Integrand size = 24, antiderivative size = 100 \[ \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx=\frac {c \cosh (x)+b \sinh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {c+\sqrt {b^2-c^2} \sinh (x)}{3 c \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))} \]
1/3*(c*cosh(x)+b*sinh(x))/(b^2-c^2)^(1/2)/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^( 1/2))^2+1/3*(-c-sinh(x)*(b^2-c^2)^(1/2))/c/(c*cosh(x)+b*sinh(x))/(b^2-c^2) ^(1/2)
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx=-\frac {-2 c \sqrt {b^2-c^2}+2 b c \cosh ^3(x)+2 c^2 \sinh (x)+c^2 \cosh ^2(x) \sinh (x)+b^2 \sinh ^3(x)}{3 c (c \cosh (x)+b \sinh (x))^3} \]
-1/3*(-2*c*Sqrt[b^2 - c^2] + 2*b*c*Cosh[x]^3 + 2*c^2*Sinh[x] + c^2*Cosh[x] ^2*Sinh[x] + b^2*Sinh[x]^3)/(c*(c*Cosh[x] + b*Sinh[x])^3)
Time = 0.38 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3595, 3042, 3593}
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.
\(\displaystyle \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cos (i x)-i c \sin (i x)\right )^2}dx\) |
\(\Big \downarrow \) 3595 |
\(\displaystyle \frac {\int \frac {1}{b \cosh (x)+c \sinh (x)+\sqrt {b^2-c^2}}dx}{3 \sqrt {b^2-c^2}}+\frac {b \sinh (x)+c \cosh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b \sinh (x)+c \cosh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac {\int \frac {1}{b \cos (i x)-i c \sin (i x)+\sqrt {b^2-c^2}}dx}{3 \sqrt {b^2-c^2}}\) |
\(\Big \downarrow \) 3593 |
\(\displaystyle \frac {b \sinh (x)+c \cosh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {\sqrt {b^2-c^2} \sinh (x)+c}{3 c \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}\) |
(c*Cosh[x] + b*Sinh[x])/(3*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (c + Sqrt[b^2 - c^2]*Sinh[x])/(3*c*Sqrt[b^2 - c^2]*(c*Cosh [x] + b*Sinh[x]))
3.8.58.3.1 Defintions of rubi rules used
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (-1), x_Symbol] :> Simp[-(c - a*Sin[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[ d + e*x])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
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]
Time = 0.56 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.47
method | result | size |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{x} b +3 c \,{\mathrm e}^{x}+\sqrt {b^{2}-c^{2}}\right )}{3 \left ({\mathrm e}^{x} b +c \,{\mathrm e}^{x}+\sqrt {b^{2}-c^{2}}\right )^{3}}\) | \(47\) |
default | \(\frac {2 \left (\sqrt {b^{2}-c^{2}}+b \right ) \left (\frac {\left (\sqrt {b^{2}-c^{2}}+b \right ) \tanh \left (\frac {x}{2}\right )^{2}}{c^{2}}+\frac {\left (2 b^{2}-c^{2}+2 \sqrt {b^{2}-c^{2}}\, b \right ) \tanh \left (\frac {x}{2}\right )}{c^{3}}+\frac {\frac {4 \sqrt {b^{2}-c^{2}}\, b^{2}}{3}-\frac {2 \sqrt {b^{2}-c^{2}}\, c^{2}}{3}+\frac {4 b^{3}}{3}-\frac {4 c^{2} b}{3}}{c^{4}}\right )}{c^{2} \left (\tanh \left (\frac {x}{2}\right )^{2}+\frac {2 \sqrt {\left (b -c \right ) \left (b +c \right )}\, \tanh \left (\frac {x}{2}\right )}{c}+\frac {2 \tanh \left (\frac {x}{2}\right ) b}{c}+\frac {2 \sqrt {\left (b -c \right ) \left (b +c \right )}\, b}{c^{2}}+\frac {2 b^{2}}{c^{2}}-1\right ) \left (\tanh \left (\frac {x}{2}\right )+\frac {\sqrt {\left (b -c \right ) \left (b +c \right )}}{c}+\frac {b}{c}\right )}\) | \(217\) |
Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (88) = 176\).
Time = 0.29 (sec) , antiderivative size = 660, normalized size of antiderivative = 6.60 \[ \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx=-\frac {2 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 12 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} + 6 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + b^{2} - c^{2}\right )} \sinh \left (x\right )^{2} - b^{2} + 2 \, b c - c^{2} + 12 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} + {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 8 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3}\right )} \sqrt {b^{2} - c^{2}}\right )}}{3 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{6} + 6 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \sinh \left (x\right )^{6} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{4} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4} - 5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} - b^{4} + 2 \, b^{3} c - 2 \, b c^{3} + c^{4} + 4 \, {\left (5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{4} + b^{4} - 2 \, b^{2} c^{2} + c^{4} - 6 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 6 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{5} - 2 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{3} + {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
-2/3*(3*(b^2 + 2*b*c + c^2)*cosh(x)^4 + 12*(b^2 + 2*b*c + c^2)*cosh(x)*sin h(x)^3 + 3*(b^2 + 2*b*c + c^2)*sinh(x)^4 + 6*(b^2 - c^2)*cosh(x)^2 + 6*(3* (b^2 + 2*b*c + c^2)*cosh(x)^2 + b^2 - c^2)*sinh(x)^2 - b^2 + 2*b*c - c^2 + 12*((b^2 + 2*b*c + c^2)*cosh(x)^3 + (b^2 - c^2)*cosh(x))*sinh(x) - 8*((b + c)*cosh(x)^3 + 3*(b + c)*cosh(x)^2*sinh(x) + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3)*sqrt(b^2 - c^2))/((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c ^3 + c^4)*cosh(x)^6 + 6*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x )*sinh(x)^5 + (b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*sinh(x)^6 - 3*(b ^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^4 - 3*(b^4 + 2*b^3*c - 2*b*c^3 - c^4 - 5*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^2)*sinh(x)^4 - b^ 4 + 2*b^3*c - 2*b*c^3 + c^4 + 4*(5*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^3 - 3*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x))*sinh(x)^3 + 3* (b^4 - 2*b^2*c^2 + c^4)*cosh(x)^2 + 3*(5*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b* c^3 + c^4)*cosh(x)^4 + b^4 - 2*b^2*c^2 + c^4 - 6*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^2)*sinh(x)^2 + 6*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^ 4)*cosh(x)^5 - 2*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^3 + (b^4 - 2*b^2* c^2 + c^4)*cosh(x))*sinh(x))
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[2,0]%%%}+%%%{2,[1,1]%%%}+%%%{1,[0,2]%%%},[4]%%%}+%% %{%%{[%%%
Timed out. \[ \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx=\int \frac {1}{{\left (b\,\mathrm {cosh}\left (x\right )+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )\right )}^2} \,d x \]