Integrand size = 18, antiderivative size = 114 \[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=-\frac {3 b \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}+\frac {3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \sinh \left (3 a+3 b x^n\right )}{4 n}-\frac {3 b \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \] Output:
-3/4*b*cosh(a)*Chi(b*x^n)/n+3/4*b*cosh(3*a)*Chi(3*b*x^n)/n+3/4*sinh(a+b*x^ n)/n/(x^n)-1/4*sinh(3*a+3*b*x^n)/n/(x^n)-3/4*b*sinh(a)*Shi(b*x^n)/n+3/4*b* sinh(3*a)*Shi(3*b*x^n)/n
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=-\frac {x^{-n} \left (3 b x^n \cosh (a) \text {Chi}\left (b x^n\right )-3 b x^n \cosh (3 a) \text {Chi}\left (3 b x^n\right )-3 \sinh \left (a+b x^n\right )+\sinh \left (3 \left (a+b x^n\right )\right )+3 b x^n \sinh (a) \text {Shi}\left (b x^n\right )-3 b x^n \sinh (3 a) \text {Shi}\left (3 b x^n\right )\right )}{4 n} \] Input:
Integrate[x^(-1 - n)*Sinh[a + b*x^n]^3,x]
Output:
-1/4*(3*b*x^n*Cosh[a]*CoshIntegral[b*x^n] - 3*b*x^n*Cosh[3*a]*CoshIntegral [3*b*x^n] - 3*Sinh[a + b*x^n] + Sinh[3*(a + b*x^n)] + 3*b*x^n*Sinh[a]*Sinh Integral[b*x^n] - 3*b*x^n*Sinh[3*a]*SinhIntegral[3*b*x^n])/(n*x^n)
Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5885, 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.
\(\displaystyle \int x^{-n-1} \sinh ^3\left (a+b x^n\right ) \, dx\) |
\(\Big \downarrow \) 5885 |
\(\displaystyle \int \left (\frac {1}{4} x^{-n-1} \sinh \left (3 a+3 b x^n\right )-\frac {3}{4} x^{-n-1} \sinh \left (a+b x^n\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 b \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}-\frac {3 b \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n}+\frac {3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n}\) |
Input:
Int[x^(-1 - n)*Sinh[a + b*x^n]^3,x]
Output:
(-3*b*Cosh[a]*CoshIntegral[b*x^n])/(4*n) + (3*b*Cosh[3*a]*CoshIntegral[3*b *x^n])/(4*n) + (3*Sinh[a + b*x^n])/(4*n*x^n) - Sinh[3*(a + b*x^n)]/(4*n*x^ n) - (3*b*Sinh[a]*SinhIntegral[b*x^n])/(4*n) + (3*b*Sinh[3*a]*SinhIntegral [3*b*x^n])/(4*n)
Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x ] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 3.58 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\left (-3 b \,{\mathrm e}^{-3 a} \operatorname {expIntegral}_{1}\left (3 b \,x^{n}\right ) x^{n}+3 b \,{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b \,x^{n}\right ) x^{n}+3 b \,{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{3 a} \operatorname {expIntegral}_{1}\left (-3 b \,x^{n}\right ) x^{n}+{\mathrm e}^{-3 a -3 b \,x^{n}}-3 \,{\mathrm e}^{-a -b \,x^{n}}+3 \,{\mathrm e}^{a +b \,x^{n}}-{\mathrm e}^{3 a +3 b \,x^{n}}\right ) x^{-n}}{8 n}\) | \(128\) |
Input:
int(x^(-1-n)*sinh(a+b*x^n)^3,x,method=_RETURNVERBOSE)
Output:
1/8*(-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( 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)+3*exp(a+b*x^n)-exp(3*a+3*b*x^n))/(x^n)/n
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (102) = 204\).
Time = 0.10 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.66 \[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=-\frac {2 \, \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{3} - 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 \, {\left (\cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - 1\right )} \sinh \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 )}} \] Input:
integrate(x^(-1-n)*sinh(a+b*x^n)^3,x, algorithm="fricas")
Output:
-1/8*(2*sinh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^3 - 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))) + 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*log( x)) + b*sinh(n*log(x))) + 3*((b*cosh(a) - b*sinh(a))*cosh(n*log(x)) + (b*c osh(a) - b*sinh(a))*sinh(n*log(x)))*Ei(-b*cosh(n*log(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*(c osh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^2 - 1)*sinh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a))/(n*cosh(n*log(x)) + n*sinh(n*log(x)))
\[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=\int x^{- n - 1} \sinh ^{3}{\left (a + b x^{n} \right )}\, dx \] Input:
integrate(x**(-1-n)*sinh(a+b*x**n)**3,x)
Output:
Integral(x**(-n - 1)*sinh(a + b*x**n)**3, x)
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=\frac {3 \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{n}\right )}{8 \, n} - \frac {3 \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{8 \, n} - \frac {3 \, b e^{a} \Gamma \left (-1, -b x^{n}\right )}{8 \, n} + \frac {3 \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{n}\right )}{8 \, n} \] Input:
integrate(x^(-1-n)*sinh(a+b*x^n)^3,x, algorithm="maxima")
Output:
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
\[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \sinh \left (b x^{n} + a\right )^{3} \,d x } \] Input:
integrate(x^(-1-n)*sinh(a+b*x^n)^3,x, algorithm="giac")
Output:
integrate(x^(-n - 1)*sinh(b*x^n + a)^3, x)
Timed out. \[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x^n\right )}^3}{x^{n+1}} \,d x \] Input:
int(sinh(a + b*x^n)^3/x^(n + 1),x)
Output:
int(sinh(a + b*x^n)^3/x^(n + 1), x)
Time = 0.17 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.43 \[ \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx=\frac {-3 x^{n} e^{3 x^{n} b +2 a} \mathit {ei} \left (-x^{n} b \right ) b +3 x^{n} e^{3 x^{n} b} \mathit {ei} \left (-3 x^{n} b \right ) b -3 x^{n} e^{3 x^{n} b +4 a} \mathit {ei} \left (x^{n} b \right ) b +3 x^{n} e^{3 x^{n} b +6 a} \mathit {ei} \left (3 x^{n} b \right ) b -e^{6 x^{n} b +6 a}+3 e^{4 x^{n} b +4 a}-3 e^{2 x^{n} b +2 a}+1}{8 x^{n} e^{3 x^{n} b +3 a} n} \] Input:
int(x^(-1-n)*sinh(a+b*x^n)^3,x)
Output:
( - 3*x**n*e**(3*x**n*b + 2*a)*ei( - x**n*b)*b + 3*x**n*e**(3*x**n*b)*ei( - 3*x**n*b)*b - 3*x**n*e**(3*x**n*b + 4*a)*ei(x**n*b)*b + 3*x**n*e**(3*x** n*b + 6*a)*ei(3*x**n*b)*b - e**(6*x**n*b + 6*a) + 3*e**(4*x**n*b + 4*a) - 3*e**(2*x**n*b + 2*a) + 1)/(8*x**n*e**(3*x**n*b + 3*a)*n)