Integrand size = 13, antiderivative size = 116 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\frac {2 (b \cosh (x)+a \sinh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}} \]
2/3*(b*cosh(x)+a*sinh(x))/(a^2-b^2)/(a*cosh(x)+b*sinh(x))^(3/2)-2/3*I*(cos (1/2*I*x-1/2*arctan(a,-I*b))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(a,-I*b))*Elli pticF(sin(1/2*I*x-1/2*arctan(a,-I*b)),2^(1/2))*((a*cosh(x)+b*sinh(x))/(a^2 -b^2)^(1/2))^(1/2)/(a^2-b^2)/(a*cosh(x)+b*sinh(x))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.43 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=-\frac {2 \left (\sqrt {1-\frac {a^2}{b^2}} b (b \cosh (x)+a \sinh (x))+\sqrt {\cosh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\sinh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right ) (a \cosh (x)+b \sinh (x))^2\right )}{3 \sqrt {1-\frac {a^2}{b^2}} b (-a+b) (a+b) (a \cosh (x)+b \sinh (x))^{3/2}} \]
(-2*(Sqrt[1 - a^2/b^2]*b*(b*Cosh[x] + a*Sinh[x]) + Sqrt[Cosh[x + ArcTanh[a /b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, -Sinh[x + ArcTanh[a/b]]^2]*Se ch[x + ArcTanh[a/b]]*(a*Cosh[x] + b*Sinh[x])^2))/(3*Sqrt[1 - a^2/b^2]*b*(- a + b)*(a + b)*(a*Cosh[x] + b*Sinh[x])^(3/2))
Time = 0.38 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3555, 3042, 3557, 3042, 3120}
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.
| \(\displaystyle \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \cos (i x)-i b \sin (i x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3555 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}}dx}{3 \left (a^2-b^2\right )}+\frac {2 (a \sinh (x)+b \cosh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (a \sinh (x)+b \cosh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a \cos (i x)-i b \sin (i x)}}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3557 |
| \(\displaystyle \frac {2 (a \sinh (x)+b \cosh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}+\frac {\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \int \frac {1}{\sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )}}dx}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (a \sinh (x)+b \cosh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}+\frac {\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \int \frac {1}{\sqrt {\sin \left (i x-\tan ^{-1}(a,-i b)+\frac {\pi }{2}\right )}}dx}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 (a \sinh (x)+b \cosh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}-\frac {2 i \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right )}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}\) |
(2*(b*Cosh[x] + a*Sinh[x]))/(3*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^(3/2)) - (((2*I)/3)*EllipticF[(I*x - ArcTan[a, (-I)*b])/2, 2]*Sqrt[(a*Cosh[x] + b *Sinh[x])/Sqrt[a^2 - b^2]])/((a^2 - b^2)*Sqrt[a*Cosh[x] + b*Sinh[x]])
3.6.95.3.1 Defintions of rubi rules used
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x _Symbol] :> Simp[(b*Cos[c + d*x] - a*Sin[c + d*x])*((a*Cos[c + d*x] + b*Sin [c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[(n + 2)/((n + 1)*(a^ 2 + b^2)) Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x _Symbol] :> Simp[(a*Cos[c + d*x] + b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*S in[c + d*x])/Sqrt[a^2 + b^2])^n Int[Cos[c + d*x - ArcTan[a, b]]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && !(GeQ[n, 1] || LeQ[n, -1]) && !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])
Time = 0.63 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.32
| method | result | size |
| default | \(-\frac {\cosh \left (x \right )}{\left (a^{2}-b^{2}\right ) \sinh \left (x \right ) \sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(37\) |
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 679, normalized size of antiderivative = 5.85 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (a^{2} - b^{2}\right )}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b + b^{2}\right )}\right )} \sqrt {a + b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{3} - {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) + {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}\right )}}{3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{3} + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
2/3*((sqrt(2)*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 4*sqrt(2)*(a^2 + 2*a*b + b^2 )*cosh(x)*sinh(x)^3 + sqrt(2)*(a^2 + 2*a*b + b^2)*sinh(x)^4 + 2*sqrt(2)*(a ^2 - b^2)*cosh(x)^2 + 2*(3*sqrt(2)*(a^2 + 2*a*b + b^2)*cosh(x)^2 + sqrt(2) *(a^2 - b^2))*sinh(x)^2 + 4*(sqrt(2)*(a^2 + 2*a*b + b^2)*cosh(x)^3 + sqrt( 2)*(a^2 - b^2)*cosh(x))*sinh(x) + sqrt(2)*(a^2 - 2*a*b + b^2))*sqrt(a + b) *weierstrassPInverse(-4*(a - b)/(a + b), 0, cosh(x) + sinh(x)) + 2*((a^2 + 2*a*b + b^2)*cosh(x)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)^2 + (a^2 + 2*a*b + b^2)*sinh(x)^3 - (a^2 - b^2)*cosh(x) + (3*(a^2 + 2*a*b + b^2)*cos h(x)^2 - a^2 + b^2)*sinh(x))*sqrt(a*cosh(x) + b*sinh(x)))/(a^5 - a^4*b - 2 *a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^ 3 - 3*a*b^4 - b^5)*cosh(x)^4 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)*sinh(x)^3 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*sinh(x)^4 + 2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^ 4 + b^5)*cosh(x)^2 + 2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + 3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^2)*sin h(x)^2 + 4*((a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x )^3 + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*cosh(x))*sinh(x) )
Timed out. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\int \frac {1}{{\left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )}^{5/2}} \,d x \]