\(\int \frac {1}{(-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{3/2}} \, dx\) [538]

Optimal result

Integrand size = 28, antiderivative size = 159 \[ \int \frac {1}{\left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt {2} \left (b^2-c^2\right )^{3/4}}-\frac {c \cosh (x)+b \sinh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \] Output:

1/4*arctanh(1/2*(b^2-c^2)^(1/4)*sinh(x+I*arctan(-I*c,b))*2^(1/2)/(-(b^2-c^ 
2)^(1/2)+(b^2-c^2)^(1/2)*cosh(x+I*arctan(-I*c,b)))^(1/2))*2^(1/2)/(b^2-c^2 
)^(3/4)-1/2*(c*cosh(x)+b*sinh(x))/(b^2-c^2)^(1/2)/(-(b^2-c^2)^(1/2)+b*cosh 
(x)+c*sinh(x))^(3/2)
 

Mathematica [F(-1)]

Timed out. \[ \int \frac {1}{\left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx=\text {\$Aborted} \] Input:

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

Output:

$Aborted
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3595, 3042, 3594, 3042, 3128, 217}

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.

\(\displaystyle \int \frac {1}{\left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3595

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {\int \frac {1}{\sqrt {b \cos (i x)-i c \sin (i x)-\sqrt {b^2-c^2}}}dx}{4 \sqrt {b^2-c^2}}\)

\(\Big \downarrow \) 3594

\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )-\sqrt {b^2-c^2}}}dx}{4 \sqrt {b^2-c^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )-\sqrt {b^2-c^2}}}dx}{4 \sqrt {b^2-c^2}}\)

\(\Big \downarrow \) 3128

\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {i \int \frac {1}{\frac {\left (b^2-c^2\right ) \sinh ^2\left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )-\sqrt {b^2-c^2}}-2 \sqrt {b^2-c^2}}d\left (-\frac {i \sqrt {b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )-\sqrt {b^2-c^2}}}\right )}{2 \sqrt {b^2-c^2}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt {2} \left (b^2-c^2\right )^{3/4}}\)

Input:

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

Output:

ArcTanh[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[-S 
qrt[b^2 - c^2] + Sqrt[b^2 - c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])]/(2*Sqrt[ 
2]*(b^2 - c^2)^(3/4)) - (c*Cosh[x] + b*Sinh[x])/(2*Sqrt[b^2 - c^2]*(-Sqrt[ 
b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))
 

Defintions of rubi rules used

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 3042

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

rule 3128

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

rule 3594

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( 
x_)]], x_Symbol] :> Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, 
c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
 

rule 3595

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(414\) vs. \(2(132)=264\).

Time = 0.25 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.61

method	result	size
default	\(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\cosh \left (x \right ) \sqrt {2}}{2}\right )}{2 \sqrt {b^{2}-c^{2}}\, \sqrt {-\frac {b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}-\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}\, \sqrt {b^{2}-c^{2}}\, \sqrt {2}\, \left (\ln \left (-\frac {2 \left (\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}\, \sinh \left (x \right )-\sqrt {b^{2}-c^{2}}\, \sinh \left (x \right )+\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}-\sqrt {b^{2}-c^{2}}-\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\right )}{\cosh \left (x \right )-\sqrt {2}}\right )-\ln \left (\frac {2 \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}\, \sinh \left (x \right )+2 \sqrt {b^{2}-c^{2}}\, \sinh \left (x \right )+2 \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}+2 \sqrt {b^{2}-c^{2}}+2 \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right )^{2}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}}{\cosh \left (x \right )+\sqrt {2}}\right )\right )}{4 \left (b -c \right ) \left (b +c \right ) \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(415\)

Input:

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

Output:

