Integrand size = 14, antiderivative size = 156 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=-\frac {2 (c \cosh (x)+b \sinh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]
-2*(c*cosh(x)+b*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^(1/2)-2*I*( cos(1/2*I*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*E llipticE(sin(1/2*I*x-1/2*arctan(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2- c^2)^(1/2)))^(1/2))*(a+b*cosh(x)+c*sinh(x))^(1/2)/(a^2-b^2+c^2)/((a+b*cosh (x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.02 (sec) , antiderivative size = 806, normalized size of antiderivative = 5.17 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\frac {2 \left (a b+\left (b^2-c^2\right ) \cosh (x)\right )-\frac {2 b^3 (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}+\frac {2 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {a+b \cosh (x)+c \sinh (x)}{a+i \sqrt {1-\frac {b^2}{c^2}} c},\frac {a+b \cosh (x)+c \sinh (x)}{a-i \sqrt {1-\frac {b^2}{c^2}} c}\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {b}{c}\right )\right ) (a+b \cosh (x)+c \sinh (x)) \sqrt {-\frac {-i \sqrt {1-\frac {b^2}{c^2}} c+b \cosh (x)+c \sinh (x)}{a+i \sqrt {1-\frac {b^2}{c^2}} c}} \sqrt {-\frac {i \sqrt {1-\frac {b^2}{c^2}} c+b \cosh (x)+c \sinh (x)}{a-i \sqrt {1-\frac {b^2}{c^2}} c}}}{\sqrt {1-\frac {b^2}{c^2}}}+\frac {b c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}}}-\frac {b c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \cosh (x)+c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}},\frac {a+b \cosh (x)+c \sinh (x)}{a-b \sqrt {1-\frac {c^2}{b^2}}}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}-b \cosh (x)-c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}+b \cosh (x)+c \sinh (x)}{-a+b \sqrt {1-\frac {c^2}{b^2}}}}}+\frac {c^2 \left (\frac {2 b^2 (a+b \cosh (x)+c \sinh (x))}{b^2-c^2}-\frac {c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}}}+\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \cosh (x)+c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}},\frac {a+b \cosh (x)+c \sinh (x)}{a-b \sqrt {1-\frac {c^2}{b^2}}}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{\sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}-b \cosh (x)-c \sinh (x)}{a+b \sqrt {1-\frac {c^2}{b^2}}}} \sqrt {\frac {b \sqrt {1-\frac {c^2}{b^2}}+b \cosh (x)+c \sinh (x)}{-a+b \sqrt {1-\frac {c^2}{b^2}}}}}\right )}{b}}{c \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}} \]
(2*(a*b + (b^2 - c^2)*Cosh[x]) - (2*b^3*(a + b*Cosh[x] + c*Sinh[x]))/(b^2 - c^2) + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, (a + b*Cosh[x] + c*Sinh[x])/(a + I*Sqrt[1 - b^2/c^2]*c), (a + b*Cosh[x] + c*Sinh[x])/(a - I*Sqrt[1 - b^2/ c^2]*c)]*Sech[x + ArcTanh[b/c]]*(a + b*Cosh[x] + c*Sinh[x])*Sqrt[-(((-I)*S qrt[1 - b^2/c^2]*c + b*Cosh[x] + c*Sinh[x])/(a + I*Sqrt[1 - b^2/c^2]*c))]* Sqrt[-((I*Sqrt[1 - b^2/c^2]*c + b*Cosh[x] + c*Sinh[x])/(a - I*Sqrt[1 - b^2 /c^2]*c))])/Sqrt[1 - b^2/c^2] + (b*c*Sinh[x + ArcTanh[c/b]])/Sqrt[1 - c^2/ b^2] - (b*c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Cosh[x] + c*Sinh[x])/(a + b*Sqrt[1 - c^2/b^2]), (a + b*Cosh[x] + c*Sinh[x])/(a - b*Sqrt[1 - c^2/b ^2])]*Sinh[x + ArcTanh[c/b]])/(Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[1 - c^2/b^2] - b*Cosh[x] - c*Sinh[x])/(a + b*Sqrt[1 - c^2/b^2])]*Sqrt[(b*Sqrt[1 - c^2/ b^2] + b*Cosh[x] + c*Sinh[x])/(-a + b*Sqrt[1 - c^2/b^2])]) + (c^2*((2*b^2* (a + b*Cosh[x] + c*Sinh[x]))/(b^2 - c^2) - (c*Sinh[x + ArcTanh[c/b]])/Sqrt [1 - c^2/b^2] + (c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Cosh[x] + c*Sinh [x])/(a + b*Sqrt[1 - c^2/b^2]), (a + b*Cosh[x] + c*Sinh[x])/(a - b*Sqrt[1 - c^2/b^2])]*Sinh[x + ArcTanh[c/b]])/(Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[1 - c ^2/b^2] - b*Cosh[x] - c*Sinh[x])/(a + b*Sqrt[1 - c^2/b^2])]*Sqrt[(b*Sqrt[1 - c^2/b^2] + b*Cosh[x] + c*Sinh[x])/(-a + b*Sqrt[1 - c^2/b^2])])))/b)/(c* (a^2 - b^2 + c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]])
Time = 0.52 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3607, 3042, 3598, 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.
