (c+d x))}} \]

input

Integrate[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(3/2),x]
 

output

(16*a*((2*I)*a + b)*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)] - (2*I)*(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*(4*a*Cosh[2*(c + d*x)] + b*S 
inh[4*(c + d*x)]))/(12*d*Sqrt[4*a + 2*b*Sinh[2*(c + d*x)]])
 

3.9.62.3 Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 3145, 3042, 3135, 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.

\(\displaystyle \int (a+b \sinh (c+d x) \cosh (c+d x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a-i b \sin (i c+i d x) \cos (i c+i d x))^{3/2}dx\)

\(\Big \downarrow \) 3145

\(\displaystyle \int \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^{3/2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a-\frac {1}{2} i b \sin (2 i c+2 i d x)\right )^{3/2}dx\)

\(\Big \downarrow \) 3135

\(\displaystyle \frac {2}{3} \int \frac {12 a^2+8 b \sinh (2 c+2 d x) a-b^2}{4 \sqrt {2} \sqrt {2 a+b \sinh (2 c+2 d x)}}dx+\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {12 a^2+8 b \sinh (2 c+2 d x) a-b^2}{\sqrt {2 a+b \sinh (2 c+2 d x)}}dx}{6 \sqrt {2}}+\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {\int \frac {12 a^2-8 i b \sin (2 i c+2 i d x) a-b^2}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{6 \sqrt {2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {8 a \int \sqrt {2 a+b \sinh (2 c+2 d x)}dx-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a+b \sinh (2 c+2 d x)}}dx}{6 \sqrt {2}}+\frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {8 a \int \sqrt {2 a-i b \sin (2 i c+2 i d x)}dx-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{6 \sqrt {2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {\frac {8 a \sqrt {2 a+b \sinh (2 c+2 d x)} \int \sqrt {\frac {2 a}{2 a-i b}+\frac {b \sinh (2 c+2 d x)}{2 a-i b}}dx}{\sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{6 \sqrt {2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {\frac {8 a \sqrt {2 a+b \sinh (2 c+2 d x)} \int \sqrt {\frac {2 a}{2 a-i b}-\frac {i b \sin (2 i c+2 i d x)}{2 a-i b}}dx}{\sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx}{6 \sqrt {2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {-\left (4 a^2+b^2\right ) \int \frac {1}{\sqrt {2 a-i b \sin (2 i c+2 i d x)}}dx-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{6 \sqrt {2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {-\frac {\left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \int \frac {1}{\sqrt {\frac {2 a}{2 a-i b}+\frac {b \sinh (2 c+2 d x)}{2 a-i b}}}dx}{\sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{6 \sqrt {2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {-\frac {\left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \int \frac {1}{\sqrt {\frac {2 a}{2 a-i b}-\frac {i b \sin (2 i c+2 i d x)}{2 a-i b}}}dx}{\sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{6 \sqrt {2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {b \cosh (2 c+2 d x) \sqrt {2 a+b \sinh (2 c+2 d x)}}{6 \sqrt {2} d}+\frac {\frac {i \left (4 a^2+b^2\right ) \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right ),\frac {2 b}{2 i a+b}\right )}{d \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {8 i a \sqrt {2 a+b \sinh (2 c+2 d x)} E\left (\frac {1}{2} \left (2 i c+2 i d x-\frac {\pi }{2}\right )|\frac {2 b}{2 i a+b}\right )}{d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}}{6 \sqrt {2}}\)

input

Int[(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(3/2),x]
 

output

(b*Cosh[2*c + 2*d*x]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(6*Sqrt[2]*d) + (((- 
8*I)*a*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)]) + (I*(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]]))/(6*Sqrt[2])
 

3.9.62.3.1 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]
 

3.9.62.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (284 ) = 568\).

Time = 0.64 (sec) , antiderivative size = 935, normalized size of antiderivative = 3.77

method	result	size
default	\(\frac {4 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}}\, \operatorname {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 +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}}\, \operatorname {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^{3}+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}}\, \operatorname {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}+6 \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}}\, \operatorname {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^{2}-32 \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}}\, \operatorname {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^{3}-8 \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}}\, \operatorname {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 \,b^{2}+\sinh \left (2 d x +2 c \right )^{3} b^{3}+2 \sinh \left (2 d x +2 c \right )^{2} a \,b^{2}+b^{3} \sinh \left (2 d x +2 c \right )+2 a \,b^{2}}{6 b \cosh \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sinh \left (2 d x +2 c \right )}\, d}\)	\(935\)

input

int((a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

output

1/6*(4*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^2*b+ 
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*sin 
h(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*b^3+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^3+6*(-(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)*((sin 
h(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^2-32*(-(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^3-8*(-(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*b^2+sinh(2*d*x+2*c)^3*b^3+2*sinh(2*d*x+2*c) 
^2*a*b^2+b^3*sinh(2*d*x+2*c)+2*a*b^2)/b/cosh(2*d*x+2*c)/(4*a+2*b*sinh(2*d* 
x+2*c))^(1/2)/d
 

3.9.62.5 Fricas [F]

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

input

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),x, algorithm="fricas")
 

output

integral((b*cosh(d*x + c)*sinh(d*x + c) + a)^(3/2), x)
 

3.9.62.6 Sympy [F]

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

input

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))**(3/2),x)
 

output

Integral((a + b*sinh(c + d*x)*cosh(c + d*x))**(3/2), x)
 

3.9.62.7 Maxima [F]

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

input

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),x, algorithm="maxima")
 

output

integrate((b*cosh(d*x + c)*sinh(d*x + c) + a)^(3/2), x)
 

3.9.62.8 Giac [F(-2)]

Exception generated. \[ \int (a+b \cosh (c+d x) \sinh (c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),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
 

3.9.62.9 Mupad [F(-1)]

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

input

int((a + b*cosh(c + d*x)*sinh(c + d*x))^(3/2),x)
 

output

int((a + b*cosh(c + d*x)*sinh(c + d*x))^(3/2), x)