1/2/(b^2-c^2)^(1/2)/(-(b^2*sinh(x)-sinh(x)*c^2+b^2-c^2)/(b^2-c^2)^(1/2))^( 
1/2)*2^(1/2)*arctanh(1/2*cosh(x)*2^(1/2))-1/4*(-(b^2-c^2)^(1/2)*(sinh(x)+1 
)*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*2^(1/2)*(ln(-2*(cosh(x)*(b^2-c^2)^(1/2) 
*2^(1/2)*sinh(x)-(b^2-c^2)^(1/2)*sinh(x)+cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)-( 
b^2-c^2)^(1/2)-(-(b^2-c^2)^(1/2)*(sinh(x)+1)*sinh(x)^2)^(1/2)*(-(b^2-c^2)^ 
(1/2)*(sinh(x)+1))^(1/2))/(cosh(x)-2^(1/2)))-ln(2*(cosh(x)*(b^2-c^2)^(1/2) 
*2^(1/2)*sinh(x)+(b^2-c^2)^(1/2)*sinh(x)+cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)+( 
b^2-c^2)^(1/2)+(-(b^2-c^2)^(1/2)*(sinh(x)+1)*sinh(x)^2)^(1/2)*(-(b^2-c^2)^ 
(1/2)*(sinh(x)+1))^(1/2))/(cosh(x)+2^(1/2))))/(b-c)/(b+c)/(-(b^2-c^2)^(1/2 
)*(sinh(x)+1))^(1/2)/sinh(x)/(-(b^2*sinh(x)-sinh(x)*c^2+b^2-c^2)/(b^2-c^2) 
^(1/2))^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2050 vs. \(2 (126) = 252\).

Time = 0.31 (sec) , antiderivative size = 2050, normalized size of antiderivative = 12.89 \[ \int \frac {1}{\left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x, algorithm="fri 
cas")
 

Output:

-1/2*(sqrt(1/2)*((b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^6 + 6*(b^3 + 3*b^ 
2*c + 3*b*c^2 + c^3)*cosh(x)*sinh(x)^5 + (b^3 + 3*b^2*c + 3*b*c^2 + c^3)*s 
inh(x)^6 - 3*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^4 - 3*(b^3 + b^2*c - b*c^ 
2 - c^3 - 5*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(b 
^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^3 - 3*(b^3 + b^2*c - b*c^2 - c^3)*co 
sh(x))*sinh(x)^3 - b^3 + 3*b^2*c - 3*b*c^2 + c^3 + 3*(b^3 - b^2*c - b*c^2 
+ c^3)*cosh(x)^2 + 3*(5*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^4 + b^3 - 
b^2*c - b*c^2 + c^3 - 6*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^2)*sinh(x)^2 + 
 6*((b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^5 - 2*(b^3 + b^2*c - b*c^2 - c 
^3)*cosh(x)^3 + (b^3 - b^2*c - b*c^2 + c^3)*cosh(x))*sinh(x))*(b^2 - c^2)^ 
(1/4)*log(-((b^2 + 2*b*c + c^2)*cosh(x)^4 + 4*(b^2 + 2*b*c + c^2)*cosh(x)^ 
3*sinh(x) + 6*(b^2 + 2*b*c + c^2)*cosh(x)^2*sinh(x)^2 + 4*(b^2 + 2*b*c + c 
^2)*cosh(x)*sinh(x)^3 + (b^2 + 2*b*c + c^2)*sinh(x)^4 - b^2 + 2*b*c - c^2 
- 2*((b + c)*cosh(x)^3 + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 + 
 (b - c)*cosh(x) + (3*(b + c)*cosh(x)^2 + b - c)*sinh(x) + 2*sqrt(b^2 - c^ 
2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))*(b^2 - c^2)^(1/4)*sqrt(((b 
 + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - 2*sqrt(b 
^2 - c^2)*(cosh(x) + sinh(x)) + b - c)/(cosh(x) + sinh(x))) + 2*((b + c)*c 
osh(x)^3 + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 - (b - c)*cosh( 
x) + (3*(b + c)*cosh(x)^2 - b + c)*sinh(x))*sqrt(b^2 - c^2))/((b^2 + 2*...
 

Sympy [F]

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

integrate(1/(-(b**2-c**2)**(1/2)+b*cosh(x)+c*sinh(x))**(3/2),x)
 

Output:

Integral((b*cosh(x) + c*sinh(x) - sqrt(b**2 - c**2))**(-3/2), x)
 

Maxima [F]

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

integrate(1/(-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x, algorithm="max 
ima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (126) = 252\).

