\(\int x^{-1-n} \sinh (a+b x^n) \, dx\) [84]

Optimal result

Integrand size = 16, antiderivative size = 45 \[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\frac {b \cosh (a) \text {Chi}\left (b x^n\right )}{n}-\frac {x^{-n} \sinh \left (a+b x^n\right )}{n}+\frac {b \sinh (a) \text {Shi}\left (b x^n\right )}{n} \] Output:

b*cosh(a)*Chi(b*x^n)/n-sinh(a+b*x^n)/n/(x^n)+b*sinh(a)*Shi(b*x^n)/n
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\frac {x^{-n} \left (b x^n \cosh (a) \text {Chi}\left (b x^n\right )-\sinh \left (a+b x^n\right )+b x^n \sinh (a) \text {Shi}\left (b x^n\right )\right )}{n} \] Input:

Integrate[x^(-1 - n)*Sinh[a + b*x^n],x]
 

Output:

(b*x^n*Cosh[a]*CoshIntegral[b*x^n] - Sinh[a + b*x^n] + b*x^n*Sinh[a]*SinhI 
ntegral[b*x^n])/(n*x^n)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {5843, 3042, 26, 3778, 3042, 3784, 26, 3042, 26, 3779, 3782}

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.

\(\displaystyle \int x^{-n-1} \sinh \left (a+b x^n\right ) \, dx\)

\(\Big \downarrow \) 5843

\(\displaystyle \frac {\int x^{-2 n} \sinh \left (b x^n+a\right )dx^n}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i x^{-2 n} \sin \left (i b x^n+i a\right )dx^n}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int x^{-2 n} \sin \left (i b x^n+i a\right )dx^n}{n}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {i \left (i b \int x^{-n} \cosh \left (b x^n+a\right )dx^n-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i \left (i b \int x^{-n} \sin \left (i b x^n+i a+\frac {\pi }{2}\right )dx^n-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {i \left (i b \left (\cosh (a) \int x^{-n} \cosh \left (b x^n\right )dx^n-i \sinh (a) \int i x^{-n} \sinh \left (b x^n\right )dx^n\right )-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \left (i b \left (\sinh (a) \int x^{-n} \sinh \left (b x^n\right )dx^n+\cosh (a) \int x^{-n} \cosh \left (b x^n\right )dx^n\right )-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i \left (i b \left (\sinh (a) \int -i x^{-n} \sin \left (i b x^n\right )dx^n+\cosh (a) \int x^{-n} \sin \left (i b x^n+\frac {\pi }{2}\right )dx^n\right )-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \left (i b \left (\cosh (a) \int x^{-n} \sin \left (i b x^n+\frac {\pi }{2}\right )dx^n-i \sinh (a) \int x^{-n} \sin \left (i b x^n\right )dx^n\right )-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {i \left (i b \left (\sinh (a) \text {Shi}\left (b x^n\right )+\cosh (a) \int x^{-n} \sin \left (i b x^n+\frac {\pi }{2}\right )dx^n\right )-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

\(\Big \downarrow \) 3782

\(\displaystyle -\frac {i \left (i b \left (\cosh (a) \text {Chi}\left (b x^n\right )+\sinh (a) \text {Shi}\left (b x^n\right )\right )-i x^{-n} \sinh \left (a+b x^n\right )\right )}{n}\)

Input:

Int[x^(-1 - n)*Sinh[a + b*x^n],x]
 

Output:

((-I)*(((-I)*Sinh[a + b*x^n])/x^n + I*b*(Cosh[a]*CoshIntegral[b*x^n] + Sin 
h[a]*SinhIntegral[b*x^n])))/n
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 5843

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x] 
)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplif 
y[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplif 
y[(m + 1)/n], 0]))
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.47

method	result	size
risch	\(\frac {\left (-b \,{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b \,x^{n}\right ) x^{n}-b \,{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b \,x^{n}\right ) x^{n}+{\mathrm e}^{-a -b \,x^{n}}-{\mathrm e}^{a +b \,x^{n}}\right ) x^{-n}}{2 n}\)	\(66\)

Input:

int(x^(-1-n)*sinh(a+b*x^n),x,method=_RETURNVERBOSE)
 

Output:

1/2*(-b*exp(-a)*Ei(1,b*x^n)*x^n-b*exp(a)*Ei(1,-b*x^n)*x^n+exp(-a-b*x^n)-ex 
p(a+b*x^n))/n/(x^n)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (45) = 90\).

Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.09 \[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\frac {{\left ({\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) + {\left ({\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right ) - 2 \, \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )}{2 \, {\left (n \cosh \left (n \log \left (x\right )\right ) + n \sinh \left (n \log \left (x\right )\right )\right )}} \] Input:

integrate(x^(-1-n)*sinh(a+b*x^n),x, algorithm="fricas")
 

Output:

1/2*(((b*cosh(a) + b*sinh(a))*cosh(n*log(x)) + (b*cosh(a) + b*sinh(a))*sin 
h(n*log(x)))*Ei(b*cosh(n*log(x)) + b*sinh(n*log(x))) + ((b*cosh(a) - b*sin 
h(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))) - 2*sinh(b*cosh(n*log(x)) + b*sinh(n*log(x)) 
 + a))/(n*cosh(n*log(x)) + n*sinh(n*log(x)))
 

Sympy [F]

\[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\int x^{- n - 1} \sinh {\left (a + b x^{n} \right )}\, dx \] Input:

integrate(x**(-1-n)*sinh(a+b*x**n),x)
 

Output:

Integral(x**(-n - 1)*sinh(a + b*x**n), x)
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\frac {b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{2 \, n} + \frac {b e^{a} \Gamma \left (-1, -b x^{n}\right )}{2 \, n} \] Input:

integrate(x^(-1-n)*sinh(a+b*x^n),x, algorithm="maxima")
 

Output:

1/2*b*e^(-a)*gamma(-1, b*x^n)/n + 1/2*b*e^a*gamma(-1, -b*x^n)/n
 

Giac [F]

\[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \sinh \left (b x^{n} + a\right ) \,d x } \] Input:

integrate(x^(-1-n)*sinh(a+b*x^n),x, algorithm="giac")
 

Output:

integrate(x^(-n - 1)*sinh(b*x^n + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x^n\right )}{x^{n+1}} \,d x \] Input:

int(sinh(a + b*x^n)/x^(n + 1),x)
 

Output:

int(sinh(a + b*x^n)/x^(n + 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.73 \[ \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx=\frac {x^{n} e^{x^{n} b} \mathit {ei} \left (-x^{n} b \right ) b +x^{n} e^{x^{n} b +2 a} \mathit {ei} \left (x^{n} b \right ) b -e^{2 x^{n} b +2 a}+1}{2 x^{n} e^{x^{n} b +a} n} \] Input:

int(x^(-1-n)*sinh(a+b*x^n),x)
 

Output:

(x**n*e**(x**n*b)*ei( - x**n*b)*b + x**n*e**(x**n*b + 2*a)*ei(x**n*b)*b - 
e**(2*x**n*b + 2*a) + 1)/(2*x**n*e**(x**n*b + a)*n)