| \(\displaystyle \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \cos (i x)-i c \sin (i x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3607 |
| \(\displaystyle \frac {\int \sqrt {a+b \cosh (x)+c \sinh (x)}dx}{a^2-b^2+c^2}-\frac {2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\int \sqrt {a+b \cos (i x)-i c \sin (i x)}dx}{a^2-b^2+c^2}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2-c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\) |
(-2*(c*Cosh[x] + b*Sinh[x]))/((a^2 - b^2 + c^2)*Sqrt[a + b*Cosh[x] + c*Sin h[x]]) - ((2*I)*EllipticE[(I*x - ArcTan[b, (-I)*c])/2, (2*Sqrt[b^2 - c^2]) /(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/((a^2 - b^2 + c^2 )*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])
3.8.65.3.1 Defintions of rubi rules used
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]
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])] Int[Sqrt[a/(a + S qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (-3/2), x_Symbol] :> Simp[2*((c*Cos[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^ 2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] + Simp[1/(a^2 - b^ 2 - c^2) Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{ a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs. \(2(177)=354\).
Time = 1.02 (sec) , antiderivative size = 888, normalized size of antiderivative = 5.69
| method | result | size |
| default | \(\frac {\sqrt {b^{2}-c^{2}}\, \operatorname {arctanh}\left (\frac {\left (b^{2}-c^{2}\right ) \cosh \left (x \right )}{\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b^{2}-c^{2}\right )}}\right )}{\sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}\, \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b^{2}-c^{2}\right )}}-\frac {\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, a \left (-\frac {\left (-b^{2}+c^{2}\right ) \ln \left (\frac {-\frac {2 \left (\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-a \right ) a^{2}}{b^{2}-c^{2}}+\frac {2 \left (b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}-a \sqrt {b^{2}-c^{2}}\right ) \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{b^{2}-c^{2}}\right )}{\left (b^{2}-c^{2}\right )^{\frac {3}{2}}}+2 \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}\, \sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}}{\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{b^{2}-c^{2}}}\right )}{2 \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (b^{2}-c^{2}\right ) \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}}+\frac {\left (b^{2}-c^{2}\right ) \ln \left (\frac {-\frac {2 \left (\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-a \right ) a^{2}}{b^{2}-c^{2}}-\frac {2 \left (b^{2} \sinh \left (x \right )-\sinh \left (x \right ) c^{2}-a \sqrt {b^{2}-c^{2}}\right ) \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{-b^{2}+c^{2}}\right )}{\left (b^{2}-c^{2}\right )^{\frac {3}{2}}}+2 \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}\, \sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}}{\cosh \left (x \right )+\frac {\sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}}{-b^{2}+c^{2}}}\right )}{2 \sqrt {\left (a^{2}+b^{2}-c^{2}\right ) \left (b -c \right ) \left (b +c \right )}\, \left (-b^{2}+c^{2}\right ) \sqrt {\frac {\left (-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+a \right ) a^{2}}{b^{2}-c^{2}}}}\right )}{\sinh \left (x \right ) \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\) | \(888\) |