Time = 0.30 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.94 \[ \int \frac {1}{\left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {2 \, {\left (\sqrt {2} b^{2} + 2 \, \sqrt {2} b c + \sqrt {2} c^{2}\right )} \sqrt {b^{2} - c^{2}} \sqrt {{\left | b + c \right |}} \arctan \left (-\frac {\sqrt {b e^{x} + c e^{x}} b + \sqrt {b e^{x} + c e^{x}} c}{\sqrt {-\sqrt {b^{2} - c^{2}} b - \sqrt {b^{2} - c^{2}} c} \sqrt {{\left | b + c \right |}}}\right )}{{\left ({\left (b + c\right )}^{2} {\left (b - c\right )} + b^{3} + b^{2} c - b c^{2} - c^{3} - 2 \, {\left (b^{2} - c^{2}\right )} {\left | b + c \right |}\right )} \sqrt {-\sqrt {b^{2} - c^{2}} b - \sqrt {b^{2} - c^{2}} c}} + \frac {{\left (\sqrt {2} b^{3} + 3 \, \sqrt {2} b^{2} c + 3 \, \sqrt {2} b c^{2} + \sqrt {2} c^{3} + {\left (\sqrt {2} b^{2} + 2 \, \sqrt {2} b c + \sqrt {2} c^{2}\right )} {\left | b + c \right |}\right )} \arctan \left (\frac {\sqrt {b^{2} - c^{2}} \sqrt {b e^{x} + c e^{x}}}{\sqrt {-\sqrt {b^{2} - c^{2}} b^{2} + \sqrt {b^{2} - c^{2}} c^{2}}}\right )}{\sqrt {-\sqrt {b^{2} - c^{2}} b^{2} + \sqrt {b^{2} - c^{2}} c^{2}} {\left ({\left (b + c\right )}^{2} + b^{2} + 2 \, b c + c^{2} - 2 \, {\left (b + c\right )} {\left | b + c \right |}\right )}} + \frac {{\left (\sqrt {2} b^{2} + 2 \, \sqrt {2} b c + \sqrt {2} c^{2}\right )} \sqrt {b e^{x} + c e^{x}}}{{\left (b e^{x} + c e^{x} - \sqrt {b^{2} - c^{2}}\right )} {\left (\sqrt {b^{2} - c^{2}} {\left (b + c\right )} - \sqrt {b^{2} - c^{2}} {\left | b + c \right |}\right )}}\right )}}{b + c} \] Input:

integrate(1/(-(b^2-c^2)^(1/2)+b*cosh(x)+c*sinh(x))^(3/2),x, algorithm="gia 
c")
 

Output:

2*(2*(sqrt(2)*b^2 + 2*sqrt(2)*b*c + sqrt(2)*c^2)*sqrt(b^2 - c^2)*sqrt(abs( 
b + c))*arctan(-(sqrt(b*e^x + c*e^x)*b + sqrt(b*e^x + c*e^x)*c)/(sqrt(-sqr 
t(b^2 - c^2)*b - sqrt(b^2 - c^2)*c)*sqrt(abs(b + c))))/(((b + c)^2*(b - c) 
 + b^3 + b^2*c - b*c^2 - c^3 - 2*(b^2 - c^2)*abs(b + c))*sqrt(-sqrt(b^2 - 
c^2)*b - sqrt(b^2 - c^2)*c)) + (sqrt(2)*b^3 + 3*sqrt(2)*b^2*c + 3*sqrt(2)* 
b*c^2 + sqrt(2)*c^3 + (sqrt(2)*b^2 + 2*sqrt(2)*b*c + sqrt(2)*c^2)*abs(b + 
c))*arctan(sqrt(b^2 - c^2)*sqrt(b*e^x + c*e^x)/sqrt(-sqrt(b^2 - c^2)*b^2 + 
 sqrt(b^2 - c^2)*c^2))/(sqrt(-sqrt(b^2 - c^2)*b^2 + sqrt(b^2 - c^2)*c^2)*( 
(b + c)^2 + b^2 + 2*b*c + c^2 - 2*(b + c)*abs(b + c))) + (sqrt(2)*b^2 + 2* 
sqrt(2)*b*c + sqrt(2)*c^2)*sqrt(b*e^x + c*e^x)/((b*e^x + c*e^x - sqrt(b^2 
- c^2))*(sqrt(b^2 - c^2)*(b + c) - sqrt(b^2 - c^2)*abs(b + c))))/(b + c)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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