\(\int (e x)^{-1+n} (a+b \cosh (c+d x^n))^p \, dx\) [45]

Optimal result

Integrand size = 22, antiderivative size = 131 \[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\frac {\sqrt {2} x^{-n} (e x)^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\cosh \left (c+d x^n\right )\right ),\frac {b \left (1-\cosh \left (c+d x^n\right )\right )}{a+b}\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac {a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} \sinh \left (c+d x^n\right )}{d e n \sqrt {1+\cosh \left (c+d x^n\right )}} \] Output:

2^(1/2)*(e*x)^n*AppellF1(1/2,-p,1/2,3/2,b*(1-cosh(c+d*x^n))/(a+b),1/2-1/2* 
cosh(c+d*x^n))*(a+b*cosh(c+d*x^n))^p*sinh(c+d*x^n)/d/e/n/(x^n)/(1+cosh(c+d 
*x^n))^(1/2)/(((a+b*cosh(c+d*x^n))/(a+b))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.13 \[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\frac {x^{-n} (e x)^n \operatorname {AppellF1}\left (1+p,\frac {1}{2},\frac {1}{2},2+p,\frac {a+b \cosh \left (c+d x^n\right )}{a+b},\frac {a+b \cosh \left (c+d x^n\right )}{a-b}\right ) \sqrt {-\frac {b \left (-1+\cosh \left (c+d x^n\right )\right )}{a+b}} \sqrt {\frac {b \left (1+\cosh \left (c+d x^n\right )\right )}{-a+b}} \left (a+b \cosh \left (c+d x^n\right )\right )^{1+p} \text {csch}\left (c+d x^n\right )}{b d e n (1+p)} \] Input:

Integrate[(e*x)^(-1 + n)*(a + b*Cosh[c + d*x^n])^p,x]
 

Output:

((e*x)^n*AppellF1[1 + p, 1/2, 1/2, 2 + p, (a + b*Cosh[c + d*x^n])/(a + b), 
 (a + b*Cosh[c + d*x^n])/(a - b)]*Sqrt[-((b*(-1 + Cosh[c + d*x^n]))/(a + b 
))]*Sqrt[(b*(1 + Cosh[c + d*x^n]))/(-a + b)]*(a + b*Cosh[c + d*x^n])^(1 + 
p)*Csch[c + d*x^n])/(b*d*e*n*(1 + p)*x^n)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5846, 5844, 3042, 3144, 156, 155}

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.

\(\displaystyle \int (e x)^{n-1} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx\)

\(\Big \downarrow \) 5846

\(\displaystyle \frac {x^{-n} (e x)^n \int x^{n-1} \left (a+b \cosh \left (d x^n+c\right )\right )^pdx}{e}\)

\(\Big \downarrow \) 5844

\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \cosh \left (d x^n+c\right )\right )^pdx^n}{e n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \sin \left (i d x^n+i c+\frac {\pi }{2}\right )\right )^pdx^n}{e n}\)

\(\Big \downarrow \) 3144

\(\displaystyle -\frac {x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \int \frac {\left (a+b \cosh \left (d x^n+c\right )\right )^p}{\sqrt {1-\cosh \left (d x^n+c\right )} \sqrt {\cosh \left (d x^n+c\right )+1}}d\cosh \left (d x^n+c\right )}{d e n \sqrt {1-\cosh \left (c+d x^n\right )} \sqrt {\cosh \left (c+d x^n\right )+1}}\)

\(\Big \downarrow \) 156

\(\displaystyle -\frac {x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac {a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} \int \frac {\left (\frac {a}{a+b}+\frac {b \cosh \left (d x^n+c\right )}{a+b}\right )^p}{\sqrt {1-\cosh \left (d x^n+c\right )} \sqrt {\cosh \left (d x^n+c\right )+1}}d\cosh \left (d x^n+c\right )}{d e n \sqrt {1-\cosh \left (c+d x^n\right )} \sqrt {\cosh \left (c+d x^n\right )+1}}\)

\(\Big \downarrow \) 155

\(\displaystyle \frac {\sqrt {2} x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac {a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\cosh \left (d x^n+c\right )\right ),\frac {b \left (1-\cosh \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt {\cosh \left (c+d x^n\right )+1}}\)

Input:

Int[(e*x)^(-1 + n)*(a + b*Cosh[c + d*x^n])^p,x]
 

Output:

(Sqrt[2]*(e*x)^n*AppellF1[1/2, 1/2, -p, 3/2, (1 - Cosh[c + d*x^n])/2, (b*( 
1 - Cosh[c + d*x^n]))/(a + b)]*(a + b*Cosh[c + d*x^n])^p*Sinh[c + d*x^n])/ 
(d*e*n*x^n*Sqrt[1 + Cosh[c + d*x^n]]*((a + b*Cosh[c + d*x^n])/(a + b))^p)
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 3042

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

rule 3144

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + 
d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]])   Subst[Int[(a + b*x 
)^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, 
d, n}, x] && NeQ[a^2 - b^2, 0] &&  !IntegerQ[2*n]
 

rule 5844

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cosh[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]))
 

rule 5846

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_), x 
_Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m])   Int[x^m*( 
a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && I 
ntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \left (e x \right )^{n -1} {\left (a +b \cosh \left (c +d \,x^{n}\right )\right )}^{p}d x\]

Input:

int((e*x)^(n-1)*(a+b*cosh(c+d*x^n))^p,x)
 

Output:

int((e*x)^(n-1)*(a+b*cosh(c+d*x^n))^p,x)
 

Fricas [F]

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

integrate((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="fricas")
 

Output:

integral((e*x)^(n - 1)*(b*cosh(d*x^n + c) + a)^p, x)
 

Sympy [F]

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

integrate((e*x)**(-1+n)*(a+b*cosh(c+d*x**n))**p,x)
 

Output:

Integral((e*x)**(n - 1)*(a + b*cosh(c + d*x**n))**p, x)
 

Maxima [F]

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

integrate((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="maxima")
 

Output:

integrate((e*x)^(n - 1)*(b*cosh(d*x^n + c) + a)^p, x)
 

Giac [F]

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

integrate((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="giac")
 

Output:

integrate((e*x)^(n - 1)*(b*cosh(d*x^n + c) + a)^p, x)
 

Mupad [F(-1)]

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

int((e*x)^(n - 1)*(a + b*cosh(c + d*x^n))^p,x)
 

Output:

int((e*x)^(n - 1)*(a + b*cosh(c + d*x^n))^p, x)
 

Reduce [F]

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

int((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x)
                                                                                    
                                                                                    
 

Output:

(e**n*int((x**n*(cosh(x**n*d + c)*b + a)**p)/x,x))/e