Integrand size = 18, antiderivative size = 113 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n}+\frac {3 b \text {Chi}\left (b x^n\right ) \sinh (a)}{4 n}+\frac {3 b \text {Chi}\left (3 b x^n\right ) \sinh (3 a)}{4 n}+\frac {3 b \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]
-3/4*cosh(a+b*x^n)/n/(x^n)-1/4*cosh(3*a+3*b*x^n)/n/(x^n)+3/4*b*cosh(a)*Shi (b*x^n)/n+3/4*b*cosh(3*a)*Shi(3*b*x^n)/n+3/4*b*Chi(b*x^n)*sinh(a)/n+3/4*b* Chi(3*b*x^n)*sinh(3*a)/n
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\frac {x^{-n} \left (-3 \cosh \left (a+b x^n\right )-\cosh \left (3 \left (a+b x^n\right )\right )+3 b x^n \text {Chi}\left (b x^n\right ) \sinh (a)+3 b x^n \text {Chi}\left (3 b x^n\right ) \sinh (3 a)+3 b x^n \cosh (a) \text {Shi}\left (b x^n\right )+3 b x^n \cosh (3 a) \text {Shi}\left (3 b x^n\right )\right )}{4 n} \]
(-3*Cosh[a + b*x^n] - Cosh[3*(a + b*x^n)] + 3*b*x^n*CoshIntegral[b*x^n]*Si nh[a] + 3*b*x^n*CoshIntegral[3*b*x^n]*Sinh[3*a] + 3*b*x^n*Cosh[a]*SinhInte gral[b*x^n] + 3*b*x^n*Cosh[3*a]*SinhIntegral[3*b*x^n])/(4*n*x^n)
Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5886, 2009}
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.
\(\displaystyle \int x^{-n-1} \cosh ^3\left (a+b x^n\right ) \, dx\) |
\(\Big \downarrow \) 5886 |
\(\displaystyle \int \left (\frac {3}{4} x^{-n-1} \cosh \left (a+b x^n\right )+\frac {1}{4} x^{-n-1} \cosh \left (3 a+3 b x^n\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 b \sinh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {3 b \sinh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}+\frac {3 b \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n}\) |
(-3*Cosh[a + b*x^n])/(4*n*x^n) - Cosh[3*(a + b*x^n)]/(4*n*x^n) + (3*b*Cosh Integral[b*x^n]*Sinh[a])/(4*n) + (3*b*CoshIntegral[3*b*x^n]*Sinh[3*a])/(4* n) + (3*b*Cosh[a]*SinhIntegral[b*x^n])/(4*n) + (3*b*Cosh[3*a]*SinhIntegral [3*b*x^n])/(4*n)
3.1.52.3.1 Defintions of rubi rules used
Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x ] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 7.61 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {\left (3 b \,{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{-3 a} \operatorname {Ei}_{1}\left (3 b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{n}\right ) x^{n}+3 b \,{\mathrm e}^{3 a} \operatorname {Ei}_{1}\left (-3 b \,x^{n}\right ) x^{n}+{\mathrm e}^{-3 a -3 b \,x^{n}}+3 \,{\mathrm e}^{-a -b \,x^{n}}+{\mathrm e}^{3 a +3 b \,x^{n}}+3 \,{\mathrm e}^{a +b \,x^{n}}\right ) x^{-n}}{8 n}\) | \(126\) |
-1/8*(3*b*exp(a)*Ei(1,-b*x^n)*x^n-3*b*exp(-3*a)*Ei(1,3*b*x^n)*x^n-3*b*exp( -a)*Ei(1,b*x^n)*x^n+3*b*exp(3*a)*Ei(1,-3*b*x^n)*x^n+exp(-3*a-3*b*x^n)+3*ex p(-a-b*x^n)+exp(3*a+3*b*x^n)+3*exp(a+b*x^n))/(x^n)/n
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (102) = 204\).
Time = 0.26 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.83 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {2 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{3} + 6 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right ) \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - 3 \, {\left ({\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (3 \, b \cosh \left (n \log \left (x\right )\right ) + 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) - 3 \, {\left ({\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \, {\left ({\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \, {\left ({\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (-3 \, b \cosh \left (n \log \left (x\right )\right ) - 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 6 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )}{8 \, {\left (n \cosh \left (n \log \left (x\right )\right ) + n \sinh \left (n \log \left (x\right )\right )\right )}} \]
-1/8*(2*cosh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^3 + 6*cosh(b*cosh(n* log(x)) + b*sinh(n*log(x)) + a)*sinh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^2 - 3*((b*cosh(3*a) + b*sinh(3*a))*cosh(n*log(x)) + (b*cosh(3*a) + b*s inh(3*a))*sinh(n*log(x)))*Ei(3*b*cosh(n*log(x)) + 3*b*sinh(n*log(x))) - 3* ((b*cosh(a) + b*sinh(a))*cosh(n*log(x)) + (b*cosh(a) + b*sinh(a))*sinh(n*l og(x)))*Ei(b*cosh(n*log(x)) + b*sinh(n*log(x))) + 3*((b*cosh(a) - b*sinh(a ))*cosh(n*log(x)) + (b*cosh(a) - b*sinh(a))*sinh(n*log(x)))*Ei(-b*cosh(n*l og(x)) - b*sinh(n*log(x))) + 3*((b*cosh(3*a) - b*sinh(3*a))*cosh(n*log(x)) + (b*cosh(3*a) - b*sinh(3*a))*sinh(n*log(x)))*Ei(-3*b*cosh(n*log(x)) - 3* b*sinh(n*log(x))) + 6*cosh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a))/(n*co sh(n*log(x)) + n*sinh(n*log(x)))
\[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\int x^{- n - 1} \cosh ^{3}{\left (a + b x^{n} \right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.62 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3 \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{n}\right )}{8 \, n} - \frac {3 \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{8 \, n} + \frac {3 \, b e^{a} \Gamma \left (-1, -b x^{n}\right )}{8 \, n} + \frac {3 \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{n}\right )}{8 \, n} \]
-3/8*b*e^(-3*a)*gamma(-1, 3*b*x^n)/n - 3/8*b*e^(-a)*gamma(-1, b*x^n)/n + 3 /8*b*e^a*gamma(-1, -b*x^n)/n + 3/8*b*e^(3*a)*gamma(-1, -3*b*x^n)/n
\[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cosh \left (b x^{n} + a\right )^{3} \,d x } \]
Timed out. \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^3}{x^{n+1}} \,d x \]