\(\int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx\) [625]

Optimal result

Integrand size = 20, antiderivative size = 158 \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=-\frac {2 \sqrt {2} b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d \sqrt {2 a+b \sinh (2 c+2 d x)}}-\frac {2 i \sqrt {2} E\left (\frac {1}{2} \left (2 i c-\frac {\pi }{2}+2 i d x\right )|\frac {2 b}{2 i a+b}\right ) \sqrt {2 a+b \sinh (2 c+2 d x)}}{\left (4 a^2+b^2\right ) d \sqrt {\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \] Output:

-2*2^(1/2)*b*cosh(2*d*x+2*c)/(4*a^2+b^2)/d/(2*a+b*sinh(2*d*x+2*c))^(1/2)+2 
*I*2^(1/2)*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)/(4*a^2+b^2)/d/((2*a+b*sinh(2*d*x+2*c))/(2*a-I* 
b))^(1/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=\frac {2 \left (-b \cosh (2 (c+d x))+(2 i a+b) E\left (\frac {1}{4} (-4 i c+\pi -4 i d x)|-\frac {2 i b}{2 a-i b}\right ) \sqrt {\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}}\right )}{\left (4 a^2+b^2\right ) d \sqrt {a+\frac {1}{2} b \sinh (2 (c+d x))}} \] Input:

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

Output:

(2*(-(b*Cosh[2*(c + d*x)]) + ((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)]))/((4*a^2 + b^2)*d*Sqrt[a + (b*Sinh[2*(c + d*x)])/2])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3042, 3145, 3042, 3143, 27, 3042, 3134, 3042, 3132}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3145

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3143

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

Input:

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

Output:

(-2*Sqrt[2]*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)*d*Sqrt[2*a + b*Sinh[2*c + 
2*d*x]]) - ((2*I)*Sqrt[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]])/((4*a^2 + b^2)*d*Sqrt[(2*a 
 + b*Sinh[2*c + 2*d*x])/(2*a - I*b)])
 

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 3143

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 + 1)*(a^2 - b^2))), x] + Simp 
[1/((n + 1)*(a^2 - b^2))   Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) 
- b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
 

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (143 ) = 286\).

Time = 0.41 (sec) , antiderivative size = 630, normalized size of antiderivative = 3.99

method	result	size
default	\(\frac {16 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {b \left (\sinh \left (2 d x +2 c \right )+i\right )}{i b -2 a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{2}+4 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {b \left (\sinh \left (2 d x +2 c \right )+i\right )}{i b -2 a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) b^{2}-16 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {b \left (\sinh \left (2 d x +2 c \right )+i\right )}{i b -2 a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) a^{2}-4 \sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}\, \sqrt {\frac {\left (-\sinh \left (2 d x +2 c \right )+i\right ) b}{i b +2 a}}\, \sqrt {\frac {b \left (\sinh \left (2 d x +2 c \right )+i\right )}{i b -2 a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {2 a +b \sinh \left (2 d x +2 c \right )}{i b -2 a}}, \sqrt {-\frac {i b -2 a}{i b +2 a}}\right ) b^{2}-4 b^{2} \sinh \left (2 d x +2 c \right )^{2}-4 b^{2}}{\left (4 a^{2}+b^{2}\right ) b \cosh \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sinh \left (2 d x +2 c \right )}\, d}\)	\(630\)

Input:

int(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

4*(4*(-(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)*(b*(sinh(2*d*x+2*c)+I)/(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+(-(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)*(b*(sinh(2*d*x+2*c)+I)/(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^2-4*(-(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)*(b*(s 
inh(2*d*x+2*c)+I)/(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-(-(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)*(b*(sinh(2*d*x+2 
*c)+I)/(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))*b^2-b^2*sinh(2*d*x+2*c)^2-b^2)/(4*a^2+b^2 
)/b/cosh(2*d*x+2*c)/(4*a+2*b*sinh(2*d*x+2*c))^(1/2)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (138) = 276\).

Time = 0.11 (sec) , antiderivative size = 1283, normalized size of antiderivative = 8.12 \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

-4*(((b^3*cosh(d*x + c)^4 + 4*b^3*cosh(d*x + c)*sinh(d*x + c)^3 + b^3*sinh 
(d*x + c)^4 + 4*a*b^2*cosh(d*x + c)^2 - b^3 + 2*(3*b^3*cosh(d*x + c)^2 + 2 
*a*b^2)*sinh(d*x + c)^2 + 4*(b^3*cosh(d*x + c)^3 + 2*a*b^2*cosh(d*x + c))* 
sinh(d*x + c))*sqrt(-b)*sqrt((4*a^2 + b^2)/b^2) + 2*(a*b^2*cosh(d*x + c)^4 
 + 4*a*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^2*sinh(d*x + c)^4 + 4*a^2*b 
*cosh(d*x + c)^2 - a*b^2 + 2*(3*a*b^2*cosh(d*x + c)^2 + 2*a^2*b)*sinh(d*x 
+ c)^2 + 4*(a*b^2*cosh(d*x + c)^3 + 2*a^2*b*cosh(d*x + c))*sinh(d*x + c))* 
sqrt(-b))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b)*elliptic_e(arcsin(sqrt 
((b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b)*(cosh(d*x + c) + sinh(d*x + c))), (4 
*a*b*sqrt((4*a^2 + b^2)/b^2) - 8*a^2 - b^2)/b^2) + (((2*a*b^2 - b^3)*cosh( 
d*x + c)^4 + 4*(2*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2 - 
b^3)*sinh(d*x + c)^4 - 2*a*b^2 + b^3 + 4*(2*a^2*b - a*b^2)*cosh(d*x + c)^2 
 + 2*(4*a^2*b - 2*a*b^2 + 3*(2*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c) 
^2 + 4*((2*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(2*a^2*b - a*b^2)*cosh(d*x + c 
))*sinh(d*x + c))*sqrt(-b)*sqrt((4*a^2 + b^2)/b^2) - 2*((2*a^2*b + a*b^2)* 
cosh(d*x + c)^4 + 4*(2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a 
^2*b + a*b^2)*sinh(d*x + c)^4 - 2*a^2*b - a*b^2 + 4*(2*a^3 + a^2*b)*cosh(d 
*x + c)^2 + 2*(4*a^3 + 2*a^2*b + 3*(2*a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh 
(d*x + c)^2 + 4*((2*a^2*b + a*b^2)*cosh(d*x + c)^3 + 2*(2*a^3 + a^2*b)*cos 
h(d*x + c))*sinh(d*x + c))*sqrt(-b))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) + ...
 

Sympy [F]

\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(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)
 

Maxima [F]

\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(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)
 

Giac [F]

\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

int(1/(a + b*cosh(c + d*x)*sinh(c + d*x))^(3/2),x)
 

Output:

int(1/(a + b*cosh(c + d*x)*sinh(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \cosh (c+d x) \sinh (c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\cosh \left (d x +c \right ) \sinh \left (d x +c \right ) b +a}}{\cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )^{2} b^{2}+2 \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) a b +a^{2}}d x \] Input:

int(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^(3/2),x)
 

Output:

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