1/((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2))^(1/2)*(b^ 2-c^2)^(1/2)/((a^2+b^2-c^2)*(b^2-c^2))^(1/2)*arctanh((b^2-c^2)*cosh(x)/((a ^2+b^2-c^2)*(b^2-c^2))^(1/2))-((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2) )/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*a*(-1/2*(-b^2+c^2)/((a^2+b^2-c^2)*(b-c) *(b+c))^(1/2)/(b^2-c^2)/((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2) *ln((-2*(sinh(x)*(b^2-c^2)^(1/2)-a)*a^2/(b^2-c^2)+2*(b^2*sinh(x)-sinh(x)*c ^2-a*(b^2-c^2)^(1/2))/(b^2-c^2)^(3/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*(c osh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(b^2-c^2))+2*((-sinh(x)*(b^2-c^2) ^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2 ))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2))/(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^ (1/2)/(b^2-c^2)))+1/2*(b^2-c^2)/((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(-b^2+c^ 2)/((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*ln((-2*(sinh(x)*(b^2 -c^2)^(1/2)-a)*a^2/(b^2-c^2)-2*(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2)) /(b^2-c^2)^(3/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*(cosh(x)+((a^2+b^2-c^2) *(b-c)*(b+c))^(1/2)/(-b^2+c^2))+2*((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c ^2))^(1/2)*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*s inh(x)^2)^(1/2))/(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(-b^2+c^2))))/ sinh(x)/((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2))^(1/ 2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 798, normalized size of antiderivative = 5.12 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\text {Too large to display} \]
2/3*((2*sqrt(2)*a^2*cosh(x) + sqrt(2)*(a*b + a*c)*cosh(x)^2 + sqrt(2)*(a*b + a*c)*sinh(x)^2 + 2*(sqrt(2)*a^2 + sqrt(2)*(a*b + a*c)*cosh(x))*sinh(x) + sqrt(2)*(a*b - a*c))*sqrt(b + c)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b ^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x) + 3*(b + c)*sinh(x) + 2*a)/( b + c)) - 3*(sqrt(2)*(b^2 + 2*b*c + c^2)*cosh(x)^2 + sqrt(2)*(b^2 + 2*b*c + c^2)*sinh(x)^2 + 2*sqrt(2)*(a*b + a*c)*cosh(x) + 2*(sqrt(2)*(b^2 + 2*b*c + c^2)*cosh(x) + sqrt(2)*(a*b + a*c))*sinh(x) + sqrt(2)*(b^2 - c^2))*sqrt (b + c)*weierstrassZeta(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), - 8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), weierstr assPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh( x) + 3*(b + c)*sinh(x) + 2*a)/(b + c))) - 6*((b^2 + 2*b*c + c^2)*cosh(x)^2 + (b^2 + 2*b*c + c^2)*sinh(x)^2 + (a*b + a*c)*cosh(x) + (a*b + a*c + 2*(b ^2 + 2*b*c + c^2)*cosh(x))*sinh(x))*sqrt(b*cosh(x) + c*sinh(x) + a))/(a^2* b^2 - b^4 - c^4 - (a^2 - 2*b^2)*c^2 + (a^2*b^2 - b^4 + a^2*c^2 + 2*b*c^3 + c^4 + 2*(a^2*b - b^3)*c)*cosh(x)^2 + (a^2*b^2 - b^4 + a^2*c^2 + 2*b*c^3 + c^4 + 2*(a^2*b - b^3)*c)*sinh(x)^2 + 2*(a^3*b - a*b^3 + a*b*c^2 + a*c^3 + (a^3 - a*b^2)*c)*cosh(x) + 2*(a^3*b - a*b^3 + a*b*c^2 + a*c^3 + (a^3 - a* b^2)*c + (a^2*b^2 - b^4 + a^2*c^2 + 2*b*c^3 + c^4 + 2*(a^2*b - b^3)*c)*...
\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \cosh {\left (x \right )} + c \sinh {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{3/2}} \,d x \]